{ "id": "6c9ec36960a43c3e18a2a469ea4f758a77e2d4ed", "text": "V. On the Calculus of Symbols, with Applications to the Theory of Differential Equations.\n\nBy W. H. L. Russell, Esq., A.B. Communicated by Arthur Cayley, Esq., F.R.S.\n\nReceived December 20, 1860,—Read January 24, 1861.\n\nThe calculus of generating functions, discovered by Laplace, was, as is well known, highly instrumental in calling the attention of mathematicians to the analogy which exists between differentials and powers. This analogy was perceived at length to involve an essential identity, and several analysts devoted themselves to the improvement of the new methods of calculation which were thus called into existence. For a long time the modes of combination assumed to exist between different classes of symbols were those of ordinary algebra; and this sufficed for investigations respecting functions of differential coefficients and constants, and consequently for the integration of linear differential equations, with constant coefficients. The laws of combination of ordinary algebraical symbols may be divided into the commutative and distributive laws; and the number of symbols in the higher branches of mathematics, which are commutative with respect to one another, is very small. It became then necessary to invent an algebra of non-commutative symbols. This important step was effected by Professor Boole, for certain classes of symbols, in his well-known and beautiful memoir published in the Transactions of this Society for the year 1844, and the object of the paper which I have now the honour to lay before the Society is to perfect and develope the methods there employed.\n\nFor this purpose I have constructed systems of multiplication and division for functions of non-commutative symbols, subject to the same laws of combination as those assumed in Professor Boole's memoir, and I thus arrive at equations of great utility in the integration of linear differential equations with variable coefficients.\n\nI then proceed to develope certain general theorems, which will, I hope, be found interesting. I have applied the methods of multiplication, as just explained, to deduce theorems for non-commutative symbols analogous to the binomial and multinomial theorems of ordinary algebra.\n\nLastly, I have shown how to employ the equations deduced in the earlier part of this paper in the integration of linear differential equations. I have, for this purpose, made use of methods closely resembling the method of divisors which has so long been used in resolving ordinary algebraical equations. The whole paper will, I hope, be found to be a step upwards in the important subject of which it treats. I shall just observe, that the symbolical combinations used in this paper may also be applied to the calculus of finite differences, as may be seen in Professor Boole's memoir.\n\nMDCCCLXI.\nSection I. On the Principles of Symbolical Algebra.\n\nLet \\( \\xi \\) and \\( \\pi \\) be two functional symbols combining according to the law \\( \\xi^n f(\\pi) u = f(\\pi - n) \\xi^n u \\), where \\( (u) \\) is the subject. We shall suppose throughout this paper that \\( \\xi = x, \\quad \\pi = x \\frac{d}{dx} \\).\n\nLet \\( P, Q, \\) and \\( R \\) be three functions of \\( (\\pi) \\) and \\( (\\xi) \\), such that \\( PQ \\) acting on any subject is equivalent to \\( R \\) acting on the same subject, or \\( PQ = R \\). We shall say that \\( P \\) externally multiplies \\( Q \\), and is an external factor of \\( R \\). In like manner we shall say that \\( Q \\) internally multiplies \\( P \\), and is an internal factor of \\( R \\). We shall also say that \\( R \\) is externally divisible by \\( P \\), internally by \\( Q \\).\n\nWe easily see the truth of the following symbolical equations depending on the laws of combination assumed above:\n\n\\[\n\\xi^a \\pi^b (\\xi^c \\pi^d) u = \\xi^{a+c} (\\pi + a)^b \\pi^d u \\\\\n\\xi^a \\pi^b (\\xi^c \\pi^d)^{-1} u = \\xi^{a-c} (\\pi - a)^b \\pi^d u.\n\\]\n\nWe shall commence with instances of symbolical multiplication and division, when the multipliers and divisors are monomials.\n\nThe following is an instance of external multiplication:\n\n\\[\n\\xi \\pi (\\xi^2 + \\xi \\pi + \\pi^2) = \\xi^3 (\\pi + 2) + \\xi^2 (\\pi^2 + \\pi) + \\xi \\pi^3;\n\\]\n\nthe following is an instance of internal multiplication:\n\n\\[\n(2 \\xi^2 - 3 \\xi \\pi^2 + (\\pi^2 + \\pi)) \\xi \\pi^2 = 2 \\xi^3 \\pi^2 - 3 \\xi^2 (\\pi^2 + \\pi)^2 + \\xi \\pi^2 (\\pi + 1)(\\pi + 2);\n\\]\n\nthe following of external division:\n\n\\[\n(\\xi^3 \\pi)^{-1} (\\xi^4 (\\pi + 2) - 2 \\xi^3 \\pi (\\pi + 1) + 3 \\xi^2 \\pi^3) = \\xi^2 - 2 \\xi \\pi + 3 \\pi^3;\n\\]\n\nthe following of internal division:\n\n\\[\n(\\xi^3 \\pi + 3 \\xi^2 (\\pi^2 + \\pi) + \\xi \\pi (\\pi + 1)^2) (\\xi \\pi)^{-1} = \\xi^2 + 3 \\xi \\pi + \\pi^2.\n\\]\n\nWe shall now consider cases where the multipliers and divisors are polynomials.\n\nThe following are instances of external multiplication:\n\n\\[\n\\xi + \\pi \\\\\n\\xi - \\pi \\\\\n\\xi^2 + \\xi \\pi \\\\\n\\xi^2 - \\xi - \\pi^2 \\\\\n\\xi \\pi^2 - (\\pi + 1) \\\\\n\\xi \\pi - \\pi^2 \\\\\n\\xi^2 (\\pi + 1) \\pi^2 - \\xi \\pi (\\pi + 1) \\\\\n\\xi^2 (\\pi + 1) \\pi^2 - \\xi \\pi (\\pi + 1)(\\pi^2 + \\pi + 1) + \\pi^2 (\\pi + 1).\n\\]\nThe following are instances of internal multiplication:\n\n\\[\n\\begin{align*}\n&\\varepsilon + \\pi \\\\\n&\\varepsilon - \\pi \\\\\n&\\varepsilon^2 + \\varepsilon(\\pi + 1) \\\\\n&- \\varepsilon \\pi \\\\\n&\\varepsilon^2 + \\varepsilon - \\pi^2 \\\\\n\\varepsilon \\pi^2 - (\\pi + 1) \\\\\n\\varepsilon \\pi - \\pi^2 \\\\\n\\varepsilon^2(\\pi + 1)^3 \\pi^2 - \\varepsilon \\pi(\\pi + 2) \\\\\n- \\varepsilon \\pi^4 + \\pi^3(\\pi + 1) \\\\\n\\varepsilon^2(\\pi + 1)^3 \\pi^2 - \\varepsilon \\pi(\\pi^3 + \\pi + 2) + \\pi^3(\\pi + 1).\n\\end{align*}\n\\]\n\nThe results of the four last examples may be written thus:\n\n\\[\n\\begin{align*}\n(\\varepsilon - \\pi)(\\varepsilon + \\pi) &= \\varepsilon^2 - \\varepsilon - \\pi^2 \\\\\n(\\varepsilon \\pi - \\pi^2)(\\varepsilon \\pi^2 - (\\pi + 1)) &= (\\varepsilon^2 \\pi - \\varepsilon(\\pi^2 + \\pi + 1) + \\pi)(\\pi + 1) \\\\\n(\\varepsilon + \\pi)(\\varepsilon - \\pi) &= \\varepsilon^2 + \\varepsilon - \\pi^2 \\\\\n(\\varepsilon \\pi^2 - (\\pi + 1))(\\varepsilon \\pi - \\pi^2) &= (\\varepsilon^2(\\pi^2 + \\pi) - \\varepsilon(\\pi^2 - \\pi + 2) + \\pi)(\\pi + 1).\n\\end{align*}\n\\]\n\nI shall now give some examples of external division, the divisor being a polynomial.\n\n\\[\n\\begin{align*}\n\\varepsilon + \\pi &\\varepsilon^3 + 2\\varepsilon^2(\\pi + 1) + \\varepsilon \\pi(2\\pi + 1) + \\pi^3(\\varepsilon^2 + \\varepsilon \\pi + \\pi^2) \\\\\n\\varepsilon^2 + \\varepsilon^2(\\pi + 2) \\\\\n\\varepsilon^2 \\pi + \\varepsilon \\pi(2\\pi + 1) \\\\\n\\varepsilon^2 \\pi + \\varepsilon \\pi(\\pi + 1) \\\\\n\\varepsilon^2 \\pi^2 + \\pi^3 \\\\\n\\varepsilon \\pi^2 + \\pi^3 \\\\\n\\varepsilon \\pi + \\pi^2 &\\varepsilon^3(\\pi + 2) + \\varepsilon^2(2\\pi + 3) - \\varepsilon(3\\pi^2 + 3\\pi + 1) + \\pi^4(\\varepsilon^3 - \\varepsilon(\\pi + 1) + \\pi^2) \\\\\n\\varepsilon^3(\\pi + 2) + \\varepsilon^2(\\pi + 2)^2 \\\\\n- \\varepsilon^2(\\pi + 1)^2 - \\varepsilon(3\\pi^2 + 3\\pi + 1) \\\\\n- \\varepsilon^2(\\pi + 1)^2 - \\varepsilon(\\pi + 1)^3 \\\\\n\\varepsilon \\pi^3 + \\pi^4 \\\\\n\\varepsilon \\pi^3 + \\pi^4\n\\end{align*}\n\\]\n\nWe shall next consider some examples of internal division.\n\n\\[\n\\begin{align*}\n\\varepsilon + \\pi &\\varepsilon^3 + \\varepsilon^2(2\\pi + 1) + \\varepsilon(2\\pi^2 + 2\\pi + 1) + \\pi^3(\\varepsilon^2 + \\varepsilon \\pi + \\pi^2) \\\\\n\\varepsilon^3 + \\varepsilon^2 \\pi \\\\\n\\varepsilon^2(\\pi + 1) + \\varepsilon(2\\pi^2 + 2\\pi + 1) \\\\\n\\varepsilon^2(\\pi + 1) + \\varepsilon \\pi^2 \\\\\n\\varepsilon(\\pi + 1)^2 + \\pi^3 \\\\\n\\varepsilon(\\pi + 1)^2 + \\pi^3\n\\end{align*}\n\\]\nThe results of the four last examples may be written thus:\n\n\\[\n(\\varepsilon + \\pi)^{-1}(\\varepsilon^3 + 2\\varepsilon^2(\\pi + 1) + \\varepsilon\\pi(2\\pi + 1) + \\pi^2) = \\varepsilon^3 + \\varepsilon\\pi + \\pi^2\n\\]\n\n\\[\n(\\varepsilon + \\pi^2)^{-1}(\\varepsilon^3(\\pi + 2) + \\varepsilon^2(2\\pi + 3) - \\varepsilon(3\\pi^2 + 3\\pi + 1) + \\pi^4) = \\varepsilon^2 - \\varepsilon(\\pi + 1) + \\pi^2\n\\]\n\n\\[\n(\\varepsilon^3 + \\varepsilon^2(2\\pi + 1) + \\varepsilon(2\\pi^2 + 2\\pi + 1) + \\pi^3)(\\varepsilon + \\pi)^{-1} = \\varepsilon^2 + \\varepsilon\\pi + \\pi^2\n\\]\n\n\\[\n(\\varepsilon^3\\pi - 2\\varepsilon^2\\pi + \\varepsilon(\\pi^2 + \\pi) + \\pi^4)(\\varepsilon + \\pi^2)^{-1} = \\varepsilon^3 - \\varepsilon(\\pi + 1) + \\pi^2.\n\\]\n\nI now come to two propositions of great importance.\n\nFirst, to determine the condition that \\( \\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi) \\) shall divide the symbolical function\n\n\\[\n\\varepsilon^n \\varphi_n(\\pi) + \\varepsilon^{n-1} \\varphi_{n-1}(\\pi) + \\varepsilon^{n-2} \\varphi_{n-2}(\\pi) + \\ldots + \\varepsilon \\varphi_1(\\pi) + \\varphi_0(\\pi)\n\\]\n\ninternally without a remainder,\n\n\\[\n\\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi) \\varepsilon^n \\varphi_n(\\pi) + \\varepsilon^{n-1} \\varphi_{n-1}(\\pi) + \\varepsilon^{n-2} \\varphi_{n-2}(\\pi) + \\ldots + \\varepsilon \\varphi_1(\\pi) + \\varphi_0(\\pi)\n\\]\n\nwhere the symbolical quotient is\n\n\\[\n\\varepsilon^{n-1} \\left\\{ \\frac{\\varphi_{n-1}(\\pi)}{\\psi_1(\\pi - 1)} \\right\\} + \\varepsilon^{n-2} \\left\\{ \\frac{\\varphi_{n-2}(\\pi)}{\\psi_1(\\pi - 1)} \\right\\} + \\ldots + \\varepsilon \\left\\{ \\frac{\\varphi_1(\\pi)}{\\psi_1(\\pi - 1)} \\right\\} + \\varphi_0(\\pi)\n\\]\n\nThe required condition is found by equating the remainder to zero; and we have\n\n\\[\n\\varphi_0(\\pi) - \\frac{\\psi_0(\\pi)}{\\psi_1(\\pi - 1)} \\varphi_1(\\pi - 1) + \\frac{\\psi_0(\\pi)\\psi_0(\\pi - 1)}{\\psi_1(\\pi - 1)\\psi_1(\\pi - 2)} \\varphi_2(\\pi - 2) - \\frac{\\psi_0(\\pi)\\psi_0(\\pi - 1)\\psi_0(\\pi - 2)}{\\psi_1(\\pi - 1)\\psi_1(\\pi - 2)\\psi_1(\\pi - 3)} \\varphi_3(\\pi - 3) + \\ldots = 0,\n\\]\n\nwhere \\( \\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi) \\) is an internal factor of \\( \\varepsilon^n \\varphi_n(\\pi) + \\varepsilon^{n-1} \\varphi_{n-1}(\\pi) + \\ldots + \\varepsilon \\varphi_1(\\pi) + \\varphi_0(\\pi) \\).\n\nHence we see how we may resolve the symbolical function\n\n\\[\n\\varepsilon^n \\varphi_n(\\pi) + \\varepsilon^{n-1} \\varphi_{n-1}(\\pi) + \\varepsilon^{n-2} \\varphi_{n-2}(\\pi) + \\ldots + \\varepsilon \\varphi_1(\\pi) + \\varphi_0(\\pi)\n\\]\n\ninto factors in all possible cases.\nPut\n\n\\[ \\psi_0(\\pi) = A_0 + B_0 \\pi + C_0 \\pi^2 + \\&c., \\]\n\\[ \\psi_1(\\pi) = A_1 + B_1 \\pi + C_1 \\pi^2 + \\&c., \\]\n\nand substitute in the above equation, and equate the resulting coefficients of \\( \\pi \\) to zero.\n\nWe shall thus be furnished with equations for determining the values of \\( A_0, B_0, \\&c., A_1, B_1, \\&c. \\) in all cases in which the above symbolical function is capable of resolution.\n\nWe thus obtain the values of \\( \\psi_0(\\pi), \\psi_1(\\pi) \\), and of the symbolical quotient. We next ascertain if the symbolical quotient admits of an internal factor, and repeating the process we at length resolve the above symbolical function into factors of the form\n\n\\[ (\\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi))(\\varepsilon \\psi_1^{(n-1)}(\\pi) + \\psi_0^{(n-1)}(\\pi)) \\cdots (\\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi)). \\]\n\nTo determine the condition that \\( \\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi) \\) shall divide the symbolical function\n\n\\[ \\varepsilon^n \\varphi_n(\\pi) + \\varepsilon^{n-1} \\varphi_{n-1}(\\pi) + \\varepsilon^{n-2} \\varphi_{n-2}(\\pi) + \\&c. + \\varepsilon \\varphi_1(\\pi) + \\varphi_0(\\pi) \\]\n\nexternally without a remainder,\n\n\\[ \\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi) \\varepsilon^n \\varphi_n(\\pi) + \\varepsilon^{n-1} \\varphi_{n-1}(\\pi) + \\varepsilon^{n-2} \\varphi_{n-2}(\\pi) + \\&c. + \\varepsilon \\varphi_1(\\pi) + \\varphi_0(\\pi) \\]\n\nwhere the symbolical quotient is\n\n\\[ \\varepsilon^{n-1} \\frac{\\varphi_n(\\pi)}{\\psi_1(\\pi + n - 1)} + \\varepsilon^{n-2} \\left\\{ \\frac{\\varphi_{n-1}(\\pi)}{\\psi_1(\\pi + n - 2)} - \\frac{\\psi_0(\\pi + n - 1)}{\\psi_1(\\pi + n - 2) \\psi_1(\\pi + n - 1)} \\varphi_n(\\pi) \\right\\} + \\&c. \\]\n\nThe required condition is found by equating the remainder to zero: whence we have\n\n\\[ \\varphi_0(\\pi) - \\frac{\\psi_0(\\pi)}{\\psi_1(\\pi)} \\varphi_1(\\pi) + \\frac{\\psi_0(\\pi) \\psi_0(\\pi + 1)}{\\psi_1(\\pi) \\psi_1(\\pi + 1)} \\varphi_2(\\pi) - \\frac{\\psi_0(\\pi) \\psi_0(\\pi + 1) \\psi_0(\\pi + 2)}{\\psi_1(\\pi) \\psi_1(\\pi + 1) \\psi_1(\\pi + 2)} \\varphi_3(\\pi) + \\&c. \\]\n\n\\[ \\pm \\frac{\\psi_0(\\pi) \\psi_0(\\pi + 1) \\psi_0(\\pi + 2) \\cdots \\psi_0(\\pi + n - 1)}{\\psi_1(\\pi) \\psi_1(\\pi + 1) \\psi_1(\\pi + 2) \\cdots \\psi_1(\\pi + n - 1)} \\varphi_n(\\pi) = 0. \\]\n\nIn the next investigation we shall suppose the symbolical function arranged in powers of \\( (\\pi) \\) instead of powers of \\( (\\varepsilon) \\). To determine the condition that \\( \\psi_1(\\varepsilon) \\pi + \\psi_0(\\varepsilon) \\) may be an internal factor of the symbolical function\n\n\\[ \\varphi_3(\\varepsilon) \\pi^3 + \\varphi_2(\\varepsilon) \\pi^2 + \\varphi_1(\\varepsilon) \\pi + \\varphi_0(\\varepsilon). \\]\nWe easily see that\n\n\\[ \\pi \\varphi(\\varepsilon) = \\varphi(\\varepsilon)\\pi + \\varepsilon \\frac{d}{d\\varepsilon} \\varphi(\\varepsilon) \\]\n\n\\[ \\pi^2 \\varphi(\\varepsilon) = \\varphi(\\varepsilon)\\pi^2 + 2\\varepsilon \\frac{d}{d\\varepsilon} \\varphi(\\varepsilon)\\pi + \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\varphi(\\varepsilon). \\]\n\nHence we shall have\n\n\\[ \\varphi_3(\\varepsilon)\\pi^3\\{\\psi_1(\\varepsilon)\\cdot\\pi\\}^{-1} = \\varphi_3(\\varepsilon)\\pi^2 \\cdot \\frac{1}{\\psi_1(\\varepsilon)} = \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi^2 + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\pi + \\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)}; \\]\n\n\\[ \\therefore \\varphi_3(\\varepsilon)\\cdot\\pi^3 = \\left\\{ \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi^2 + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\pi + \\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\right\\} \\{ \\psi_1(\\varepsilon)\\cdot\\pi + \\psi_0(\\varepsilon) - \\psi_0(\\varepsilon) \\}; \\]\n\n\\[ = \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi^2 \\{ \\psi_1(\\varepsilon)\\cdot\\pi + \\psi_0(\\varepsilon) \\} + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\pi \\psi_1(\\varepsilon)\\pi \\]\n\n\\[ + \\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon)\\pi - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi^2 \\psi_0(\\varepsilon) \\]\n\n\\[ = \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi^2 \\{ \\psi_1(\\varepsilon)\\cdot\\pi + \\psi_0(\\varepsilon) \\} + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\{ \\psi_1(\\varepsilon)\\pi + \\varepsilon \\frac{d}{d\\varepsilon} \\psi_1(\\varepsilon)\\cdot\\pi \\}\n\n\\[ + \\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon)\\pi \\]\n\n\\[ - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\{ \\psi_0(\\varepsilon)\\cdot\\pi^2 + 2\\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\psi_0(\\varepsilon)\\cdot\\pi + \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\psi_0(\\varepsilon) \\}; \\]\n\n\\[ \\therefore \\varphi_3(\\varepsilon)\\cdot\\pi^3 + \\varphi_2(\\varepsilon)\\pi^2 + \\varphi_1(\\varepsilon)\\cdot\\pi + \\varphi_0(\\varepsilon) \\]\n\n\\[ = \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi^2 \\{ \\psi_1(\\varepsilon)\\cdot\\pi + \\psi_0(\\varepsilon) \\} \\]\n\n\\[ + \\left\\{ \\varphi_2(\\varepsilon) + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\psi_0(\\varepsilon) \\right\\} \\pi^2 \\]\n\n\\[ + \\left\\{ \\varphi_1(\\varepsilon) + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\varepsilon \\frac{d}{d\\varepsilon} \\psi_1(\\varepsilon) + \\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) - 2 \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\psi_0(\\varepsilon) \\right\\} \\pi \\]\n\n\\[ + \\varphi_0(\\varepsilon) - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\psi_0(\\varepsilon); \\]\n\nwhere we may put\n\n\\[ \\theta_a(\\varepsilon) = \\varphi_a(\\varepsilon) + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\psi_1(\\varepsilon) - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\psi_0(\\varepsilon) \\]\n\n\\[ \\theta_b(\\varepsilon) = \\varphi_1(\\varepsilon) + 2\\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\varepsilon \\frac{d}{d\\varepsilon} \\psi_1(\\varepsilon) + \\varphi_3(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) - 2 \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\psi_0(\\varepsilon) \\]\n\n\\[ \\theta_c(\\varepsilon) = \\varphi_0(\\varepsilon) - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\psi_0(\\varepsilon) \\]\n\n\\[ \\theta_d(\\varepsilon)\\pi^2 \\{ \\psi_1(\\varepsilon)\\cdot\\pi \\}^{-1} = \\theta_d(\\varepsilon)\\cdot\\pi \\cdot \\frac{1}{\\psi_1(\\varepsilon)} = \\frac{\\theta_d(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi + \\theta_a(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)}; \\]\n\n\\[ \\theta_e(\\varepsilon)\\pi^2 = \\frac{\\theta_e(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi \\{ \\psi_1(\\varepsilon)\\cdot\\pi + \\psi_0(\\varepsilon) \\} \\]\n\n\\[ - \\frac{\\theta_e(\\varepsilon)}{\\psi_1(\\varepsilon)} \\{ \\psi_0(\\varepsilon)\\cdot\\pi + \\varepsilon \\frac{d}{d\\varepsilon} \\psi_0(\\varepsilon) \\} + \\theta_a(\\varepsilon) \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon)\\pi \\]\n\n\\[ = \\frac{\\theta_e(\\varepsilon)}{\\psi_1(\\varepsilon)} \\pi \\{ \\psi_1(\\varepsilon)\\cdot\\pi + \\psi_0(\\varepsilon) \\} - \\theta_e(\\varepsilon) \\left\\{ \\frac{\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} - \\varepsilon \\frac{d}{d\\varepsilon} \\left( \\frac{1}{\\psi_1(\\varepsilon)} \\right) \\psi_1(\\varepsilon) \\right\\} \\pi - \\frac{\\theta_e(\\varepsilon)}{\\psi_1(\\varepsilon)} \\varepsilon \\frac{d}{d\\varepsilon} \\psi_0(\\varepsilon). \\]\nThen\n\n\\[ \\theta_3(\\varepsilon)\\pi^3 + \\theta_1(\\varepsilon)\\pi + \\theta_0(\\varepsilon) = \\frac{\\theta_3(\\varepsilon)}{\\psi_1(\\varepsilon)}\\pi \\{ \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon) \\} \\]\n\n\\[ + \\left\\{ \\theta_1(\\varepsilon) - \\frac{\\theta_3(\\varepsilon)\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} + \\theta_2(\\varepsilon)\\varepsilon \\frac{d}{d\\varepsilon} \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) \\right\\} \\pi \\]\n\n\\[ + \\left\\{ \\theta_0(\\varepsilon) - \\frac{\\theta_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\varepsilon \\frac{d}{d\\varepsilon} \\psi_0(\\varepsilon) \\right\\}. \\]\n\nPut, for the sake of simplicity,\n\n\\[ \\omega_1(\\varepsilon) = \\theta_1(\\varepsilon) - \\frac{\\theta_3(\\varepsilon)\\psi_0(\\varepsilon)}{\\psi_1(\\varepsilon)} + \\theta_2(\\varepsilon)\\varepsilon \\frac{d}{d\\varepsilon} \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) \\]\n\n\\[ \\omega_0(\\varepsilon) = \\theta_0(\\varepsilon) - \\frac{\\theta_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\varepsilon \\frac{d}{d\\varepsilon} \\psi_0(\\varepsilon) \\]\n\n\\[ \\omega_1(\\varepsilon) \\cdot \\pi \\cdot \\{ \\psi_1(\\varepsilon)\\pi \\}^{-1} = \\frac{\\omega_1(\\varepsilon)}{\\psi_1(\\varepsilon)}, \\]\n\n\\[ \\therefore \\omega_1(\\varepsilon)\\pi = \\frac{\\omega_1(\\varepsilon)}{\\psi_1(\\varepsilon)} \\{ \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon) \\} - \\frac{\\omega_1(\\varepsilon)}{\\psi_1(\\varepsilon)} \\psi_0(\\varepsilon), \\]\n\n\\[ \\omega_0(\\varepsilon)\\pi + \\omega_0(\\varepsilon) = \\frac{\\omega_1(\\varepsilon)}{\\psi_1(\\varepsilon)} \\{ \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon) \\} + \\omega_0(\\varepsilon) - \\frac{\\omega_1(\\varepsilon)}{\\psi_1(\\varepsilon)} \\psi_0(\\varepsilon). \\]\n\nHence the condition that \\( \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon) \\) may be an internal factor of\n\n\\[ \\varphi_3(\\varepsilon)\\pi^3 + \\varphi_1(\\varepsilon)\\pi + \\varphi_0(\\varepsilon) \\]\n\nis equivalent to the equation\n\n\\[ \\omega_1(\\varepsilon)\\psi_0(\\varepsilon) - \\omega_0(\\varepsilon)\\psi_1(\\varepsilon) = 0. \\]\n\nHence, substituting for \\( \\omega_1(\\varepsilon) \\) and \\( \\omega_0(\\varepsilon) \\) their values, we have\n\n\\[ \\theta_1(\\varepsilon)\\psi_0(\\varepsilon) - \\frac{\\theta_3(\\varepsilon)}{\\psi_1(\\varepsilon)} (\\psi_0(\\varepsilon))^2 + \\theta_2(\\varepsilon)\\psi_0(\\varepsilon)\\varepsilon \\frac{d}{d\\varepsilon} \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) - \\theta_0(\\varepsilon)\\psi_1(\\varepsilon) + \\theta_2(\\varepsilon)\\varepsilon \\frac{d}{d\\varepsilon} \\psi_0(\\varepsilon) = 0; \\]\n\nor, again substituting for \\( \\theta_0(\\varepsilon) \\), \\( \\theta_1(\\varepsilon) \\), \\( \\theta_2(\\varepsilon) \\), we have\n\n\\[ \\left\\{ \\varphi_3(\\varepsilon) + 2\\varphi_0(\\varepsilon)\\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\psi_0(\\varepsilon) \\right\\} \\left\\{ \\varepsilon \\frac{d}{d\\varepsilon} \\psi_0(\\varepsilon) + \\psi_0(\\varepsilon)\\varepsilon \\frac{d}{d\\varepsilon} \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) - \\frac{(\\psi_0(\\varepsilon))^2}{\\psi_1(\\varepsilon)} \\right\\} \\]\n\n\\[ + \\psi_0(\\varepsilon) \\left\\{ \\varphi_1(\\varepsilon) + 2\\varphi_0(\\varepsilon)\\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) + \\varphi_0(\\varepsilon)\\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\frac{1}{\\psi_1(\\varepsilon)} \\cdot \\psi_1(\\varepsilon) - 2\\varphi_0(\\varepsilon)\\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right) \\psi_0(\\varepsilon) \\right\\} \\]\n\n\\[ - \\psi_1(\\varepsilon) \\left\\{ \\varphi_0(\\varepsilon) - \\frac{\\varphi_3(\\varepsilon)}{\\psi_1(\\varepsilon)} \\left( \\varepsilon \\frac{d}{d\\varepsilon} \\right)^2 \\psi_0(\\varepsilon) \\right\\} = 0. \\]\n\nHad we wished to ascertain the condition that \\( \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon) \\) may be an internal factor 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), \\]\n\nwe must have calculated the value of \\( \\pi^3\\varphi(\\varepsilon) \\). It is evident that for every increase in the degree of the highest power of \\( \\pi \\), the labour of the investigation becomes immensely greater, and the result far more complicated. It is, however, of considerable utility in the integration of differential equations, and we shall refer to it again at the close of this paper.\nSection II. On some General Theorems.\n\nI shall now give some theorems in general differentiation and expansion.\n\nSince\n\n\\[\n\\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n = \\left(\\frac{1}{g^2} \\cdot \\pi\\right) \\left(\\frac{1}{g^2} \\cdot \\pi\\right) \\cdots\n\\]\n\nto \\( n \\) factors, we have\n\n\\[\n\\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n = \\frac{1}{g^{2n}} \\pi(\\pi-2)(\\pi-4) \\cdots (\\pi-2n+2),\n\\]\n\n\\[\n\\therefore \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n \\cdot \\frac{1}{g} = \\frac{1}{g^{2n+1}} (\\pi-1)(\\pi-3) \\cdots (\\pi-2n+1);\n\\]\n\nwhence we easily see that\n\n\\[\n\\pi(\\pi-1)(\\pi-2)(\\pi-3) \\cdots (\\pi-2n+1) = g^{2n+1} \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n g^{2n-1} \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n,\n\\]\n\n\\[\n\\frac{1}{g^{2n}} \\pi(\\pi-1)(\\pi-2)(\\pi-3) \\cdots (\\pi-2n+1) = g \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n g^{2n-1} \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n,\n\\]\n\n\\[\n\\therefore \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^{2n} = g \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n g^{2n-1} \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^n;\n\\]\n\nwhence we shall have\n\n\\[\n\\frac{d^{2n}u}{dx^{2n}} = x \\left(\\frac{1}{x} \\cdot \\frac{d}{dx}\\right)^n x^{2n-1} \\left(\\frac{1}{x} \\cdot \\frac{d}{dx}\\right)^n u.\n\\]\n\nIf we equate the coefficients of \\( z^n \\) in \\( (1+z)^{2n} = (1+z)^n(z+1)^n \\), we have\n\n\\[\n\\frac{2n(2n-1)(2n-2) \\cdots (2n-r+1)}{1.2.3 \\cdots r} = \\frac{n(n-1)(n-2) \\cdots (n-r+1)}{1.2.3 \\cdots r}\n\\]\n\n\\[\n+ \\frac{n(n-1)(n-2) \\cdots (n-r+2)}{1.2.3 \\cdots r-1} + \\frac{n(n-1) \\cdots (n-r+3)}{1.2.3 \\cdots r-2} + \\cdots\n\\]\n\n\\[\n\\therefore 2\\pi(2\\pi-1)(2\\pi-2) \\cdots (2\\pi-r+1)\n\\]\n\n\\[\n= \\pi(\\pi-1) \\cdots (\\pi-r+1) + r\\pi(\\pi-1)(\\pi-2) \\cdots (\\pi-r+2)\\pi\n\\]\n\n\\[\n+ r\\frac{r-1}{2} \\pi(\\pi-1)(\\pi-2) \\cdots (\\pi-r+3).\\pi(\\pi-1).\n\\]\n\nHence, since\n\n\\[\n\\left(\\frac{1}{g^2} \\cdot \\pi\\right)^r = \\frac{1}{g^r} \\pi(\\pi-\\frac{1}{2})(\\pi-1) \\cdots (\\pi-\\frac{r-1}{2}),\n\\]\n\nwe have\n\n\\[\n2^r g^{\\frac{r}{2}} \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^r = g^r \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^r + r g^{r-1} \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^{r-1} g \\left(\\frac{1}{g^2} \\cdot \\pi\\right)\n\\]\n\n\\[\n+ r\\frac{r-1}{2} g^{r-2} \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^{r-2} g^2 \\left(\\frac{1}{g^2} \\cdot \\pi\\right)^2 + \\text{&c.};\n\\]\n\nwhence we find\n\n\\[\nx^{\\frac{r}{2}} \\left(\\frac{d}{dx}\\right)^r u = x^{\\frac{r}{2}} \\frac{du}{dx} + rx^{\\frac{r}{2}-1} \\frac{d^{r-1}u}{dx^{r-1}} + r\\frac{r-1}{2} x^{\\frac{r}{2}-2} \\frac{d^{r-2}u}{dx^{r-2}} + \\text{&c.}\n\\]\n\n* It has been pointed out to me that this theorem might be more shortly proved by applying Vandermonde's theorem to the equation \\((2D)^r = (D+D)^r\\). I have retained the demonstration in the text merely for the sake of the method.\nI now come to the theorems respecting expansion, which I mentioned in the beginning of the paper as analogous to the binomial and multinomial theorems in ordinary algebra.\n\nTo expand \\((\\varepsilon^2 + \\varepsilon \\theta(\\pi))^n\\) in powers of \\(\\pi\\), where \\(\\theta(\\pi)\\) is a function of \\((\\pi)\\), and \\((n)\\) is a positive integer.\n\nLet us assume\n\n\\[\n(\\varepsilon^2 + \\varepsilon \\theta(\\pi))^n = \\varphi_n^{(0)}(\\varepsilon) + \\varphi_n^{(1)}(\\varepsilon) \\cdot \\pi + \\varphi_n^{(2)}(\\varepsilon) \\pi^2 + &c.,\n\\]\n\nwhere\n\n\\[\n\\varphi_n^{(0)}(\\varepsilon) = \\varepsilon^{2n} + A_n^{(1)} \\varepsilon^{2n-1} + A_n^{(2)} \\varepsilon^{2n-2} + &c.\n\\]\n\n\\[\n\\varphi_n^{(1)}(\\varepsilon) = B_n^{(0)} \\varepsilon^{2n-1} + B_n^{(1)} \\varepsilon^{2n-2} + B_n^{(2)} \\varepsilon^{2n-3} + &c.;\n\\]\n\n\\[\n\\therefore \\quad \\varphi_{n+1}^{(0)}(\\varepsilon) = \\varepsilon^{2n+2} + A_{n+1}^{(1)} \\varepsilon^{2n+1} + A_{n+1}^{(2)} \\varepsilon^{2n} + &c.\n\\]\n\n\\[\n= \\varepsilon^{2n+2} + A_{n+1}^{(1)} \\varepsilon^{2n+1} + A_{n+1}^{(2)} \\varepsilon^{2n} + &c.\n\\]\n\n\\[\n+ \\theta(2n) \\varepsilon^{2n+1} + A_{n+1}^{(1)} \\theta(2n-1) \\varepsilon^{2n} + &c.;\n\\]\n\n\\[\n\\therefore \\quad A_{n+1}^{(1)} = A_n^{(1)} + \\theta(2n), \\quad A_{n+1}^{(2)} = A_n^{(2)} + A_n^{(1)} \\theta(2n-1);\n\\]\n\nor\n\n\\[\nA_n^{(1)} = \\Sigma \\theta(2n), \\quad A_n^{(2)} = \\Sigma (\\theta(2n-1) \\Sigma \\theta(2n)).\n\\]\n\nSimilarly,\n\n\\[\nA_n^{(3)} = \\Sigma \\{ \\theta(2n-2) \\Sigma (\\theta(2n-1) \\Sigma \\theta(2n)) \\} \\ldots &c.\n\\]\n\nAgain, we shall have\n\n\\[\n\\varphi_{n+1}^{(1)}(\\varepsilon) = B_{n+1}^{(0)} \\varepsilon^{2n+1} + B_{n+1}^{(1)} \\varepsilon^{2n} + B_{n+1}^{(2)} \\varepsilon^{2n-1} + &c.\n\\]\n\n\\[\n= \\theta(2n) \\varepsilon^{2n+1} + A_n^{(1)} \\theta(2n-1) \\varepsilon^{2n} + A_n^{(2)} \\theta(2n-2) \\varepsilon^{2n-1} + \\ldots\n\\]\n\n\\[\n+ B_n^{(0)} \\varepsilon^{2n+1} + B_n^{(1)} \\varepsilon^{2n} + B_n^{(2)} \\varepsilon^{2n-1} + \\ldots\n\\]\n\n\\[\n+ B_n^{(0)} \\theta(2n-1) \\varepsilon^{2n} + B_n^{(1)} \\theta(2n-2) \\varepsilon^{2n-1} + &c.\n\\]\n\nConsequently\n\n\\[\nB_{n+1}^{(0)} = B_n^{(0)} + \\theta(2n); \\quad \\therefore \\quad B_n^{(0)} = \\Sigma \\theta(2n)\n\\]\n\n\\[\nB_{n+1}^{(1)} = B_n^{(1)} + B_n^{(0)} \\theta(2n-1) + A_n^{(1)} \\theta(2n-1);\n\\]\n\n\\[\n\\therefore \\quad B_n^{(1)} = \\Sigma (\\theta(2n-1) \\Sigma \\theta(2n)) + \\Sigma \\theta(2n-1) \\Sigma \\theta(2n)).\n\\]\n\nHence we shall have\n\n\\[\n(\\varepsilon^2 + \\varepsilon \\theta(\\pi))^n = \\varepsilon^{2n} + \\Sigma \\theta(2n) \\varepsilon^{2n-1}\n\\]\n\n\\[\n+ \\Sigma \\theta(2n-1) \\Sigma \\theta(2n) \\varepsilon^{2n-2} + \\Sigma \\theta(2n-2) \\Sigma \\theta(2n-1) \\Sigma \\theta(2n) \\varepsilon^{2n-3} + &c.\n\\]\n\n\\[\n+ \\{ \\Sigma \\theta(2n) \\varepsilon^{2n-1} + (\\Sigma \\theta(2n-1) \\Sigma \\theta(2n)) + \\Sigma \\theta(2n-1) \\Sigma \\theta(2n)) \\varepsilon^{2n-2} + &c. \\} \\pi + &c.\n\\]\n\nWhen \\(\\theta(\\pi)\\) is a rational and entire function of \\((\\pi)\\),\n\n\\[\n\\Sigma \\theta(2n), \\quad \\Sigma (\\theta(2n-1) \\Sigma (2n)) &c.; \\quad \\text{and} \\quad \\Sigma \\theta(2n) &c.\n\\]\n\ncan always be obtained in finite terms, as manifestly ought to be the case.\n\nIn like manner we shall have\n\n\\[\n(\\varepsilon + \\frac{1}{\\varepsilon} \\theta(\\pi))^n = \\varepsilon^n + \\Sigma \\theta(n) \\varepsilon^{n-2}\n\\]\n\\[ + \\Sigma (\\theta(n-2) \\Sigma \\theta(n)) \\varepsilon^{n-4} + \\Sigma (\\theta(n-4) \\Sigma \\theta(n-2) \\Sigma \\theta(n)) \\varepsilon^{n-6} + \\&c. \\]\n\\[ + \\{ \\Sigma \\theta'(n) \\varepsilon^{n-2} + (\\Sigma \\theta'(n-2) \\Sigma \\theta(n) + \\Sigma \\theta(n-2) \\Sigma \\theta'(n)) \\varepsilon^{n-4} + \\&c. \\} \\pi + \\&c.; \\]\n\nand also\n\n\\[ \\left( \\varepsilon^2 + \\frac{1}{\\varepsilon} \\theta(\\pi) \\right)^n = \\varepsilon^{2n} + \\Sigma \\theta(2n) \\varepsilon^{2n-3} \\]\n\\[ + \\Sigma (\\theta(2n-3) \\Sigma \\theta(2n)) \\varepsilon^{2n-6} + \\Sigma (\\theta(2n-6) \\Sigma \\theta(2n-3) \\Sigma \\theta(2n)) \\varepsilon^{2n-9} + \\&c. \\]\n\\[ + \\{ \\Sigma \\theta'(2n) \\varepsilon^{2n-3} + (\\Sigma \\theta'(2n-3) \\Sigma \\theta(2n) + \\Sigma \\theta(2n-3) \\Sigma \\theta'(2n)) \\varepsilon^{2n-6} + \\&c. \\} \\pi + \\&c. \\]\n\nIf we put \\( \\theta(\\pi) = \\pi^2 \\), it is obvious that the three last theorems will give us the expansions of\n\n\\[ \\left( x^2 + x^2 \\frac{d}{dx} + x^3 \\frac{d^2}{dx^2} \\right)^n, \\quad \\left( x + \\frac{d}{dx} + x \\frac{d^2}{dx^2} \\right)^n, \\quad \\text{and of} \\quad \\left( x^2 + \\frac{d}{dx} + x \\frac{d^2}{dx^2} \\right)^n \\]\n\nin terms of \\( x \\frac{d}{dx} \\).\n\nThe same methods of course will apply to all binomials included under the form \\( (\\varepsilon^a + \\varepsilon^b \\theta(\\pi))^n \\). I have found that there is no difficulty in calculating the forms of the coefficients, beyond the labour expended in performing the finite integrations.\n\nTo determine that part of the expansion of \\( (\\varepsilon^a + \\varepsilon^b \\theta_1(\\pi) + \\varepsilon^c \\theta_2(\\pi) + \\varepsilon^d \\theta_3(\\pi) + \\&c.)^n \\)\n\nwhich is independent of \\( \\pi \\).\n\nLet us assume\n\n\\[ (\\varepsilon^a + \\varepsilon^b \\theta_1(\\pi) + \\varepsilon^c \\theta_2(\\pi) + \\varepsilon^d \\theta_3(\\pi) + \\&c.)^n = \\varphi_n^{(0)}(\\varepsilon) + \\varphi_n^{(1)}(\\varepsilon) \\cdot \\pi + \\varphi_n^{(2)}(\\varepsilon) \\cdot \\pi^2 + \\&c., \\]\n\nwhere\n\n\\[ \\varphi_n^{(0)}(\\varepsilon) = \\varepsilon^{an} + A_n^{(1)} \\varepsilon^{an+1} + A_n^{(2)} \\varepsilon^{an+2} + A_n^{(3)} \\varepsilon^{an+3} + \\&c. \\]\n\nThen we shall have\n\n\\[ \\varphi_n^{(0)}(\\varepsilon) = \\varepsilon^{an+a} + A_n^{(1)} \\varepsilon^{an+a-1} + A_n^{(2)} \\varepsilon^{an+a-2} + A_n^{(3)} \\varepsilon^{an+a-3} + \\&c. \\]\n\n\\[ = \\varepsilon^{an+a} + A_n^{(1)} \\varepsilon^{an+a-1} + A_n^{(2)} \\varepsilon^{an+a-2} + A_n^{(3)} \\varepsilon^{an+a-3} + \\&c. \\]\n\n\\[ + \\theta_1(an) \\varepsilon^{an+a-1} + A_n^{(1)} \\theta_1(an-1) \\varepsilon^{an+a-2} + A_n^{(2)} \\theta_1(an-2) \\varepsilon^{an+a-3} + \\&c. \\]\n\n\\[ + \\theta_2(an) \\varepsilon^{an+a-2} + A_n^{(1)} \\theta_2(an-1) \\varepsilon^{an+a-3} + \\&c. \\]\n\n\\[ + \\theta_3(an) \\varepsilon^{an+a-3} + \\&c. \\]\n\n\\[ \\therefore A_n^{(1)} = A_n^{(1)} + \\theta_1(an) \\]\n\n\\[ A_n^{(2)} = A_n^{(2)} + A_n^{(1)} \\theta_1(an-1) + \\theta_2(an) \\]\n\n\\[ A_n^{(3)} = A_n^{(3)} + A_n^{(2)} \\theta_1(an-2) + A_n^{(1)} \\theta_2(an-1) + \\theta_3(an) \\]\n\n\\[ \\&c. = \\&c.; \\]\n\n\\[ \\therefore A_n^{(1)} = \\Sigma \\theta_1(an) \\]\n\n\\[ A_n^{(2)} = \\Sigma \\theta_1(an-1) \\Sigma \\theta_1(an) + \\Sigma \\theta_2(an) \\]\n\n\\[ A_n^{(3)} = \\Sigma \\theta_1(an-2) \\Sigma \\theta_1(an-1) \\Sigma \\theta_1(an) \\]\n\n\\[ + \\Sigma \\theta_1(an-2) \\Sigma \\theta_2(an) + \\Sigma \\theta_2(an-1) \\Sigma \\theta_1(an) + \\Sigma \\theta_3(an); \\]\n\nand consequently the part of\n\n\\[ (\\varepsilon^a + \\varepsilon^b \\theta_1(\\pi) + \\varepsilon^c \\theta_2(\\pi) + \\&c.)^n; \\]\nwhich is independent of $(\\pi)$, is\n\n$$\\varepsilon^{an} + \\sum_{i=1}^{n-1} (\\varepsilon^i(\\alpha n - i) \\varepsilon^i(\\alpha n - i)) \\varepsilon^{an-i} + (\\varepsilon^i(\\alpha n - i) \\varepsilon^i(\\alpha n - i)) \\varepsilon^{an-i} + \\ldots$$\n\nSection III. On the Solution of Linear Differential Equations with Variable Coefficients.\n\nThe general linear differential equation\n\n$$X_r \\frac{d^r u}{dx^r} + X_{r-1} \\frac{d^{r-1} u}{dx^{r-1}} + X_{r-2} \\frac{d^{r-2} u}{dx^{r-2}} + \\ldots = X,$$\n\nwhere $X_r$, $X_{r-1}$ are rational and entire functions of $(x)$, may, as Professor Boole has shown, be always expressed in the symbolical form\n\n$$\\varepsilon^n \\varphi_n(\\pi) u + \\varepsilon^{n-1} \\varphi_{n-1}(\\pi) u + \\ldots + \\varphi_1(\\pi) u + \\varphi_0(\\pi) u = X,$$\n\nwhere\n\n$$\\varepsilon = x, \\quad \\pi = x \\frac{d}{dx},$$\n\nand $\\varphi_n(\\pi), \\varphi_{n-1}(\\pi), \\ldots$ are rational and entire functions of $(\\pi)$.\n\nSuppose that by using the methods explained in this paper, we are able to reduce this equation to the form\n\n$$(\\varepsilon \\psi_1^{(n)}(\\pi) + \\psi_0^{(n)}(\\pi))(\\varepsilon \\psi_1^{(n-1)}(\\pi) + \\psi_0^{(n-1)}(\\pi)) \\ldots (\\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi)) u = X.$$*\n\nAssume\n\n$$\\varepsilon \\psi_1^{(n)}(\\pi) u_{n-1} + \\psi_0^{(n)}(\\pi) u_{n-1} = X,$$\n$$\\varepsilon \\psi_1^{(n-1)}(\\pi) u_{n-2} + \\psi_0^{(n-1)}(\\pi) u_{n-2} = u_{n-1},$$\n$$\\varepsilon \\psi_1^{(n-2)}(\\pi) u_{n-3} + \\psi_0^{(n-2)}(\\pi) u_{n-3} = u_{n-2},$$\n$$\\ldots$$\n\nWe thus reduce the proposed differential equation to forms already treated of by Professor Boole.\n\nWe may much simplify the process already explained for treating the symbolical quantity $\\varepsilon^n \\varphi_n(\\pi) + \\ldots + \\varphi_0(\\pi)$, by remarking that $\\psi_1(\\pi)$ must be sought among the divisors of $\\varphi_n(\\pi), \\psi_0(\\pi)$ among the divisors of $\\varphi_0(\\pi)$; and we shall make use of this principle in the following application of the preceding theory to the solution of differential equations.\n\nWe shall denominate the equation deduced in the former part of this memoir,\n\n$$\\varphi_0(\\pi) - \\frac{\\psi_0(\\pi)}{\\psi_1(\\pi - 1)} \\varphi_1(\\pi - 1) + \\frac{\\psi_0(\\pi) \\psi_0(\\pi - 1)}{\\psi_1(\\pi - 1) \\psi_1(\\pi - 2)} \\varphi_2(\\pi - 2) - \\ldots = 0,$$\n\nthe criterion of the factor $\\varepsilon \\psi_1(\\pi) + \\psi_0(\\pi)$.\n\n* It may be proper to remind the reader that $\\psi_1^{(n)}(\\pi), \\psi^{(n-1)}(\\pi), \\ldots$ have no reference to the functions derived from $\\psi_1 \\pi$ by differentiation.\nTo integrate the differential equation,\n\n\\[ x^2(x+1)^3 \\frac{d^3u}{dx^3} + 3x(x+1)^3 \\frac{d^2u}{dx^2} + (x^3 + 4x^2 + 3x) \\frac{du}{dx} - (2x-3)u = X. \\]\n\nThe symbolical form of this equation is\n\n\\[ g^3\\pi^3u + g^2(3\\pi^3 + \\pi - 1)u + 3g(\\pi^3 + 1)u + \\pi(\\pi^2 - 1)u = Xx. \\]\n\nThe divisor of \\( \\pi^3 \\) is \\( \\pi \\) only, the divisors of \\( \\pi(\\pi^2 - 1) \\) are \\( \\pi - 1, \\pi, \\pi + 1 \\); hence putting \\( \\psi_1(\\pi) = \\pi, \\psi_0\\pi = \\pi - 1 \\), we find the criterion of the symbolical quantity \\( g\\psi_1(\\pi) + \\psi_0(\\pi) \\) to become\n\n\\[ \\pi(\\pi^2 - 1) - 3\\{(\\pi - 1)^2 + 1\\} + \\{3(\\pi - 2)^3 + (\\pi - 2) - 1\\} - (\\pi - 3)^3 = 0, \\]\n\nan identical equation.\n\nHence \\( g\\pi + (\\pi - 1) \\) is an internal factor of\n\n\\[ g^3\\pi^3 + g^2(3\\pi^3 + \\pi - 1) + 3g(\\pi^3 + 1) + \\pi(\\pi^2 - 1); \\]\n\nand the equation may be written, effecting the internal division,\n\n\\[ \\{g^2(\\pi - 1)^2 + g(\\pi + 1)(2\\pi - 3) + \\pi(\\pi + 1)\\}(g\\pi + (\\pi - 1))u = Xx; \\]\n\nor if \\( g\\pi + (\\pi - 1)u = u_1 \\),\n\n\\[ \\{g^2(\\pi - 1)^2 + g(\\pi + 1)(2\\pi - 3) + \\pi(\\pi + 1)\\}u_1 = Xx. \\]\n\nThe only divisor of \\( (\\pi - 1)^2 \\) is \\( \\pi - 1 \\), the divisors of \\( \\pi(\\pi + 1) \\) are \\( \\pi \\) and \\( \\pi + 1 \\); and by trial it is found that the divisor \\( g(\\pi - 1) + (\\pi + 1) \\) satisfies the criterion, and is therefore an internal factor. Hence, effecting the internal division,\n\n\\[ (g(\\pi - 2) + \\pi)(g(\\pi - 1) + (\\pi + 1))u_1 = Xx, \\]\n\nand the differential equation becomes\n\n\\[ (g(\\pi - 2) + \\pi)(g(\\pi - 1) + (\\pi + 1))(g\\pi + (\\pi - 1))u = Xx, \\]\n\nor\n\n\\[ \\left\\{(x^2 + x) \\frac{d}{dx} - 2x\\right\\}\\left\\{(x^2 + x) \\frac{d}{dx} - (x - 1)\\right\\}\\left\\{(x^2 + x) \\frac{d}{dx} - 1\\right\\}u = Xx. \\]\n\nHence, performing the inverse calculations, we find for the complete integral;\n\n\\[ u = \\frac{x}{x + 1} \\int \\frac{dx}{x^3} \\int \\frac{dx}{x + 1} \\int \\frac{Xdx}{(x + 1)^3}, \\]\n\nthe three arbitrary constants being included under the signs of integration.\n\nIn case this method does not succeed, we may sometimes resolve the symbolical function into factors by assuming \\( u = (\\pi + \\xi)v \\) and proceeding as before, determining \\( \\alpha \\) from the criterion, as will be shown in the following examples:\n\nTo integrate the differential equation\n\n\\[ x^2(x+1)^3 \\frac{d^3u}{dx^3} + x(4x^3 + 11x^2 + 10x + 3) \\frac{du}{dx} + 2x^3 + 10x^2 + 5x - 3 = X. \\]\n\nThe symbolical form of the equation is\n\n\\[ g^3(\\pi^3 + 3\\pi + 2) + g^2(3\\pi^3 + 8\\pi + 10) + g(3\\pi^3 + 7\\pi + 5) + \\pi^2 + 2\\pi - 3 = X. \\]\nLet \\( u = (\\pi + \\xi)v \\), and the equation becomes\n\n\\[\n\\begin{align*}\n&\\varepsilon^3(\\pi + 1)(\\pi + 2)(\\pi + \\xi)v + \\varepsilon^3(3\\pi^2 + 8\\pi + 10)(\\pi + \\xi)v \\\\\n&+ \\varepsilon(3\\pi^2 + 7\\pi + 5)(\\pi + \\xi)v + (\\pi - 1)(\\pi + 3)(\\pi + \\xi)v = X.\n\\end{align*}\n\\]\n\nLet \\( \\psi_1(\\pi) = \\pi + 2 \\), \\( \\psi_0(\\pi) = \\pi + \\xi \\), then the criterion of \\( \\varepsilon(\\pi + 2) + (\\pi + \\xi) \\) become\n\n\\[\n\\begin{align*}\n&(\\pi^2 + 2\\pi - 3)(\\pi + \\xi) - \\frac{\\pi + \\xi}{\\pi + 1}(3\\pi^2 + \\pi + 1)(\\pi + \\xi - 1) \\\\\n&+ \\frac{(\\pi + \\xi)(\\pi + \\xi - 1)}{(\\pi + 1)\\pi}(3\\pi^2 - 4\\pi + 6)(\\pi + \\xi - 2) \\\\\n&- \\frac{(\\pi + \\xi)(\\pi + \\xi - 1)(\\pi + \\xi - 2)}{(\\pi + 1)\\pi(\\pi - 1)}(\\pi^2 - 3\\pi + 2)(\\pi + \\xi - 3) = 0.\n\\end{align*}\n\\]\n\nPut \\( \\pi = 0 \\) to determine \\( \\xi \\), and we have \\( \\xi = 0 \\) as one value of \\( \\xi \\), which on trial is found to satisfy the proposed.\n\nHence \\( \\varepsilon(\\pi + 2) + \\pi \\) is an internal factor of the symbolical function\n\n\\[\n\\varepsilon^3(\\pi + 1)(\\pi + 2) + \\varepsilon^3(3\\pi^2 + 8\\pi^2 + 10\\pi)\n+ \\varepsilon(3\\pi^2 + 7\\pi^2 + 5\\pi) + \\pi(\\pi - 1)(\\pi + 3).\n\\]\n\nTherefore, effecting the internal division, the equation becomes\n\n\\[\n(\\varepsilon^3\\pi + \\varepsilon(2\\pi + 3) + \\pi + 3)(\\pi - 1)(\\varepsilon(\\pi + 2) + \\pi)v = X,\n\\]\n\nwhence performing the inverse calculations, we have\n\n\\[\nv = \\frac{1}{(\\pi + 1)^2} \\int dx(x + 1)^3 \\int dx x^3 \\cdot X;\n\\]\n\n\\[\n\\therefore \\quad u = x \\frac{d}{dx} \\left\\{ \\frac{1}{(\\pi + 1)^2} \\int dx(x + 1)^3 \\int dx x^2 X \\right\\},\n\\]\n\nwhere the arbitrary constants must be reduced to two.\n\nNext consider the differential equation\n\n\\[\n(x^4 + 2x^3 + x^2) \\frac{d^2u}{dx^2} - 6(x^2 + x) \\frac{du}{dx} + 6(x + 2)u = X;\n\\]\n\nthe symbolical form of this equation is\n\n\\[\n\\varepsilon^3(\\pi - 1)u + 2\\varepsilon(\\pi - 1)(\\pi - 3)u + (\\pi - 3)(\\pi - 4)u = X.\n\\]\n\nLet \\( u = (\\pi + \\xi)v \\), and the equation becomes\n\n\\[\n\\varepsilon^3(\\pi - 1)(\\pi + \\xi)v + 2\\varepsilon(\\pi - 1)(\\pi - 3)(\\pi + \\xi)v + (\\pi - 3)(\\pi - 4)(\\pi + \\xi)v = X.\n\\]\n\nLet \\( \\psi_1(\\pi) = \\pi - 1 \\), \\( \\psi_0(\\pi) = \\pi - 3 \\), and the criterion becomes\n\n\\[\n(\\pi - 3)(\\pi - 4)(\\pi + \\xi) - \\frac{\\pi - 3}{\\pi - 2} \\{2(\\pi - 2)(\\pi - 4)\\}(\\pi + \\xi - 1) \\\\\n+ \\frac{(\\pi - 3)(\\pi - 4)}{(\\pi - 2)(\\pi - 3)} \\{(\\pi - 2)(\\pi - 3)\\}(\\pi + \\xi - 2) = 0.\n\\]\n\nPutting \\( \\pi = 0 \\) in this equation, we have \\( \\xi = 0 \\), and this value renders the above equation identical,\n\n\\[\n\\therefore \\quad \\varepsilon(\\pi - 1) + (\\pi - 3)\n\\]\nis an internal factor of the symbolical function\n\n\\[ \\varepsilon^2 \\pi^2 (\\pi - 1) + 2 \\varepsilon \\pi (\\pi - 1)(\\pi - 3) + \\pi (\\pi - 3)(\\pi - 4); \\]\n\nwherefore, effecting the internal division, the equation becomes\n\n\\[ \\{ \\varepsilon (\\pi - 1)^2 + \\pi (\\pi - 4) \\} \\{ \\varepsilon (\\pi - 1) + (\\pi - 3) \\} u = X. \\]\n\nThis equation may be written\n\n\\[ \\left( \\frac{(\\pi - 2)(\\pi - 3)(\\pi - 4)}{\\pi - 1} \\right) (\\varepsilon + 1) \\left( \\frac{\\pi (\\pi - 1)}{(\\pi - 2)(\\pi - 3)} \\right) (\\varepsilon (\\pi - 1) + (\\pi - 3)) = X, \\]\n\nin which the inverse calculations are all practicable.\n\nAs a final example we take the differential equation\n\n\\[ (x^5 + 4x^4 + 5x^3 + 2x^2) \\frac{d^2u}{dx^2} + (2x^4 + 3x^3 + 5x^2 - 6x) \\frac{du}{dx} + (x + 1)^2 u = X. \\]\n\nThe symbolical form of this equation is\n\n\\[ \\varepsilon^3 \\pi (\\pi + 1) u + \\varepsilon^2 (4\\pi^2 - \\pi + 1) u + \\varepsilon (5\\pi^2 - 5\\pi + 2) u + (\\pi - 1)(2\\pi - 1) u = X. \\]\n\nIf we put \\( u = \\pi v \\), the equation becomes\n\n\\[ (\\varepsilon + 1)(\\pi - 1)(\\varepsilon (\\pi - 1) + \\pi)(\\varepsilon + (2\\pi - 1)) v = X, \\]\n\nin which the inverse calculations necessary for the solution of the equation are all practicable.\n\nIn cases where the assumption \\( u = (\\pi + \\xi)v \\) does not lead to the solution of the equation, we may assume \\( u = (\\pi + \\xi_1)(\\pi + \\xi_2)v \\), and proceed as before.\n\nWe may also treat linear differential equations by ascertaining the condition that \\( \\psi_1(\\varepsilon)\\pi + \\psi_0(\\varepsilon) \\) may be an internal factor of this symbolical expression,\n\n\\[ \\varphi_n(\\varepsilon) \\cdot \\pi^n + \\varphi_{n-1}(\\varepsilon) \\pi^{n-1} + \\ldots + \\varphi_1(\\varepsilon) \\pi + \\varphi_0(\\varepsilon). \\]\n\nI have shown how this is to be effected when \\( n = 2 \\) or \\( 3 \\).\n\nFor higher degrees the investigation would be very laborious. 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