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  "text": "(3) $\\Psi$ is defined by the equation $\\Psi = Z - \\frac{dP}{dt}$, in which (after the explicit differentiation of $P$ with respect to $t$), $x_1$, &c., $y_1$, &c. are to be expressed in terms of the new variables. $y_1$, &c. are thus expressible by the help of the $m$ equations $\\frac{dP}{d\\xi_i} = \\eta_i$ and the $n-m$ equations $\\frac{dL}{dt} + \\sum_i \\left( \\frac{dL}{dx_i} \\frac{dZ}{dy_i} \\right) = 0$.\n\nIf $(x_1)$, &c., do not contain $t$ explicitly, then $\\frac{dP}{dt} = 0$, and $\\Psi$ is obtained merely by expressing $Z$ in terms of the new variables.\n\nIt may be observed that the whole of the above reasoning would apply to the case in which the new variables $\\xi_1$, ... $\\xi_m$ are more in number than the independent variables of the problem (or $m > n-r$), with this exception; that the $m$ equations $\\frac{dP}{d\\xi_i} = \\eta_i$, together with the $r$ equations obtained by differentiating the equations of condition totally with respect to $t$, would be more than sufficient to express $y_1$, ... $y_n$ in terms of the new variables; consequently $y_1$, &c. might be so expressed in different ways, and therefore, although the value of $\\Psi$ obtained by the above rule would certainly be the same as that obtained by recurring to the original formula (D.), the form of $\\Psi$ might be different, and therefore the resulting formula erroneous.\n\nThere must doubtless exist some rule for choosing $n-m$ combinations of the equations of condition in such a way as to lead to the correct forms of $y_1$, ... $y_n$ as functions of the new variables; but I have not at present attempted to investigate it, and perhaps it would be hardly worth while. The theorem in the case in which the new coordinates are independent, may, I believe, be practically useful.\n\nERRATA IN PART I.\n\nArt. 1. equation (4.), for $dx$ read $dx_i$.\nArt. 10. In paragraph preceding equation (26.) omit the words \"not containing $t$ explicitly.\"\nArt. 18. equation (5), for $y_i$ read $y'_i$.\nArt. 19. equation (29.), for $h_i$ read $b_i$.\nArt. 24. second line after equation (L.), for \"such as $h$, $k$\" read \"such as $f$, $g$.\"\nArt. 30. The expressions equated to $h$, $k$, $c$, and the three terms in the left-hand column of the table of elements, should each be multiplied by $m$.\nArt. 42. near the end, for \"according as $\\Theta$ is between $o$ and $\\pi$, or not\" read \"according as $\\Theta$ is between $\\pi$ and $2\\pi$, or between $o$ and $\\pi$.\"",
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    "identifier": "jstor-108524",
    "title": "Errata",
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    "year": 1855,
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