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By Hamilton W.R.

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9. The general principles of investigation, respecting the connexions between the partial differential coefficients of the second order, of the characteristic and auxiliary functions, having been sufficiently explained by the remarks made at the beginning of the seventh number, and by the details into which we have since entered; we shall confine ourselves, in the remaining research of such connexions, for the new auxiliary function T , to the case of extreme uniform media. And having already treated of the mutual connexions between the coefficients of the two functions V and W , it will be sufficient now to connect the coefficients of either of these two, for example, the coefficients of W , with those of T , of the first and second orders: since the connexions between the coefficients of all three functions will thus be sufficiently known.

1 = σ δσ δτ δυ ∇1 = σ (M5 ) More generally, if we denote by Tn,n the function deduced from T by the homogeneous preparation mentioned in the sixth number, which coincides with T when the variables σ τ υ σ τ υ χ are connected by the relations Ω = 0, Ω = 0, and which is, for arbitrary values of those variables, homogeneous of the dimension n with respect to σ, τ , υ, and of the dimension n with respect to σ , τ , υ , we have the following expressions, analogous to (U4 ), δTn,n = δT − δΩ . ∇n T − δΩ .

Z δy 47 2 (K5 ) And if we denote by δ 2 T1,1 the value of the second differential δ 2 T assigned by the formula (K5 ), and determined on the supposition that T has been made, before differentiation, homogeneous of the first dimension with respect to σ, τ , υ, and also with respect to σ , τ , υ , and denote by δT1,1 the corresponding value of δT , determined by the coefficients (Z4 ), we may generalise thse values by means of the following relations, analogous to (S4 ); δT1,1 = δT − δΩ . ∇1 T − δΩ .

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