{ "id": "40f38fb8f29a43bffff6867cdbded69c06a8f5f2", "text": "VI. On the Calculus of Symbols.\n\nBy William Spottiswoode, M.A., F.R.S.\n\nMade up of two Memoirs, one received November 4, Read November 21, 1861, the other received January 21, read January 30, 1862.\n\nIn a paper published in the Philosophical Transactions for 1861, p. 79, Mr. W. H. L. Russell has constructed systems of multiplication and division for functions of non-commutative symbols, subject to the same laws of combination as those in Professor Boole's memoir \"On a General Method in Analysis,\" Philosophical Transactions, 1844. In this calculus there are four systems of multiplication and division, viz. internal and external, both (1) when the functions are arranged in powers of $x$, (2) when in powers of $\\pi$, or, as they may be termed, the $x$-arrangement, and the $\\pi$-arrangement. In the paper in question the author has confined himself, so far as division is concerned, to the case most useful in practice, in which the divisor is linear. And of this he has discussed in full only the $x$-arrangement.\n\n§ 1. Internal division of the $\\pi$-arrangement by a linear factor.\n\nAdopting the same notation as Mr. Russell, I propose here to investigate, in the first place, the condition that $\\psi_1(x)\\pi + \\psi_0(x)$ may be an internal factor of\n\n$$\\varphi_n(x)\\pi^n + \\varphi_{n-1}(x)\\pi^{n-1} + \\ldots + \\varphi_0(x),$$\n\nand to determine the quotient. This is partially discussed in pp. 73–75.\n\nLet\n\n$$\\frac{d}{dx}\\psi = \\psi',$$\n\nthen performing the actual divisions, for brevity writing $\\psi$ for $\\psi(x)$, and $\\varphi$ for $\\varphi(x)$,\n\n$$\\psi_1(x)\\pi + \\psi_0(x)$$\n\n$$\\varphi_1(x)\\pi + \\varphi_0(x)$$\n\n$$\\psi_1(x)\\pi - \\varphi_0(x)\\psi_1(x).$$\n\nHence the condition that $\\psi_1(x)\\pi + \\psi_0(x)$ may be an internal factor of $\\varphi_1(x)\\pi + \\varphi_0(x)$ will be\n\n$$\\varphi_0(x) - \\varphi_1(x)\\frac{\\psi_0(x)}{\\psi_1(x)} = 0.$$... (1.)\nAgain,\n\n\\[ \\psi_1 \\pi + \\psi_0 \\phi_2 \\pi^2 + \\phi_1 \\pi + \\phi_0 \\left( \\frac{\\phi_2}{\\psi_1} \\pi + \\left( \\phi_1 - \\phi_2 \\frac{\\psi_1}{\\psi_1} \\right) \\frac{1}{\\psi_1} \\right) \\]\n\n\\[ \\phi_2 \\pi^2 + \\phi_2 \\frac{\\psi_1}{\\psi_1} \\pi + \\phi_2 \\frac{\\psi_0}{\\psi_1} \\pi + \\phi_2 \\frac{\\psi_0}{\\psi_1} \\]\n\n\\[ \\left( \\phi_1 - \\phi_2 \\frac{\\psi_1}{\\psi_1} + \\psi_0 \\right) \\pi + \\phi_0 - \\phi_2 \\frac{\\psi_0}{\\psi_1} \\]\n\n\\[ \\left( \\phi_1 - \\phi_2 \\frac{\\psi_1}{\\psi_1} + \\psi_0 \\right) \\pi + \\left( \\phi_1 - \\phi_2 \\frac{\\psi_1}{\\psi_1} + \\psi_0 \\right) \\frac{\\psi_0}{\\psi_1} \\]\n\n\\[ \\phi_0 - \\phi_1 \\frac{\\psi_0}{\\psi_1} + \\phi_2 \\left( \\frac{\\psi_0}{\\psi_1} \\right)^2 - \\frac{\\psi_0 \\psi_1 - \\psi_0 \\psi_1}{\\psi_1^2}. \\]\n\nHence the condition that \\( \\psi_1(\\varepsilon) \\pi + \\psi_0(\\varepsilon) \\) may be an internal factor of \\( \\phi_2(\\varepsilon) \\pi^2 + \\phi_1(\\varepsilon) \\pi + \\phi_0(\\varepsilon) \\)\n\nwill be\n\n\\[ \\phi_0(\\varepsilon) - \\phi_1(\\varepsilon) \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} + \\phi_2 \\left( \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\right)^2 - \\left( \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\right)' = 0. \\] (2.)\n\nBefore proceeding further, we may remark that the remainder, after internally dividing \\( \\phi_2(\\varepsilon) \\pi^3 + \\phi_1(\\varepsilon) \\pi^2 + \\phi_0(\\varepsilon) \\) by \\( \\psi_1(\\varepsilon) \\pi + \\psi_0(\\varepsilon) \\), can differ from that last above found only in respect of the remainder arising from the division of \\( \\phi_2(\\varepsilon) \\pi^3 \\) by the factor in question; hence we have now only to divide the term \\( \\phi_2(\\varepsilon) \\pi^3 \\) by \\( \\psi_1(\\varepsilon) \\pi + \\psi_0(\\varepsilon) \\), and add the remainder so found to (2.), in order to have the condition required for the third degree. Proceeding to the division, writing \\( \\chi = \\frac{\\psi_0}{\\psi_1} \\), and omitting for the present \\( \\phi_2 \\), which, since the division is internal, can be replaced as an external factor in the remainder, we have\n\n\\[ \\pi + \\chi \\pi^3 (\\pi^2 - \\chi \\pi + \\chi^2 - 2 \\chi' \\]\n\n\\[ \\pi^3 + \\chi \\pi^2 + 2 \\chi' \\pi + \\chi'' \\]\n\n\\[ - \\chi \\pi^2 - 2 \\chi' \\pi - \\chi'' \\]\n\n\\[ - \\chi \\pi^2 - \\chi^2 \\pi - \\chi \\chi' \\]\n\n\\[ (\\chi^2 - 2 \\chi') \\pi + \\chi \\chi' - \\chi'' \\]\n\n\\[ (\\chi^2 - 2 \\chi') \\pi + \\chi^3 - 2 \\chi \\chi' \\]\n\n\\[ - \\chi^3 + 3 \\chi \\chi' - \\chi''. \\]\n\nHence the condition that \\( \\psi_1(\\varepsilon) \\pi + \\psi_0(\\varepsilon) \\) may be an internal factor of \\( \\phi_2(\\varepsilon) \\pi^3 + \\phi_2(\\varepsilon) \\pi^2 + \\phi_1(\\varepsilon) \\pi + \\phi_0(\\varepsilon) \\) will be\n\n\\[ \\phi_0(\\varepsilon) - \\phi_1(\\varepsilon) \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} + \\phi_2(\\varepsilon) \\left( \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\right)^2 - \\left( \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\right)' - \\phi_2(\\varepsilon) \\left( \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\right)^3 - 3 \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\left( \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\right)' + \\left( \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} \\right)'' = 0, \\] (3.)\n\nthe identity of which with Mr. Russell's condition, given in p. 75 of his paper, I have verified.\nFor the fourth degree,\n\n\\[ \\pi + \\chi (\\pi^3 - \\chi \\pi^2 + (\\chi^2 - 3\\chi')\\pi - (\\chi^3 - 5\\chi \\chi' + 3\\chi'')) \\]\n\n\\[ \\pi^4 + \\chi \\pi^3 + 3\\chi' \\pi^2 + 3\\chi'' \\pi + \\chi''' \\]\n\n\\[ - \\chi \\pi^3 - 3\\chi' \\pi^2 - 3\\chi'' \\pi - \\chi''' \\]\n\n\\[ - \\chi \\pi^3 - \\chi^2 \\pi^2 - 2\\chi \\chi' \\pi - \\chi'' \\]\n\n\\[ (\\chi^2 - 3\\chi') \\pi^2 + (2\\chi \\chi' - 3\\chi'') \\pi + (\\chi \\chi'' - \\chi'''') \\]\n\n\\[ (\\chi^2 - 3\\chi') \\pi^2 + (\\chi^3 - 3\\chi \\chi') \\pi + (\\chi^2 \\chi' - 3\\chi'') \\]\n\n\\[ - (\\chi^3 - 5\\chi \\chi' + 3\\chi'') \\pi - \\chi^2 \\chi' + 3\\chi'' + \\chi''' - \\chi'''' \\]\n\n\\[ - (\\chi^3 - 5\\chi \\chi' + 3\\chi'') \\pi - \\chi^4 + 5\\chi^2 \\chi' - 3\\chi''' \\]\n\n\\[ \\chi^4 - 6\\chi^3 \\chi' - 4\\chi \\chi' + 3\\chi \\chi'' - \\chi''' \\]\n\nand\n\n\\[ \\chi^4 - 6\\chi^3 \\chi' - 4\\chi \\chi' + 3\\chi \\chi'' - \\chi''' = -\\chi(-\\chi^3 + 3\\chi \\chi' - \\chi'') + (-\\chi^3 + 3\\chi \\chi' - \\chi'')'; \\]\n\nor if \\( R_1, R_2, \\ldots \\) be the remainders of \\( \\pi(\\pi+\\chi)^{-1}, \\pi^2(\\pi+\\chi)^{-1}, \\ldots \\), we have\n\n\\[ R_2 = -\\chi R_1 + R'_1; \\]\n\n\\[ R_3 = -\\chi R_2 + R'_2; \\]\n\n\\[ R_4 = -\\chi R_3 + R'_3. \\]\n\nAnd generally, if\n\n\\[ \\pi^n (\\pi+\\chi)^{-1} = Q_n + R_n(\\pi+\\chi)^{-1}, \\]\n\nthen\n\n\\[ \\pi^{n+1}(\\pi+\\chi)^{-1} = \\pi Q_n + \\pi R_n(\\pi+\\chi)^{-1}, \\]\n\nthe remainder of which must be contained in the last term. Performing the actual division, and remembering that \\( \\pi R_n = R_n \\pi + R'_n \\),\n\n\\[ \\pi + \\chi R_n \\pi + R'_n R_n \\]\n\n\\[ R_n \\pi + R'_n \\chi \\]\n\n\\[ - \\chi R_n + R'_n. \\]\n\nHence we have generally,\n\n\\[ R_{n+1} = -\\chi R_n + R'_n, \\]\n\nand consequently, remembering that \\( R_0 = 1 \\), we have the condition that \\( \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon) \\) may be an internal factor of \\( \\varphi_n(\\varepsilon)\\pi^n + \\varphi_{n-1}(\\varepsilon)\\pi^{n-1} + \\ldots \\varphi_0(\\varepsilon) \\),\n\n\\[ \\varphi_0(\\varepsilon)R_0 + \\varphi_1(\\varepsilon)R_1 + \\ldots \\varphi_n(\\varepsilon)R_n = 0, \\ldots \\ldots \\ldots \\ldots \\quad (4.) \\]\n\nwhere\n\n\\[ R_{i+1} = -\\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)}R_i + \\varepsilon \\frac{d}{d\\varepsilon} R_i = \\left( -\\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} + \\varepsilon \\frac{d}{d\\varepsilon} \\right) R_i. \\]\n\nThe law of the quotients is best seen by actual division. In case of \\( \\varphi_2 \\pi^2 + \\varphi_1 \\pi + \\varphi_0 \\), given above, the quotient may be written\nFor the case of a cubic function of $\\pi$,\n\n$$\\psi_1 \\pi + \\psi_0 \\varphi_3 \\pi^3 + \\varphi_2 \\pi^2 + \\varphi_1 \\pi + \\varphi_0 \\left( \\frac{\\varphi_3}{\\psi_1} \\pi^2 + \\left( \\varphi_2 - \\varphi_3 \\right) \\frac{2\\psi_1 + \\psi_0}{\\psi_1} \\right) \\frac{1}{\\psi_1} \\pi + \\left( \\varphi_1 - \\varphi_2 \\right) \\frac{\\psi_1 + \\psi_0}{\\psi_1} + \\varphi_3 \\frac{2\\psi_1^2 + 3\\psi_0 \\psi_1 + \\psi_0^2 - \\psi_1 \\psi_0^2 - 2\\psi_1 \\psi_0}{\\psi_1^2}$$\n\nThe quotient of which may be written\n\nSimilarly, if the division be performed in the case of the quartic function, we shall find for the quotient of $(\\varphi_4 \\pi^4 + \\varphi_3 \\pi^3 + \\varphi_2 \\pi^2 + \\varphi_1 \\pi + \\varphi_0)(\\psi_1 \\pi + \\psi_0)^{-1}$,\nAnd likewise in general the quotient of \\((\\varphi_n x^n + \\varphi_{n-1} x^{n-1} + \\ldots \\varphi_0)(\\psi_1 x + \\psi_0)^{-1}\\) will be represented by a square table giving for the coefficient of \\(\\varphi_n\\)\n\n\\[\n(-)^{n-1} \\frac{1}{\\psi_1^{n-1}} \\times \\begin{array}{cccc}\ni - 1 & \\psi_1 & + & \\psi_0 \\\\\n1 & \\psi_1 & + & \\psi_0 \\\\\n(i-1)(i-2) & \\psi_1 & + & \\psi_0 \\\\\n1.2 & \\psi_1 & + & \\psi_0 \\\\\n& \\vdots & \\vdots & \\vdots \\\\\n\\psi_1^{(i-1)} + i - 1 & \\psi_1^{(i-2)} & + & \\psi_0\n\\end{array}\n\\]\n\n§ 2. External division of the \\(x\\)-arrangement by a linear factor.\n\nI next investigate the condition that \\(\\psi_1(x) x + \\psi_0(x)\\) may be an external factor of\n\n\\[\n\\varphi_n(x) x^n + \\varphi_{n-1}(x) x^{n-1} + \\ldots \\varphi_0(x).\n\\]\n\nPerforming the actual divisions, we have in the case of \\(n=1\\),\n\n\\[\n\\psi_1 x + \\psi_0 \\varphi_1 x + \\varphi_0 \\left( \\frac{\\psi_1}{\\psi_1} \\right)\n\\]\n\nor, as the remainder may be more conveniently written,\n\n\\[\n\\varphi_0 - \\varphi_1 + \\varphi_1 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots (1.)\n\\]\n\nAgain, in the case of \\(n=2\\),\n\n\\[\n\\psi_1 x + \\psi_0 \\varphi_2 x^2 + \\varphi_1 x + \\varphi_0 \\left( \\frac{\\varphi_2}{\\psi_1} x + \\frac{1}{\\psi_1} \\left[ \\varphi_1 - \\varphi_2 + \\varphi_1 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right] \\right)\n\\]\n\nor, transforming the remainder as in the former case, and continuing the division,\n\n\\[\n\\left\\{ \\varphi_1 - \\varphi_2 + \\varphi_1 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right\\} x + \\varphi_0\n\\]\n\n\\[\n\\left\\{ \\varphi_1 - \\varphi_2 + \\varphi_2 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right\\} x + \\psi_1 \\left[ \\frac{1}{\\psi_1} \\left\\{ \\varphi_1 - \\varphi_2 + \\varphi_2 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right\\} \\right] + \\psi_0 \\left\\{ \\varphi_1 - \\varphi_2 + \\varphi_2 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right\\}\n\\]\n\n\\[\n\\varphi_0 - \\psi_1 \\left[ \\frac{1}{\\psi_1} \\left\\{ \\varphi_1 - \\varphi_2 + \\varphi_2 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right\\} \\right] - \\psi_0 \\left\\{ \\varphi_1 - \\varphi_2 + \\varphi_2 \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right\\},\n\\]\nwhich also may be transformed as follows:\n\n\\[ \\varphi_0 - \\varphi_1 + \\varphi_2 + (\\varphi_1 - 2\\varphi_2)\\frac{\\psi_1 - \\psi_0}{\\psi_1} + \\varphi_3 \\left[ \\left( \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right)^2 - \\left( \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right) \\right]. \\quad (2.) \\]\n\nA similar process of division will be found, in the case of \\( n = 3 \\), to lead to the following remainder:\n\n\\[ \\varphi_0 - \\varphi_1 + \\varphi_2 - \\varphi_3 + (\\varphi_1 - 2\\varphi_2 + 3\\varphi_3)\\frac{\\psi_1 - \\psi_0}{\\psi_1} + (\\varphi_2 - 3\\varphi_3)\\left[ \\left( \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right)^3 - 3\\left( \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right)^2 + \\left( \\frac{\\psi_1 - \\psi_0}{\\psi_1} \\right) \\right]. \\quad (3.) \\]\n\nIf \\( \\Psi_1, \\Psi_2, \\ldots \\) represent the \\( \\psi \\)-functions, coefficients of the \\( \\varphi \\)'s in this expression, the law of their formation will be found to be as follows:\n\n\\[ \\Psi_1 = \\frac{\\psi_1 - \\psi_0}{\\psi_1}, \\]\n\\[ \\Psi_2 = \\Psi_2 - \\Psi_1, \\]\n\\[ \\Psi_3 = \\Psi_1 \\Psi_2 - \\Psi_2, \\]\n\\[ \\ldots \\ldots \\ldots \\]\n\nAnd generally we may write\n\n\\[ \\Psi_{i+1} = \\Psi_i \\Psi_i - \\Psi_i. \\]\n\nAnd if \\( R_1, R_2, \\ldots \\) represent the remainders in the cases of \\( n = 1, n = 2, \\ldots \\) respectively, we have\n\n\\[ R_1 = \\varphi_0 - \\varphi_1 + \\varphi_1 \\Psi_1, \\]\n\\[ R_2 = \\varphi_0 - \\varphi_1 + \\varphi_2 + (\\varphi_1 - 2\\varphi_2)\\Psi_1 + \\varphi_2 \\Psi_2, \\]\n\\[ R_3 = \\varphi_0 - \\varphi_1 + \\varphi_2 - \\varphi_3 + (\\varphi_1 - 2\\varphi_2 + 3\\varphi_3)\\Psi_1 + (\\varphi_2 - 3\\varphi_3)\\Psi_2 + \\varphi_3 \\Psi_3; \\]\n\nwhence\n\n\\[ R_2 = R_1 + \\varphi_2 - 2\\varphi_2 \\Psi_1 + \\varphi_2 \\Psi_2, \\]\n\\[ R_3 = R_2 - \\varphi_3 + 3\\varphi_3 \\Psi_1 - 3\\varphi_3 \\Psi_2 + \\varphi_3 \\Psi_3. \\]\n\nWith a view to forming the expression for \\( R_n \\), let the symbol \\( \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\) affixed to \\( R_i \\) signify that in the expression for \\( R_n \\) the suffixes of the \\( \\varphi \\)'s have all been increased by unity.\n\nThen, by the principle of division,\n\n\\[ R_{n+1} = \\varphi_0 - \\psi_1 \\left[ \\frac{1}{\\psi_1} R_n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\right] - \\frac{\\psi_0}{\\psi_1} R_n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\]\n\\[ = \\varphi_0 - R_n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) + \\psi_1 R_n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\]\n\\[ = \\varphi_0 - \\left( R_{n-1} - \\varphi_n^{(n)} + \\frac{n}{1} \\varphi_n^{(n-1)} \\psi_1 - \\frac{n(n-1)}{1.2} \\varphi_n^{(n-2)} \\psi_2 + \\ldots \\right) \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\]\n\\[ + \\psi_1 \\left( R_{n-1} - \\varphi_n^{(n)} + \\frac{n}{1} \\varphi_n^{(n-1)} \\psi_1 - \\frac{n(n-1)}{1.2} \\varphi_n^{(n-2)} \\psi_2 + \\ldots \\right) \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\]\n\\[ = \\varphi_0 - \\left( R_{n-1} - \\psi_1 R_{n-1} \\right) \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) + \\varphi_n^{(n+1)} - \\frac{n+1}{1} \\varphi_n^{(n+1)} + \\ldots; \\]\nand in this expression\n\n\\[ \\varphi_0 - (R_{n-1} - \\Psi_1 R_{n-1}) \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = R_n, \\]\n\nwhile the general term of the \\( \\varphi \\)-series, i.e. the coefficient of \\( \\varphi^{(n-r)}_{n+1} \\), will be\n\n\\[\n(-)^{r-1} \\frac{n(n-1)\\ldots(n-r+1)}{1\\cdot2\\ldots r} \\Psi_r - \\frac{n(n-1)\\ldots(n-r)}{1\\cdot2\\ldots(r+1)} \\Psi_{r+1} - \\frac{n(n-1)\\ldots(n-r+1)}{1\\cdot2\\ldots r} \\Psi_1 \\Psi_r\n\\]\n\n\\[\n= (-)^{r-1} \\frac{n(n-1)\\ldots(n-r+1)}{1\\cdot2\\ldots r} \\left( \\Psi_r - \\Psi_1 \\Psi_r - \\frac{n-r}{r+1} \\Psi_{r+1} \\right)\n\\]\n\n\\[\n= (-)^{r} \\frac{(n+1)(n-1)\\ldots(n-r+1)}{1\\cdot2\\ldots(r+1)} \\Psi_{r+1},\n\\]\n\nwhich proves the general case; so that generally\n\n\\[ R_{n+1} = R_n \\pm \\varphi^{(n+1)}_{n+1} \\mp \\frac{n+1}{1} \\varphi^{(n)}_{n+1} \\Psi_1 \\pm \\frac{(n+1)n}{1\\cdot2} \\varphi^{(n)}_{n+1} \\Psi_2 \\mp \\ldots \\ldots \\ldots \\ldots \\quad (4.) \\]\n\nthe upper or lower sign being taken according as \\((n+1)\\) is even or odd; where \\( R_{n+1} \\) is the remainder after external division \\( \\varphi_{n+1}(x) \\pi^{n+1} + \\varphi_n(x) \\pi^n + \\ldots \\varphi_0 \\) by \\( \\Psi_1(x) \\pi + \\Psi_0(x) \\).\n\nFor the quotients \\( Q_1, Q_2, \\ldots \\) we have immediately\n\n\\[ Q_1 = \\frac{1}{\\Psi_1} \\varphi_1, \\]\n\\[ Q_2 = \\frac{1}{\\Psi_1} \\left\\{ \\varphi_2 \\pi + R_1 \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\right\\}, \\]\n\\[ Q_3 = \\frac{1}{\\Psi_1} \\left\\{ \\varphi_3 \\pi^2 + R_1 \\left( \\begin{array}{c} 2 \\\\ 0 \\end{array} \\right) \\pi + R_2 \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\right\\}. \\]\n\\[ \\ldots \\ldots \\]\n\\[ Q_n = \\frac{1}{\\Psi_1} \\left\\{ \\varphi_n \\pi^{n-1} + R_1 \\left( \\begin{array}{c} n-1 \\\\ 0 \\end{array} \\right) \\pi^{n-2} + R_2 \\left( \\begin{array}{c} n-2 \\\\ 0 \\end{array} \\right) \\pi^{n-3} + \\ldots R_{n-1} \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\right\\}. \\]\n\nThis completes the solution of the problem of division by a linear factor, both internal and external.\n\n§ 3. To divide \\( \\sum_{n=0}^{N} \\varphi_n \\pi^n \\) internally by \\( \\sum_{m=0}^{M} \\Psi_m \\pi^m \\).\n\nThe first term in the quotient will obviously be\n\n\\[ \\frac{\\varphi_N}{\\Psi_M} \\pi^{N-M}, \\quad \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\quad (1.) \\]\n\nand the product of this into the divisor may, by means of Leibnitz's theorem, be written thus:\n\n\\[ \\frac{\\varphi_N}{\\Psi_m} \\sum_{p=0}^{N-M} [N-M, p] \\sum_{m=0}^{M} \\Psi_m^{(N-M-p)} \\pi^{m+p}, \\quad \\ldots \\ldots \\ldots \\ldots \\quad (2.) \\]\n\nwhere \\( \\Psi_m^{(N-M-p)} \\) means the result of the operation \\( \\pi^{N-M-p} \\) or \\( \\Psi_m \\) alone, and\n\n\\[ [N-M, p] = \\frac{(N-M)(N-M-1)\\ldots(N-M-p+1)}{1\\cdot2\\ldots p}. \\]\nThen the remainder after subtraction from the dividend may be written thus:\n\n$$\\sum_{m+p=0}^{m+p=N-1} \\left\\{ \\varphi_{m+p} - \\frac{1}{\\psi_M} \\sum_{p=0}^{p=N-M} \\sum_{m=0}^{m=M} \\varphi_N[N-M, p] \\psi_m^{(N-M-p)} \\right\\} \\pi^{m+p}, \\ldots \\quad (3.)$$\n\nsince the coefficient of $\\pi^N$ vanishes. With a view to the second term in the dividend, the first term of the remainder (3.) is\n\n$$\\left\\{ \\varphi_{N-1} - \\frac{1}{\\psi_M} \\sum_{p=0}^{p=N-M} \\sum_{m=0}^{m=M} \\varphi_N[N-M, p] \\psi_m^{(N-M-p)} \\right\\} \\pi^{N-1}, \\ldots \\quad (4.)$$\n\nin which the limits of $p$ and $m$ are subject to the further condition $p+m=N-1$. The terms under the sign of summation will be evaluated hereafter. Putting the expression\n\n$$\\Phi_1 \\pi^{N-1}, \\ldots \\quad (5.)$$\n\nand, for the sake of symmetry,\n\n$$\\varphi_N = \\Phi_0, \\ldots \\quad (6.)$$\n\nthe first and second terms in the quotient will be $\\frac{\\Phi_0}{\\psi_M} \\pi^{N-M}$, and $\\frac{\\Phi_1}{\\psi_M} \\pi^{N-M-1}$, respectively; and, in the same manner as (2.), the product of the second term of the quotient into the divisor may be written thus,\n\n$$\\frac{\\Phi_1}{\\psi_M} \\sum_{p_i=0}^{p_i=N-M-1} \\sum_{m=0}^{m=M} [N-M-1, p] \\psi_m^{(N-M-p-1)} \\pi^{m+p}, \\ldots \\quad (7.)$$\n\nand the remainder thus:\n\n$$\\sum_{m+p=0}^{m+p=N-2} \\left\\{ \\varphi_{m+p} - \\frac{1}{\\psi_M} \\sum_{p=0}^{p=N-M} \\sum_{m=0}^{m=M} \\varphi_0[N-M, p] \\psi_m^{(N-M-p)} \\right\\} \\pi^{m+p}$$\n\n$$- \\sum_{m+p_i=0}^{m+p_i=N-2} \\left\\{ \\frac{1}{\\psi_M} \\sum_{p_i=0}^{p_i=N-M-1} \\sum_{m=0}^{m=M} \\varphi_1[N-M-1, p_i] \\psi_m^{(N-M-p_i-1)} \\right\\} \\pi^{m+p_i}. \\quad (8.)$$\n\nBut since, when $p_i=N-M$, $[N-M-1, p_i]=0$, we may, without altering the value of (8.), change the superior limit of $p_i$ from $N-M-1$, to $N-M$; and by this means we may write the remainder (8.) in the following form:\n\n$$\\sum_{m+p=0}^{m+p=N-2} \\left\\{ \\varphi_{m+p} - \\frac{1}{\\psi_M} \\sum_{p=0}^{p=N-M} \\sum_{m=0}^{m=M} (\\varphi_0[N-M, p] \\psi_m^{(N-M-p)} + \\varphi_1[N-M-1, p] \\psi_m^{(N-M-p-1)}) \\right\\} \\pi^{m+p}. \\quad (9.)$$\n\nSimilarly, calling the first term of (9.) $\\Phi_2 \\pi^{N-2}$, the third term in the quotient will be $\\frac{\\Phi_2}{\\psi_M} \\pi^{N-M-2}$, and the corresponding remainder\n\n$$\\sum_{m+p=0}^{m+p=N-3} \\left\\{ \\varphi_{m+p} - \\frac{1}{\\psi_M} \\sum_{p=0}^{p=N-M} \\sum_{m=0}^{m=M} (\\varphi_0[N-M, p] \\psi_m^{(N-M-p)} + \\varphi_1[N-M-1, p] \\psi_m^{(N-M-p-1)} + \\varphi_2[N-M-2, p] \\psi_m^{(N-M-p-2)}) \\right\\} \\pi^{m+p}; \\quad (10.)$$\n\nand so generally the $(r+1)$th term in the quotient will be $\\frac{\\Phi_r}{\\psi_M} \\pi^{N-M-r}$, where\n\n$$\\Phi_r = \\varphi_{N-r} - \\frac{1}{\\psi_M} \\sum_{p=0}^{p=N-M} \\sum_{m=0}^{m=M} \\sum_{q=0}^{q=r-1} \\varphi_q[N-M-q, p] \\psi_m^{(N-M-p-q)}, \\ldots \\quad (11.)$$\nand the \\((r+1)\\)th remainder\n\n\\[\n\\sum_{m+p=N-r-1} \\left\\{ \\varphi_{m+p} - \\sum_{p=0}^{N-M} \\sum_{m=0}^{M} \\sum_{q=0}^{r} \\Phi_q[N-M-q,p] \\psi_{m+N-M-p-q} \\right\\} x^{m+p}.\n\\]\n\nThe final remainder is the \\((N-M+1)\\)th; and the expression will be derived from (10.) by replacing \\(r\\) by \\(N-M\\).\n\nIt remains to develope the terms under the sign of summation in the expressions for the \\(\\Phi_s\\). In the first place \\(\\Phi_0 = \\varphi_N\\) simply. In the case of \\(\\Phi_1\\), the limiting values of \\(p\\) and \\(m\\) are\n\n\\[\np = 0, 1, \\ldots N-M,\n\\]\n\\[\nm = 0, 1, \\ldots M,\n\\]\n\\[\np + m = N-1.\n\\]\n\nThese give as the only admissible values\n\n\\[\np = N-M, \\quad N-M-1,\n\\]\n\\[\nm = M-1, \\quad M,\n\\]\n\nand consequently\n\n\\[\n\\Phi_1 = \\varphi_{N-1} - \\frac{\\Phi_0}{\\psi_M} (\\psi_{M-1} + (N-M)\\psi_M).\n\\]\n\nIn the case of \\(\\Phi_2\\), the only admissible values are\n\n\\[\np = N-M, \\quad N-M-1, \\quad N-M-2,\n\\]\n\\[\nm = M-2, \\quad M-1, \\quad M,\n\\]\n\ngiving\n\n\\[\n\\Phi_2 = \\varphi_{N-2} - \\frac{1}{\\psi_M} \\left\\{ \\Phi_0 \\left( \\psi_{M-2} + (N-M)\\psi_{M-1} + \\frac{(N-M)(N-M-1)}{1.2} \\psi_M \\right) + \\Phi_1 \\left( \\psi_{M-1} + (N-M-1)\\psi_M \\right) \\right\\}.\n\\]\n\nBefore proceeding further, it may be well to illustrate these formulæ by an example.\n\nTaking the case of \\(N=4, M=3\\), we may determine the quotient and last remainder of internal division of\n\n\\[\n\\varphi_4(x)x^4 + \\varphi_3(x)x^3 + \\varphi_2(x)x^2 + \\varphi_1(x)x + \\varphi_0(x)\n\\]\n\nby\n\n\\[\n\\psi_3(x)x^3 + \\psi_2(x)x^2 + \\psi_1(x)x + \\psi_0(x),\n\\]\n\nand thence the conditions that the latter may be an internal factor of the former.\n\nBy the formulæ given above, we have\n\n\\[\n\\Phi_0 = \\varphi_4,\n\\]\n\\[\n\\Phi_1 = \\varphi_3 - \\frac{\\varphi_4}{\\psi_2} (\\psi_1 + 2\\psi_2),\n\\]\n\\[\n\\Phi_2 = \\varphi_2 - \\frac{1}{\\psi_2} \\left\\{ \\varphi_4 (\\psi_0 + 2\\psi_1 + \\psi_2) + \\left( \\varphi_3 - \\frac{\\varphi_4}{\\psi_2} (\\psi_1 + 2\\psi_2) \\right) (\\psi_1 + \\psi_2) \\right\\},\n\\]\n\nwhich will determine the quotient\n\n\\[\n\\frac{1}{\\psi_2} (\\Phi_0x^2 + \\Phi_1x + \\Phi_2).\n\\]\nThe last remainder is\n\n\\[ \\sum_{m+p=0}^{\\infty} \\left[ \\varphi_{m+p} - \\frac{1}{\\psi_2} \\sum_{p=0}^{m} \\sum_{q=0}^{m-p} \\Phi_q [N-M-q,p] \\psi_m^{(N-M-p-q)} \\right] \\pi^{m+p}; \\]\n\nwhence for\n\n\\[ m+p=1, \\text{i.e. } p=0, m=1, \\text{ or } p=1, m=0, \\]\n\nwe have\n\n| \\( q=0 \\) | \\( q=1 \\) | \\( q=2 \\) |\n|---|---|---|\n| \\( p=0, \\quad [2,0]=1, \\quad [1,0]=1, \\quad [0,0]=1, \\) |\n| \\( p=1, \\quad [2,1]=2, \\quad [1,1]=1, \\quad [0,1]=0. \\) |\n\nHence for \\( m=1 \\) the above expression gives\n\n\\[ \\Phi_0 \\psi_1 + \\Phi_1 \\psi_1 + \\Phi_2 \\psi_1, \\]\n\nand for \\( m=0 \\) it gives\n\n\\[ 2\\Phi_0 \\psi_0 + \\Phi_1 \\psi_0. \\]\n\nAgain, for \\( m+p=0, \\text{i.e. } p=0, m=0, \\) we have\n\n\\[ \\Phi_0 \\psi_0 + \\Phi_1 \\psi_0 + \\Phi_2 \\psi_0. \\]\n\nHence the total remainder is\n\n\\[ \\left\\{ \\varphi_1 - \\frac{1}{\\psi_2} (\\Phi_0 \\psi_1 + \\Phi_1 \\psi_1 + \\Phi_2 \\psi_1 + 2\\Phi_0 \\psi_0 + \\Phi_1 \\psi_0) \\right\\} \\pi + \\left\\{ \\varphi_0 - \\frac{1}{\\psi_2} (\\Phi_0 \\psi_0 + \\Phi_1 \\psi_0 + \\Phi_2 \\psi_0) \\right\\}. \\]\n\nIt may be useful to compare these results with the actual division, in the above example.\n\n\\[ \\psi_2 \\pi^2 + \\psi_1 \\pi + \\psi_0 \\right) \\varphi_4 \\pi^4 + \\varphi_3 \\pi^3 + \\varphi_2 \\pi^2 + \\varphi_1 \\pi + \\varphi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\pi^2 + \\frac{\\Phi_1}{\\psi_2} \\pi + \\frac{\\Phi_2}{\\psi_2} \\right) \\]\n\n\\[ \\varphi_4 \\pi^4 + 2\\varphi_3 \\frac{\\psi_1}{\\psi_2} \\pi^3 + \\varphi_2 \\frac{\\psi_2}{\\psi_2} \\pi^2 \\]\n\n\\[ + \\varphi_1 \\frac{\\psi_1}{\\psi_2} \\pi^2 + 2\\varphi_0 \\frac{\\psi_0}{\\psi_2} \\pi + \\varphi_0 \\frac{\\psi_0}{\\psi_2} \\]\n\n\\[ + \\varphi_4 \\frac{\\psi_0}{\\psi_2} \\pi^2 + 2\\varphi_3 \\frac{\\psi_0}{\\psi_2} \\pi + \\varphi_2 \\frac{\\psi_0}{\\psi_2} \\]\n\n\\[ \\varphi_1 \\pi^3 + \\left( \\varphi_2 - \\frac{\\varphi_4}{\\psi_2} (\\psi_2 + 2\\psi_1 + \\psi_0) \\right) \\pi^2 + \\left( \\varphi_1 - \\frac{\\varphi_4}{\\psi_2} (\\psi_1 + 2\\psi_0) \\right) \\pi + \\left( \\varphi_0 - \\frac{\\varphi_4}{\\psi_2} \\psi_0 \\right) \\]\n\n\\[ \\varphi_1 \\pi^3 + \\frac{\\psi_1}{\\psi_2} \\pi^2 + \\frac{\\psi_1}{\\psi_2} \\pi + \\frac{\\psi_0}{\\psi_2} \\pi + \\frac{\\psi_0}{\\psi_2} \\]\n\n\\[ \\varphi_2 \\pi^2 + \\left\\{ \\varphi_1 - \\frac{1}{\\psi_2} \\Phi_0 (\\psi_1 + 2\\psi_0) - \\frac{1}{\\psi_2} \\Phi_1 (\\psi_1 + \\psi_0) \\right\\} \\pi + \\left\\{ \\varphi_0 - \\frac{1}{\\psi_2} \\Phi_0 \\psi_0 - \\frac{1}{\\psi_2} \\Phi_1 \\psi_0 \\right\\} \\]\n\n\\[ \\varphi_2 \\pi^2 + \\frac{1}{\\psi_2} \\Phi_2 \\psi_1 \\pi + \\frac{1}{\\psi_2} \\Phi_2 \\psi_0 \\]\n\n\\[ \\left\\{ \\varphi_1 - \\frac{1}{\\psi_2} \\Phi_0 (\\psi_1 + 2\\psi_0) - \\frac{1}{\\psi_2} \\Phi_1 (\\psi_1 + \\psi_0) - \\frac{1}{\\psi_2} \\Phi_2 \\psi_1 \\right\\} \\pi + \\left\\{ \\varphi_0 - \\frac{1}{\\psi_2} \\Phi_0 \\psi_0 - \\frac{1}{\\psi_2} \\Phi_1 \\psi_0 - \\frac{1}{\\psi_2} \\Phi_2 \\psi_0 \\right\\}, \\]\n\nwhich agrees with the result before found.\nReturning to the $\\Phi$ functions, and writing for convenience the symbolical expression\n\n$$\\sum_{p=0}^{N-M-r} \\sum_{m=0}^{M} [N-M-r, p] \\psi_m^{(N-M-p-r)} = S_s[N-M-r, p] \\psi_m^{(N-M-p-r)},$$\n\nwhere the suffix $s$ indicates the number of units whereby the sum $p+m$ is less than $N$, we have\n\n$$\\Phi_0 = \\phi_N,$$\n\n$$\\psi_M \\Phi_1 = J_M \\phi_{N-1} - S_1 \\Phi_0[N-M, p] \\psi_m^{(N-M-p)}$$\n\n$$= \\begin{vmatrix}\n\\phi_{N-1} & S_1[N-M, p] \\psi_m^{(N-M-p)} \\\\\n\\phi_N & \\psi_M\n\\end{vmatrix},$$\n\n$$\\psi_M^2 \\Phi_2 = J_M^2 \\phi_{N-2} - S_2(\\Phi_0[N-M, p] \\psi_m^{(N-M-p)} + \\Phi_1[N-M-1, p] \\psi_m^{(N-M-p-1)})$$\n\n$$= \\begin{vmatrix}\nJ_M^2 \\phi_{N-2} & -S_2[N-M, p] \\psi_m^{(N-M-p)} - \\Phi_1 S_2[N-M-1, p] \\psi_m^{(N-M-p-1)} \\\\\n\\phi_{N-2} & S_2[N-M-1, p] \\psi_m^{(N-M-p-1)} \\\\\n\\phi_{N-1} & S_1[N-M-1, p] \\psi_m^{(N-M-p-1)} \\\\\n\\phi_N & 0\n\\end{vmatrix},$$\n\nand generally,\n\n$$\\psi_M^r \\Phi_r = \\begin{vmatrix}\n\\phi_{N-r} & S_r[N-M-r+1, p] \\psi_m^{(N-M-p-r+1)} & \\ldots & S_r[N-M-1, p] \\psi_m^{(N-M-p-1)} & S_r[N-M, p] \\psi_m^{(N-M-p)} \\\\\n\\phi_{N-r+1} & S_{r-1}[N-M-r+1, p] \\psi_m^{(N-M-p-r+1)} & \\ldots & S_{r-1}[N-M-1, p] \\psi_m^{(N-M-p-1)} & S_{r-1}[N-M, p] \\psi_m^{(N-M-p)} \\\\\n\\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\\n\\phi_N & 0 & \\ldots & 0 & S_0[N-M, p] \\psi_m^{(N-M-p)}\n\\end{vmatrix}.$$\n\nThese formulae give, for the example discussed above,\n\n$$\\Phi_0 = \\phi_4,$$\n\n$$\\psi_2 \\Phi_1 = \\begin{vmatrix}\n\\phi_3 & \\psi_1 + 2\\psi_2 \\\\\n\\phi_4 & \\psi_2\n\\end{vmatrix},$$\n\n$$\\psi_2^2 \\Phi_2 = \\begin{vmatrix}\n\\phi_2 & \\psi_1 + \\psi_2 & \\psi_0 + 2\\psi_1 + \\psi_2 \\\\\n\\phi_3 & \\psi_2 & \\psi_1 + 2\\psi_2 \\\\\n\\phi_4 & 0 & \\psi_2\n\\end{vmatrix},$$\n\nAnd for the final remainder, the coefficient of $\\pi$,\n\n$$\\psi_2^3 \\left\\{ \\phi_1 - \\frac{1}{\\psi_2} \\Phi_0(\\psi_1 + 2\\psi_0) - \\frac{1}{\\psi_2} \\Phi_1(\\psi_1 + \\psi_0) - \\frac{1}{\\psi_2} \\Phi_2 \\psi_1 \\right\\} = \\begin{vmatrix}\n\\phi_1 & \\psi_1 & \\psi_0 + \\psi_1 & 2\\psi_0 + \\psi_1 \\\\\n\\phi_2 & \\psi_2 & \\psi_1 + \\psi_2 & \\psi_0 + 2\\psi_1 + \\psi_2 \\\\\n\\phi_3 & 0 & \\psi_2 & \\psi_1 + 2\\psi_2 \\\\\n\\phi_4 & 0 & 0 & \\psi_2\n\\end{vmatrix}.$$\nand the terms independent of \\( \\pi \\) are\n\n\\[\n\\psi_2^3 \\left( \\phi_0 - \\frac{1}{\\psi_2} (\\Phi_0 \\psi_3 + \\Phi_1 \\psi_0 + \\Phi_2 \\psi_0) \\right)\n\\]\n\n\\[\n= \\begin{vmatrix}\n\\phi_0 & \\psi_0 & \\psi_0 & \\psi_0 \\\\\n\\phi_2 & \\psi_2 & \\psi_1 + \\psi_2 & \\psi_0 + 2\\psi_1 + \\psi_2 \\\\\n\\phi_3 & 0 & \\psi_1 & \\psi_1 + 2\\psi_2 \\\\\n\\phi_4 & 0 & 0 & \\psi_2\n\\end{vmatrix}\n\\]\n\nboth of which may be comprised under the single formula\n\n\\[\n\\begin{vmatrix}\n\\pi & \\phi_0 & \\psi_0 & \\psi_0 & \\psi_0 \\\\\n1 & \\phi_1 & \\psi_0 + \\psi_1 & 2\\psi_0 + \\psi_1 \\\\\n0 & \\phi_2 & \\psi_1 + \\psi_2 & \\psi_0 + 2\\psi_1 + \\psi_2 \\\\\n0 & \\phi_3 & 0 & \\psi_1 + 2\\psi_2 \\\\\n0 & \\phi_4 & 0 & 0 & \\psi_2\n\\end{vmatrix}\n\\]\n\n§ 4. To divide \\( \\sum_{n=0}^{N} \\phi_n \\pi^n \\) externally by \\( \\sum_{m=0}^{M} \\psi_m \\pi^m \\).\n\nThe first term in the quotient will be\n\n\\[\n\\frac{\\phi_N}{\\psi_M} \\pi^{N-M}, \\quad \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots \\ldots (1.)\n\\]\n\nthe product of this into the divisor\n\n\\[\n\\sum_{m=0}^{M} \\psi_m \\sum_{p=m}^{m} [m, p] \\left( \\frac{\\phi_N}{\\psi_M} \\right)^{(m-p)} \\pi^{N-M+p}, \\quad \\ldots \\ldots \\ldots \\ldots (2.)\n\\]\n\nand the remainder\n\n\\[\n\\sum_{n=0}^{N} \\phi_n \\pi^n - \\sum_{m=0}^{M} \\psi_m \\sum_{p=m}^{m} [m, p] \\left( \\frac{\\phi_N}{\\psi_M} \\right)^{(m-p)} \\pi^{N-M+p}; \\quad \\ldots \\ldots \\ldots \\ldots (3.)\n\\]\n\nthe first term of which is\n\n\\[\n\\left\\{ \\phi_{N-1} - \\sum_{m=0}^{M} \\psi_m [m, M-1] \\left( \\frac{\\phi_N}{\\psi_M} \\right)^{(m-M+1)} \\right\\} \\pi^{N-1}\n\\]\n\n\\[\n= \\left\\{ \\phi_{N-1} - \\psi_{M-1} \\left( \\frac{\\phi_N}{\\psi_M} \\right) - M \\psi_M \\left( \\frac{\\phi_N}{\\psi_M} \\right)' \\right\\} \\pi^{N-1}. \\quad \\ldots \\ldots \\ldots \\ldots (4.)\n\\]\n\nLet \\( \\phi_N = \\Phi_0 \\), and let the coefficient of \\( \\pi^{N-1} \\) above written \\( = \\Phi_1 \\); then the next term in the quotient will be \\( \\frac{\\Phi_1}{\\psi_M} \\pi^{N-M-1} \\), and the product of this into the divisor will take the same form as (2.), writing only \\( \\Phi_1 \\) for \\( \\Phi_0 \\), or \\( \\phi_N \\). The remainder will be the same as (3.), with the addition of the term\n\n\\[\n-\\sum_{m=0}^{M} \\psi_m \\sum_{p=m}^{m} [m, p] \\left( \\frac{\\Phi_1}{\\psi_M} \\right)^{m-p} \\pi^{N-M+p-1}, \\quad \\ldots \\ldots \\ldots \\ldots (5.)\n\\]\n\nand the first term of the entire remainder will be\n\n\\[\n\\left\\{ \\phi_{N-2} - \\sum_{m=0}^{M} \\psi_m [m, M-2] \\left( \\frac{\\phi_0}{\\psi_M} \\right)^{(m-M+2)} + [m, M-1] \\left( \\frac{\\Phi_1}{\\psi_M} \\right)^{(m-M+1)} \\right\\} \\pi^{N-2}\n\\]\n\n\\[\n= \\left\\{ \\phi_{N-2} - \\psi_{M-2} \\left( \\frac{\\phi_0}{\\psi_M} \\right) - (M-1) \\psi_{M-1} \\left( \\frac{\\phi_0}{\\psi_M} \\right)' - \\frac{M(M-1)}{1.2} \\psi_M \\left( \\frac{\\phi_0}{\\psi_M} \\right)'' - \\psi_{M-1} \\left( \\frac{\\Phi_1}{\\psi_M} \\right) - M \\psi_M \\left( \\frac{\\Phi_1}{\\psi_M} \\right)' \\right\\} \\pi^{N-2}; \\quad \\ldots \\ldots \\ldots \\ldots (6.)\n\\]\nand if we make this expression \\( = \\Phi_2 \\pi^{N-2} \\), the next remainder will be\n\n\\[\n\\sum_{n=0}^{N} \\varphi_n \\pi^n - \\sum_{m=0}^{M} \\psi_m \\sum_{p=m}^{m} [m, p] \\left\\{ \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{(m-p)} \\pi^2 + \\left( \\frac{\\Phi_1}{\\psi_m} \\right)^{(m-p)} \\pi + \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{(m-p)} \\right\\} \\pi^{N-M+p-3},\n\\]\n\nthe first term of which is\n\n\\[\n\\varphi_{N-3} - \\sum_{m=0}^{M} \\psi_m \\left[ m, M-3 \\right] \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{m-M+3} + \\left[ m, M-2 \\right] \\left( \\frac{\\Phi_1}{\\psi_m} \\right)^{m-M+2} + \\left[ m, M-1 \\right] \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{m-M+1} \\right] \\pi^{N-2}\n\\]\n\n\\[\n= \\left\\{ \\varphi_{N-3} - \\psi_{M-3} \\left( \\frac{\\Phi_0}{\\psi_m} \\right) - (M-2) \\psi_{M-2} \\left( \\frac{\\Phi_0}{\\psi_m} \\right)' - \\frac{(M-1)(M-2)}{1.2} \\psi_{M-1} \\left( \\frac{\\Phi_0}{\\psi_m} \\right)\" - \\frac{M(M-1)(M-2)}{1.2.3} \\psi_M \\left( \\frac{\\Phi_0}{\\psi_m} \\right)\"'\n\\]\n\n\\[\n- \\psi_{M-2} \\left( \\frac{\\Phi_1}{\\psi_m} \\right) - (M-1) \\psi_{M-1} \\left( \\frac{\\Phi_1}{\\psi_m} \\right)' - \\frac{M(M-1)}{1.2} \\psi_M \\left( \\frac{\\Phi_1}{\\psi_m} \\right)\"\n\\]\n\n\\[\n- \\psi_{M-1} \\left( \\frac{\\Phi_2}{\\psi_m} \\right) - M \\psi_M \\left( \\frac{\\Phi_2}{\\psi_m} \\right)' \\right\\} \\pi^{N-3};\n\\]\n\nand generally the \\((r+1)\\)th term in the quotient will be\n\n\\[\n\\frac{\\Phi_r}{\\psi_m} \\pi^{N-M-r},\n\\]\n\nand the \\((r+1)\\)th remainder\n\n\\[\n\\sum_{n=0}^{N} \\varphi_n \\pi^n - \\sum_{m=0}^{M} \\psi_m \\sum_{p=m}^{m} [m, p] \\sum_{q=r}^{q} \\left( \\frac{\\Phi_q}{\\psi_m} \\right)^{(m-p)} \\pi^{N-M+p-q}.\n\\]\n\nThe formation of the \\(\\Phi_s\\)s is as follows:\n\n\\[\n\\Phi_0 = \\varphi_N,\n\\]\n\n\\[\n\\Phi_1 = \\varphi_{N-1} - \\sum_{m=0}^{M} \\psi_m \\left[ m, M-1 \\right] \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{(m-M+1)},\n\\]\n\n\\[\n\\Phi_2 = \\varphi_{N-2} - \\sum_{m=0}^{M} \\psi_m \\left[ m, M-1 \\right] \\left( \\frac{\\Phi_1}{\\psi_m} \\right)^{(m-M+1)} + \\left[ m, M-2 \\right] \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{(m-M+2)}\n\\]\n\n\\[\n+ \\left[ m, M-3 \\right] \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{(m-M+3)},\n\\]\n\n\\[\n\\Phi_3 = \\varphi_{N-3} - \\sum_{m=0}^{M} \\psi_m \\left[ m, M-1 \\right] \\left( \\frac{\\Phi_2}{\\psi_m} \\right)^{(m-M+1)} + \\left[ m, M-2 \\right] \\left( \\frac{\\Phi_1}{\\psi_m} \\right)^{(m-M+2)}\n\\]\n\n\\[\n+ \\left[ m, M-3 \\right] \\left( \\frac{\\Phi_0}{\\psi_m} \\right)^{(m-M+3)},\n\\]\n\n\\[\n\\Phi_s = \\varphi_{N-s} - \\sum_{m=0}^{M} \\psi_m \\left[ m, M-1 \\right] \\left( \\frac{\\Phi_{s-1}}{\\psi_m} \\right)^{(m-M+1)} + \\left[ m, M-2 \\right] \\left( \\frac{\\Phi_{s-2}}{\\psi_m} \\right)^{(m-M+2)}\n\\]\n\n\\[\n+ \\left[ m, 1 \\right] \\left( \\frac{\\Phi_{s-M+1}}{\\psi_m} \\right)^{(M-1)} + \\left[ m, 0 \\right] \\left( \\frac{\\Phi_{s-M}}{\\psi_m} \\right)^{(M)}.\n\\]\n\nThe final remainder is given by the formula\n\n\\[\n\\sum_{n=0}^{N} \\varphi_n \\pi^n - \\sum_{m=0}^{M} \\psi_m \\sum_{p=m}^{m} [m, p] \\sum_{q=0}^{N-M} \\left( \\frac{\\Phi_q}{\\psi_m} \\right)^{(m-p)} \\pi^{N-M+p-q};\n\\]\n\nand the general term of this \\(\\pi^{N-s}\\) is to be found as follows:\n\n\\[\nN-M+p-q=N-s,\n\\]\n\ni.e.\n\n\\[\np-q=M-s.\n\\]\nThen we have for\n\n\\[ p = M, \\quad q = s, \\]\n\n\\[ \\sum_{m=0}^{M} \\psi_m[m, M] \\left( \\frac{\\Phi_s}{\\psi_M} \\right)^{(m-M)} = \\Phi_s; \\]\n\n\\[ p = M-1, \\quad q = s-1, \\]\n\n\\[ \\sum_{m=0}^{M} \\psi_m[m, M-1] \\left( \\frac{\\Phi_{s-1}}{\\psi_M} \\right)^{(m-M+1)} = M \\psi_M \\left( \\frac{\\Phi_{s-1}}{\\psi_M} \\right) + \\psi_{M-1} \\left( \\frac{\\Phi_{s-1}}{\\psi_M} \\right); \\]\n\n\\[ p = M-2, \\quad q = s-2, \\]\n\n\\[ \\sum_{m=0}^{M} \\psi_m[m, M-2] \\left( \\frac{\\Phi_{s-2}}{\\psi_M} \\right)^{(m-M+2)} = \\frac{M(M-1)}{1 \\cdot 2} \\psi_M \\left( \\frac{\\Phi_{s-2}}{\\psi_M} \\right) + (M-1) \\psi_{M-1} \\left( \\frac{\\Phi_{s-2}}{\\psi_M} \\right) + \\psi_{M-2} \\left( \\frac{\\Phi_{s-2}}{\\psi_M} \\right); \\]\n\n\\[ p = 1, \\quad q = s-M+1, \\]\n\n\\[ \\sum_{m=0}^{M} \\psi_m[m, 1] \\left( \\frac{\\Phi_{s-M+1}}{\\psi_M} \\right)^{(m-1)} = M \\psi_M \\left( \\frac{\\Phi_{s-M+1}}{\\psi_M} \\right) + (M-1) \\psi_{M-1} \\left( \\frac{\\Phi_{s-M+1}}{\\psi_M} \\right) + \\ldots \\psi_1 \\left( \\frac{\\Phi_{s-M+1}}{\\psi_M} \\right); \\]\n\n\\[ p = 0, \\quad q = s-M, \\]\n\n\\[ \\sum_{m=0}^{M} \\psi_m[m, 0] \\left( \\frac{\\Phi_{s-M}}{\\psi_M} \\right)^{(m)} = \\psi_M \\left( \\frac{\\Phi_{s-M}}{\\psi_M} \\right) + \\psi_{M-1} \\left( \\frac{\\Phi_{s-M}}{\\psi_M} \\right) + \\ldots \\psi_0 \\left( \\frac{\\Phi_{s-M}}{\\psi_M} \\right); \\]\n\nthe sum of all which will be found, on reference to the expressions for the formation of the \\( \\Phi_s \\), to be equal to the first term of \\( \\Phi_s \\), viz. \\( \\phi_{N-s} \\); and consequently the coefficient of \\( \\pi^{N-s} \\) vanishes for all values of \\( s \\) not exceeding the greatest value of \\( q \\), viz. \\( N-M \\).\n\nIf, however, \\( s \\) is greater than \\( N-M \\), by any number \\( t \\), so that \\( s = N-M+t \\), then the pairs of values\n\n\\[ p = M, \\quad q = s, \\]\n\\[ p = M-1, \\quad q = s-1, \\]\n\\[ p = M-t+1, \\quad q = s-t+1 \\]\n\nare inadmissible, and the pairs\n\n\\[ p = M-t, \\quad q = s-t, \\]\n\\[ p = M-t-1, \\quad q = s-t-1, \\]\n\\[ p = 0, \\quad q = s-M \\]\n\nalone remain; and consequently the coefficients of the powers of \\( \\pi \\), for \\( s > N-M \\), do not vanish, and the remainder consists of a series of terms, the index of the highest power of \\( \\pi \\) being\n\n\\[ N-M+p-q = N-s = N-N+M-1 = M-1, \\]\n\nas it should be.\n\nAs an example, we may calculate by means of the formulæ given above, the final remainder in the external division of\n\n\\[ \\varphi_4(\\varepsilon)\\pi^4 + \\varphi_3(\\varepsilon)\\pi^3 + \\varphi_2(\\varepsilon)\\pi^2 + \\varphi_1(\\varepsilon)\\pi + \\varphi_0(\\varepsilon) \\]\nby\n\n\\[ \\psi_3(\\varepsilon)\\pi^3 + \\psi_2(\\varepsilon)\\pi^2 + \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon), \\]\n\nviz.\n\n\\[ \\sum_{n=0}^{m-4} q_n \\pi^n - \\sum_{m=0}^{m-2} \\sum_{p=0}^{m-p} \\sum_{q=0}^{m-p} \\psi_m[m, p] \\left( \\frac{\\Phi_q}{\\psi_m} \\right)^{m-p} \\pi^{m+p-q}. \\]\n\nThe conditions\n\n\\[ p=2, q=0 \\text{ give } \\Phi_0\\pi^4 \\]\n\\[ p=2, q=1 \\text{ --- } \\Phi_1\\pi^3 \\]\n\\[ p=2, q=2 \\text{ --- } \\Phi_2\\pi^2 \\]\n\\[ p=1, q=0 \\text{ --- } \\left\\{ 2\\psi_2 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) + \\psi_1 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\right\\} \\pi^2 \\]\n\\[ p=1, q=1 \\text{ --- } \\left\\{ 2\\psi_2 \\left( \\frac{\\Phi_1}{\\psi_2} \\right) + \\psi_1 \\left( \\frac{\\Phi_1}{\\psi_2} \\right) \\right\\} \\pi^2 \\]\n\\[ p=1, q=2 \\text{ --- } \\left\\{ 2\\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) + \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) \\right\\} \\pi \\]\n\\[ p=0, q=0 \\text{ --- } \\left\\{ \\psi_2 \\left( \\frac{\\Phi_0}{\\psi_2} \\right)'' + \\psi_1 \\left( \\frac{\\Phi_0}{\\psi_2} \\right)' + \\psi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\right\\} \\pi^2 \\]\n\\[ p=0, q=1 \\text{ --- } \\left\\{ \\psi_2 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)'' + \\psi_1 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)' + \\psi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\right\\} \\pi \\]\n\\[ p=0, q=2 \\text{ --- } \\left\\{ \\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)'' + \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)' + \\psi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\right\\}. \\]\n\nHence taking the sum of all the terms, the coefficients of \\( \\pi^4, \\pi^3, \\pi^2 \\) vanish, and the final remainder is\n\n\\[ \\left\\{ \\phi_1 - 2\\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)' - \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) - \\psi_2 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)'' - \\psi_1 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)' - \\psi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\right\\} \\pi + \\phi_0 - \\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)' - \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) - \\psi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\right). \\]\n\nThese results may be compared with the actual division,\n\n\\[ \\psi_2\\pi^3 + \\psi_1\\pi + \\psi_0 \\left( \\phi_1\\pi^4 + \\phi_3\\pi^3 + \\phi_2\\pi^2 + \\phi_1\\pi + \\phi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\pi^2 + \\frac{\\Phi_1}{\\psi_2} \\pi + \\frac{\\Phi_2}{\\psi_2} \\right) \\right. \\]\n\\[ \\left. + \\psi_2 \\left( \\frac{\\Phi_0}{\\psi_2} \\right)' \\pi^2 + \\psi_2 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)'' \\pi^2 \\right) \\]\n\\[ + \\psi_1 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\pi^2 + \\psi_1 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)' \\pi^2 \\]\n\\[ + \\psi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\pi^2 \\]\n\\[ \\phi_1\\pi^3 + \\left\\{ \\phi_2 - \\psi_2 \\left( \\frac{\\Phi_0}{\\psi_2} \\right)'' - \\psi_1 \\left( \\frac{\\Phi_0}{\\psi_2} \\right)' - \\psi_0 \\left( \\frac{\\Phi_0}{\\psi_2} \\right) \\right\\} \\pi^2 + \\phi_1\\pi + \\phi_0 \\]\n\\[ \\phi_1\\pi^3 + \\left\\{ 2\\psi_2 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)' \\pi^2 + \\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)'' \\pi \\right. \\]\n\\[ + \\psi_1 \\left( \\frac{\\Phi_1}{\\psi_2} \\right) \\pi^2 + \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)' \\pi \\]\n\\[ + \\psi_0 \\left( \\frac{\\Phi_1}{\\psi_2} \\right) \\pi \\]\n\\[ \\Phi_2 \\pi^2 + \\left\\{ \\varphi_1 - \\psi_2 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)'' - \\psi_1 \\left( \\frac{\\Phi_1}{\\psi_2} \\right)' - \\psi_0 \\left( \\frac{\\Phi_1}{\\psi_2} \\right) \\right\\} \\pi + \\varphi_0, \\]\n\n\\[ \\Phi_3 \\pi^2 + 2 \\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)' \\pi + \\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) + \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) \\pi + \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) + \\psi_0 \\left( \\frac{\\Phi_2}{\\psi_2} \\right), \\]\n\n\\[ \\Phi_3 \\pi + \\left\\{ \\varphi_0 - \\psi_2 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)'' - \\psi_1 \\left( \\frac{\\Phi_2}{\\psi_2} \\right)' - \\psi_0 \\left( \\frac{\\Phi_2}{\\psi_2} \\right) \\right\\}, \\]\n\nwhich agrees with the results found above.\n\n§ 5. To divide \\( \\sum_{n=0}^{N} \\xi^n \\varphi_n(\\pi) \\) internally by \\( \\sum_{m=0}^{M} \\xi^m \\psi_m(\\pi) \\).\n\nThe first term of the quotient will be\n\n\\[ \\xi^{N-M} \\frac{\\varphi_N(\\pi-M)}{\\psi_M(\\pi-M)}, \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad (1.) \\]\n\nand the product of this into the divisor,\n\n\\[ \\sum_{m=0}^{M} \\xi^{N-M+m} \\frac{\\psi_m(\\pi)}{\\psi_M(\\pi-M+m)} \\varphi_N(\\pi-M+m). \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad (2.) \\]\n\nThe first term of the remainder will then be\n\n\\[ \\xi^{N-1} \\left\\{ \\varphi_{N-1}(\\pi) - \\frac{\\psi_{M-1}(\\pi)}{\\psi_M(\\pi-1)} \\varphi_N(\\pi-1) \\right\\} = \\xi^{N-1} \\frac{1}{\\psi_M(\\pi-1)} \\begin{vmatrix} \\varphi_{N-1}(\\pi) & \\psi_{M-1}(\\pi) \\\\ \\varphi_N(\\pi-1) & \\psi_M(\\pi-1) \\end{vmatrix}, \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad (3.) \\]\n\nand consequently the second term in the quotient will be\n\n\\[ \\xi^{N-M-1} \\left\\{ \\varphi_{N-1}(\\pi-M) - \\frac{\\psi_{M-1}(\\pi-M)}{\\psi_M(\\pi-M-1)} \\varphi_N(\\pi-M-1) \\right\\} \\frac{1}{\\psi_M(\\pi-M)} \\begin{vmatrix} \\varphi_{N-1}(\\pi-M) & \\psi_{M-1}(\\pi-M) \\\\ \\varphi_N(\\pi-M-1) & \\psi_M(\\pi-M-1) \\end{vmatrix}, \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad (4.) \\]\n\nThe first term of the second remainder will then be\n\n\\[ \\xi^{N-2} \\left\\{ \\varphi_{N-2}(\\pi) - \\frac{\\psi_{M-1}(\\pi)}{\\psi_M(\\pi-1)\\psi_M(\\pi-2)} \\varphi_N(\\pi-2) \\right\\} \\frac{1}{\\psi_M(\\pi-1)\\psi_M(\\pi-2)} \\begin{vmatrix} \\varphi_{N-2}(\\pi) & \\psi_{M-1}(\\pi) \\\\ \\varphi_{N-1}(\\pi-1) & \\psi_M(\\pi-1) \\\\ \\varphi_N(\\pi-2) & \\psi_M(\\pi-2) \\end{vmatrix}, \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad \\ldots \\quad (5.) \\]\nAnd it is not difficult to see that the first term of the \\( r \\)th remainder will be\n\n\\[\n\\begin{vmatrix}\n\\varphi_{N-r}(\\pi) & \\psi_{M-1}(\\pi) & \\ldots & 0 \\\\\n\\varphi_{N-r+1}(\\pi-1) & \\psi_{M}(\\pi-1) & \\ldots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\varphi_{N}(\\pi-r) & 0 & \\ldots & \\psi_{M}(\\pi-r),\n\\end{vmatrix}\n\\]\n\nin which determinant every column after the first consists of only two terms, viz. \\( \\psi_{M-1}(\\pi-s) \\) and \\( \\psi_{M}(\\pi-s-1) \\). Hence also the \\((r+1)\\)th term of the quotient will be\n\n\\[\n\\begin{vmatrix}\n\\varphi_{N-r}(\\pi-M) & \\psi_{M-1}(\\pi) & \\ldots & 0 \\\\\n\\varphi_{N-r+1}(\\pi-M-1) & \\psi_{M}(\\pi-1) & \\ldots & 0 \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\varphi_{N}(\\pi-r) & 0 & \\ldots & \\psi_{M}(\\pi-r),\n\\end{vmatrix}\n\\]\n\nAs to the other terms, than the first, of the various remainders. In the first remainder, the first term of which is given by (3.), the \\((s+1)\\)th term will be found by making \\( n=N-s, m=M-s \\), in the expression\n\n\\[\n\\sum_{n=0}^{N} \\xi^n \\varphi_n(\\pi) - \\sum_{m=0}^{M} \\xi^{N-M+m} \\frac{\\psi_m(\\pi)}{\\psi_{M}(\\pi-M+m)} \\varphi_{N}(\\pi-M+m),\n\\]\n\nwhich gives\n\n\\[\n\\xi^{N-s} \\left\\{ \\varphi_{N-s}(\\pi) - \\frac{\\psi_{M-s}(\\pi)}{\\psi_{M}(\\pi-s)} \\varphi_{N}(\\pi-s) \\right\\}\n\\]\n\n\\[\n= \\xi^{N-s} \\frac{1}{\\psi_{M}(\\pi-s)} \\begin{vmatrix}\n\\varphi_{N-s}(\\pi) & \\psi_{M-s}(\\pi) \\\\\n\\varphi_{N}(\\pi-s) & \\psi_{M}(\\pi-s).\n\\end{vmatrix}\n\\]\n\nHence the entire first remainder may be expressed thus:\n\n\\[\n\\sum_{s=0}^{M} \\xi^{N-s} \\frac{1}{\\psi_{M}(\\pi-s)} \\begin{vmatrix}\n\\varphi_{N-s}(\\pi) & \\psi_{M-s}(\\pi) \\\\\n\\varphi_{N}(\\pi-s) & \\psi_{M}(\\pi-s).\n\\end{vmatrix}\n\\]\n\nSimilarly, the general expression for the second remainder is\n\n\\[\n\\sum_{n=0}^{N} \\xi^n \\varphi_n(\\pi) - \\sum_{m=0}^{M} \\xi^{N-M+m} \\frac{\\psi_m(\\pi)}{\\psi_{M}(\\pi-M+m)} \\varphi_{N}(\\pi-M+m),\n\\]\n\nwhich may be transformed thus:\n\n\\[\n\\sum_{s=0}^{M} \\xi^{N-s-1} \\left\\{ \\varphi_{N-s-1}(\\pi) - \\frac{\\psi_{M-s}(\\pi)}{\\psi_{M}(\\pi-s)} \\varphi_{N}(\\pi-s) \\right\\}\n\\]\n\n\\[\n= \\sum_{s=0}^{M} \\xi^{N-s-1} \\frac{1}{\\psi_{M}(\\pi-s)} \\begin{vmatrix}\n\\varphi_{N-s-1}(\\pi) & \\psi_{M-s}(\\pi) & 0 \\\\\n\\varphi_{N-1}(\\pi-s) & \\psi_{M}(\\pi-s) & \\psi_{M-1}(\\pi-s) \\\\\n\\varphi_{N}(\\pi-s-1) & 0 & \\psi_{M}(\\pi-s-1).\n\\end{vmatrix}\n\\]\nAnd generally the expression for the \\( t \\)th remainder may be written\n\n\\[\n\\sum_{s=0}^{M} \\xi^{N-s-t+1} \\frac{1}{\\psi_M(\\pi-s)\\psi_M(\\pi-s-1)\\ldots\\psi_M(\\pi-s-t+1)} \\begin{vmatrix}\n\\phi_{N-s+t}(\\pi) & \\psi_{M-s}(\\pi) & \\ldots & 0 \\\\\n\\phi_{N-t+1}(\\pi-s) & \\psi_{M-s}(\\pi-s) & \\ldots & \\psi_{M-t+1}(\\pi-s) \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\phi_N(\\pi-s-t+1) & 0 & \\ldots & \\psi_M(\\pi-s-t+1)\n\\end{vmatrix}\n\\]\n\n(10.)\n\nThe last remainder is the \\((N-M+1)\\)th. Then\n\n\\[\nt = N - M + 1,\n\\]\n\n\\[\nN - s - t + 1 = N - s - N + M - 1 + 1 = M - s\n\\]\n\n\\[\nN - t + 1 = N - N + M - 1 + 1 = M\n\\]\n\n\\[\nM - t + 1 = M - N + M - 1 + 1 = 2M - N,\n\\]\n\nand the remainder in question\n\n\\[\n\\sum_{s=0}^{M} \\xi^{M-s} \\frac{1}{\\psi_M(\\pi-s)\\psi_M(\\pi-s-1)\\ldots\\psi_M(\\pi-s-N+M)} \\begin{vmatrix}\n\\phi_{M-s}(\\pi) & \\psi_{M-s}(\\pi) & \\ldots & 0 \\\\\n\\phi_M(\\pi-s) & \\psi_M(\\pi-s) & \\ldots & \\psi_{2M-N}(\\pi-s) \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\phi_N(\\pi-s-N+M) & 0 & \\ldots & \\psi_M(\\pi-s-N+M),\n\\end{vmatrix}\n\\]\n\n(11.)\n\nin which the coefficient of \\( \\xi^M \\) vanishes, as it should. The last term, viz. that independent of \\( \\xi \\),\n\n\\[\n\\frac{1}{\\psi_M(\\pi-M)\\psi_M(\\pi-M-1)\\ldots\\psi_M(\\pi-N)} \\begin{vmatrix}\n\\phi_0(\\pi) & \\psi_0(\\pi) & \\ldots & 0 \\\\\n\\phi_M(\\pi-M) & \\psi_M(\\pi-M) & \\ldots & \\psi_{2M-N}(\\pi-M) \\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\phi_N(\\pi-N) & 0 & \\ldots & \\psi_M(\\pi-N);\n\\end{vmatrix}\n\\]\n\n(12.)\n\nand if \\( N = 1 \\), the result agrees with that given by Mr. Russell*.\n\n§ 6. To divide \\( \\sum_{n=0}^{N} \\xi^n \\phi_n(\\pi) \\) externally by \\( \\sum_{m=0}^{M} \\xi^m \\psi_m(\\pi) \\).\n\nThe first term of the quotient will be\n\n\\[\n\\xi^{N-M} \\frac{\\phi_N(\\pi)}{\\psi_M(\\pi+N-M)}.\n\\]\n\n(1.)\n\nThe first remainder\n\n\\[\n\\sum_{n=0}^{N} \\xi^n \\phi_n(\\pi) - \\sum_{m=0}^{M} \\xi^{N-M+m} \\frac{\\psi_m(\\pi+N-M)}{\\psi_M(\\pi+N-M)} \\phi_N(\\pi)\n\\]\n\n\\[\n= \\sum_{s=0}^{M} \\xi^{N-s} \\left\\{ \\phi_{N-s}(\\pi) - \\frac{\\psi_{M-s}(\\pi+N-M)}{\\psi_M(\\pi+N-M)} \\phi_N(\\pi) \\right\\},\n\\]\n\n(2.)\n\n* Philosophical Transactions, vol. cli. p. 72.\nthe first term of which is\n\n\\[ \\varepsilon^{N-1} \\left\\{ \\varphi_{N-1}(\\pi) - \\frac{\\psi_{M-1}(\\pi+N-M)}{\\psi_M(\\pi+N-M)} \\varphi_N(\\pi) \\right\\}, \\]\n\nwhence the second term of the quotient will be\n\n\\[ \\varepsilon^{N-M-1} \\frac{1}{\\psi_M(\\pi+N-M-1)} \\left\\{ \\varphi_{N-1}(\\pi) - \\frac{\\psi_{M-1}(\\pi+N-M)}{\\psi_M(\\pi+N-M)} \\varphi_N(\\pi) \\right\\}. \\]\n\nSimilarly, the second remainder will be\n\n\\[ \\sum_{n=0}^{N-1} \\varepsilon^n \\varphi_n(\\pi) - \\sum_{m=0}^{M-N-M+1} \\frac{\\psi_m(\\pi+N-M-1)}{\\psi_M(\\pi+N-M-1)\\psi_M(\\pi+N-M)} \\begin{vmatrix} \\varphi_{N-1}(\\pi) & \\psi_{M-1}(\\pi+N-M) \\\\ \\varphi_N(\\pi) & \\psi_M(\\pi+N-M) \\end{vmatrix}. \\]\n\nThe \\( t \\)th remainder\n\n\\[ = \\sum_{s=0}^{M} \\varepsilon^{N-s-t+1} \\frac{1}{\\psi_M(\\pi+N-M-t+1)\\psi_M(\\pi+N-M-t+2) \\ldots \\psi_M(\\pi+N-M)} \\begin{vmatrix} \\varphi_{N-s-t+1}(\\pi) & \\psi_{M-s}(\\pi+N-M-t+1) & \\ldots & 0 \\\\ \\varphi_{N-t+1}(\\pi) & \\psi_M(\\pi+N-M-t+1) & \\ldots & \\psi_{M-t+1}(\\pi+N-M) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\varphi_N(\\pi) & 0 & \\ldots & \\psi_M(\\pi+N-M) \\end{vmatrix}; \\]\n\nand the last remainder, viz. the \\((N-M+1)\\)th,\n\n\\[ = \\sum_{s=0}^{M} \\varepsilon^{M-s} \\frac{1}{\\psi_M(\\pi)\\psi_M(\\pi+1) \\ldots \\psi_M(\\pi+N-M)} \\begin{vmatrix} \\varphi_{M-s}(\\pi) & \\psi_{M-s}(\\pi) & \\ldots & 0 \\\\ \\varphi_M(\\pi) & \\psi_M(\\pi) & \\ldots & \\psi_{2M-N}(\\pi+N-M) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\varphi_N(\\pi) & 0 & \\ldots & \\psi_M(\\pi+N-M) \\end{vmatrix}. \\]\n\nIn the case considered by Mr. Russell, viz. \\( M=1 \\), (6.) gives only the single term\n\n\\[ (\\psi_1(\\pi)\\psi_1(\\pi+1) \\ldots \\psi_1(\\pi+N-1))^{-1} \\begin{vmatrix} \\varphi_0(\\pi) & \\psi_0(\\pi) & \\ldots & 0 \\\\ \\varphi_1(\\pi) & \\psi_1(\\pi) & \\ldots & 0 \\\\ \\varphi_{N-1}(\\pi) & 0 & \\ldots & \\psi_0(\\pi+N-1) \\\\ \\varphi_N(\\pi) & 0 & \\ldots & \\psi_1(\\pi+N-1) \\end{vmatrix}. \\]\nand in this the coefficient of $\\phi_i(\\pi)$ is $(-)^i \\times$\n\n$$\\begin{array}{ccccccc}\n\\Psi_0(\\pi) & 0 & \\ldots & 0 & 0 & \\ldots & 0 \\\\\n\\Psi_1(\\pi) & \\Psi_0(\\pi+1) & \\ldots & 0 & 0 & \\ldots & 0 \\\\\n& \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & \\ldots & \\Psi_0(\\pi+i-1) & 0 & \\ldots & 0 \\\\\n0 & 0 & \\ldots & 0 & \\Psi_1(\\pi+i) & \\ldots & 0 \\\\\n& \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & \\ldots & 0 & 0 & \\ldots & \\Psi_1(\\pi+N-2) \\Psi_0(\\pi+N-1) \\\\\n0 & 0 & \\ldots & 0 & 0 & \\ldots & \\Psi_1(\\pi+N-1)\n\\end{array}$$\n\n$$= \\Psi_0(\\pi)\\Psi_0(\\pi+1) \\ldots \\Psi_0(\\pi+i-1)\\Psi_1(\\pi+i) \\ldots \\Psi_1(\\pi+N-2)\\Psi_1(\\pi+N-1);$$\n\nwhence the whole expression\n\n$$\\sum_{i=0}^{N} (-)^i \\phi_i(\\pi) \\frac{\\Psi_0(\\pi)\\Psi_0(\\pi+1) \\ldots \\Psi_0(\\pi+i-1)}{\\Psi_1(\\pi)\\Psi_1(\\pi+1) \\ldots \\Psi_1(\\pi+i-1)},$$\n\nwhich agrees with the result given in the Philosophical Transactions, vol. cli. p. 73.\n\nIn the particular case of $N=4$, $M=2$, the final remainder in internal division is\n\n$$\\frac{1}{\\Psi_2(\\pi-1)\\Psi_2(\\pi-2)\\Psi_2(\\pi-3)} \\begin{array}{cccc}\n\\phi_1(\\pi) & \\psi_1(\\pi) & 0 & 0 \\\\\n\\phi_2(\\pi-1) & \\psi_2(\\pi-1) & \\psi_1(\\pi-1) & \\psi_0(\\pi-1) \\\\\n\\phi_3(\\pi-2) & 0 & \\psi_2(\\pi-2) & \\psi_1(\\pi-2) \\\\\n\\phi_4(\\pi-3) & 0 & 0 & \\psi_2(\\pi-3)\n\\end{array}$$\n\n$$+ \\frac{1}{\\Psi_2(\\pi-2)\\Psi_2(\\pi-3)\\Psi_2(\\pi-4)} \\begin{array}{cccc}\n\\phi_0(\\pi) & \\psi_0(\\pi) & 0 & 0 \\\\\n\\phi_2(\\pi-2) & \\psi_2(\\pi-2) & \\psi_1(\\pi-2) & \\psi_0(\\pi-2) \\\\\n\\phi_3(\\pi-3) & 0 & \\psi_2(\\pi-3) & \\psi_1(\\pi-3) \\\\\n\\phi_4(\\pi-4) & 0 & 0 & \\psi_2(\\pi-4)\n\\end{array};$$\n\nand in external division it is\n\n$$\\frac{1}{\\Psi_2(\\pi)\\Psi_2(\\pi+1)\\Psi_2(\\pi+2)} \\begin{array}{cccc}\n\\phi_1(\\pi) & \\psi_1(\\pi) & 0 & 0 \\\\\n\\phi_2(\\pi) & \\psi_2(\\pi) & \\psi_1(\\pi+1) & \\psi_0(\\pi+2) \\\\\n\\phi_3(\\pi) & 0 & \\psi_2(\\pi+1) & \\psi_1(\\pi+2) \\\\\n\\phi_4(\\pi) & 0 & 0 & \\psi_2(\\pi+2)\n\\end{array}$$\n\n$$+ \\frac{1}{\\Psi_2(\\pi)\\Psi_2(\\pi+1)\\Psi_2(\\pi+2)} \\begin{array}{cccc}\n\\phi_0(\\pi) & \\psi_0(\\pi) & 0 & 0 \\\\\n\\phi_2(\\pi) & \\psi_2(\\pi) & \\psi_1(\\pi+1) & \\psi_0(\\pi+2) \\\\\n\\phi_3(\\pi) & 0 & \\psi_2(\\pi+1) & \\psi_1(\\pi+1) \\\\\n\\phi_4(\\pi) & 0 & 0 & \\psi_2(\\pi).\n\\end{array}$$\nThe expressions (10.) for the formation of the $\\Phi$s admit of further development thus:\n\n$$\\Phi_0 = \\phi_N$$\n\n$$\\Phi_1 = \\phi_{N-1} - M \\psi_M \\left( \\frac{\\phi_N}{\\psi_M} \\right)' - \\psi_{M-1} \\left( \\frac{\\phi_N}{\\psi_M} \\right)$$\n\n$$\\Phi_2 = \\phi_{N-2} - \\frac{M(M-1)}{1.2} \\psi_M \\left( \\frac{\\phi_N}{\\psi_M} \\right)'' - (M-1) \\psi_{M-1} \\left( \\frac{\\phi_N}{\\psi_M} \\right)' - \\psi_{M-2} \\left( \\frac{\\phi_N}{\\psi_M} \\right)$$\n\n$$+ M^2 \\psi_M \\left( \\frac{\\phi_N}{\\psi_M} \\right)'' + M \\psi_{M-1} \\left( \\frac{\\phi_N}{\\psi_M} \\right)' + M \\psi_M \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)' \\left( \\frac{\\phi_N}{\\psi_M} \\right) - M \\psi_M \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)'$$\n\n$$+ M \\psi_{M-1} \\left( \\frac{\\phi_N}{\\psi_M} \\right)' + \\psi_{M-1} \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right) \\left( \\frac{\\phi_N}{\\psi_M} \\right) - \\psi_{M-1} \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)$$\n\n$$= \\frac{(M+1)M}{1.2} \\psi_M \\left( \\frac{\\phi_N}{\\psi_M} \\right)'' + (M+1) \\psi_{M-1} \\left( \\frac{\\phi_N}{\\psi_M} \\right)' + \\left\\{ - \\psi_{M-2} + M \\psi_M \\left( \\frac{\\phi_{M-1}}{\\psi_M} \\right)' + \\psi_{M-1} \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right) \\right\\} \\left( \\frac{\\phi_N}{\\psi_M} \\right)$$\n\n$$- M \\psi_M \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)' - \\psi_{M-1} \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)$$\n\n$$+ \\phi_{N-2};$$\n\nor writing, by analogy to the $\\Phi$s,\n\n$$\\Psi_0 = \\psi_{M-1}$$\n\n$$- \\Psi_1 = \\psi_{M-2} - M \\psi_M \\left( \\frac{\\psi_0}{\\psi_M} \\right)' - \\psi_{M-1} \\left( \\frac{\\psi_0}{\\psi_M} \\right),$$\n\nthe expression for $\\Phi_2$ becomes\n\n$$\\Phi_2 = \\frac{(M+1)M}{1.2} \\psi_M \\left( \\frac{\\phi_N}{\\psi_M} \\right)'' + (M+1) \\psi_0 \\left( \\frac{\\phi_N}{\\psi_M} \\right)' + \\psi_1 \\left( \\frac{\\phi_N}{\\psi_M} \\right)$$\n\n$$- M \\psi_M \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)' - \\psi_0 \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)$$\n\n$$+ \\psi_M \\left( \\frac{\\phi_{N-2}}{\\psi_M} \\right).$$\n\nAnd so likewise writing\n\n$$\\Psi_2 = \\psi_{M-3} - \\frac{M(M-1)}{1.2} \\psi_M \\left( \\frac{\\psi_0}{\\psi_M} \\right)'' - (M-1) \\psi_{M-1} \\left( \\frac{\\psi_0}{\\psi_M} \\right)' - \\psi_{M-2} \\left( \\frac{\\psi_0}{\\psi_M} \\right)$$\n\n$$- M \\psi_M \\left( \\frac{\\psi_1}{\\psi_M} \\right)' - \\psi_{M-1} \\left( \\frac{\\psi_1}{\\psi_M} \\right),$$\n\nit will be found that\n\n$$\\Phi_3 = \\frac{M(M-1)(M-2)}{1.2.3} \\psi_M \\left( \\frac{\\phi_0}{\\psi_M} \\right)''' - \\frac{(M-1)(M-2)}{1.2} \\psi_{M-1} \\left( \\frac{\\phi_0}{\\psi_M} \\right)'' - (M-2) \\psi_{M-2} \\left( \\frac{\\phi_0}{\\psi_M} \\right)' - \\psi_{M-3} \\left( \\frac{\\phi_0}{\\psi_M} \\right)$$\n\n$$- \\frac{M(M-1)}{1.2} \\psi_M \\left( \\frac{\\phi_1}{\\psi_M} \\right)'' - (M-1) \\psi_{M-1} \\left( \\frac{\\phi_1}{\\psi_M} \\right)' - \\psi_{M-2} \\left( \\frac{\\phi_1}{\\psi_M} \\right)$$\n\n$$- M \\psi_M \\left( \\frac{\\phi_2}{\\psi_M} \\right)' - \\psi_{M-1} \\left( \\frac{\\phi_2}{\\psi_M} \\right).$$\n\\[\n= - \\frac{(M+2)(M+1)M}{1.2.3} \\psi_M \\left( \\frac{\\phi_N}{\\psi_M} \\right)^{'''} - \\frac{(M+2)(M+1)}{1.2} \\psi_0 \\left( \\frac{\\phi_N}{\\psi_M} \\right)^{''} - (M+2) \\psi_1 \\left( \\frac{\\phi_N}{\\psi_M} \\right)' - \\psi_2 \\left( \\frac{\\phi_N}{\\psi_M} \\right)\n\\]\n\\[\n+ \\frac{(M+1)M}{1.2} \\psi_M \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)^{''} + (M+1) \\psi_0 \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)' + \\psi_1 \\left( \\frac{\\phi_{N-1}}{\\psi_M} \\right)\n\\]\n\\[\n- M \\psi_M \\left( \\frac{\\phi_{N-2}}{\\psi_M} \\right)^{''} - \\psi_0 \\left( \\frac{\\phi_{N-2}}{\\psi_M} \\right)\n\\]\n\\[\n+ \\psi_M \\left( \\frac{\\phi_{N-3}}{\\psi_M} \\right).\n\\]\n\nBut the law of the expressions in the first form having been established above, it is unnecessary to pursue these latter formulae further.", "source": "olmocr", "added": "2026-01-12", "created": "2026-01-12", "metadata": { "Source-File": "/home/jic823/projects/def-jic823/royalsociety/pdfs/108824.pdf", "olmocr-version": "0.3.4", "pdf-total-pages": 23, "total-input-tokens": 37628, "total-output-tokens": 23814, "total-fallback-pages": 0 }, "attributes": { "pdf_page_numbers": [ [ 0, 0, 1 ], [ 0, 1865, 2 ], [ 1865, 5059, 3 ], [ 5059, 7378, 4 ], [ 7378, 8079, 5 ], [ 8079, 10108, 6 ], [ 10108, 13037, 7 ], [ 13037, 15634, 8 ], [ 15634, 18342, 9 ], [ 18342, 20325, 10 ], [ 20325, 23008, 11 ], [ 23008, 25072, 12 ], [ 25072, 27856, 13 ], [ 27856, 30802, 14 ], [ 30802, 33232, 15 ], [ 33232, 36251, 16 ], [ 36251, 38641, 17 ], [ 38641, 40851, 18 ], [ 40851, 42853, 19 ], [ 42853, 44639, 20 ], [ 44639, 46777, 21 ], [ 46777, 49579, 22 ], [ 49579, 50334, 23 ] ], "primary_language": [ "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en", "en" ], "is_rotation_valid": [ true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true ], "rotation_correction": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "is_table": [ false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, true, false, false ], "is_diagram": [ false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false ] }, "jstor_metadata": { "identifier": "jstor-108824", "title": "On the Calculus of Symbols", "authors": "William Spottiswoode", "year": 1862, "volume": "152", "journal": "Philosophical Transactions of the Royal Society of London", "page_count": 23, "jstor_url": "https://www.jstor.org/stable/108824" } }