Under this form, the solution is presented by Lagrange in : Mécanique Analytique, Tom. II. p. 320, Generalization of the foregoing theory. 9. All equations, whatsoever their nature or subje which are expressible in the form (π”+A ̧π” ̄1+A ̧π” ̃3 ..... + A„) u = X ...... (!) where π is an operative symbol subject to the laws a being a constant and u and v functions of x, admit of traformations analogous to those of Art. 5. 2 Thus, since u = (π” + A‚π” ̄1 + A„π” ̃2 .....+A„) ̃1 X, we shall have, when the roots a,, a,,...an of the auxi equation n-1 n-2 m” +А ̧m”-1 +А ̧m” ̃2.....+ An are real and unequal, the transformation = 0 the coefficients N1, N,,... N, being determined as in Art. The legitimacy of this transformation is proved by opera on both sides of (2) with "+4"...+4„, and shew t (1) is reproduced with the same conditions for detering N1, N,...N, as if π were a symbol of quantity. But question of its completeness, of its conducting, through the formance of the inverse operations (π-a,), &c., to the st general solution of (1), is one that we are not called upon determine a priori. In all the cases we shall have to coner, its completeness will be obvious. reducible to the form π (-1) u = 0 where π = Let (-1)-10=y, then, since (π-1) y = 0, we have A very interesting application of the same theory to the lution of partial differential equations is afforded by what r Carmichael has termed the index symbol of homogeneous nctions. Cambridge and Dublin Math. Journal, Vol. VI. 277. Since, if u represent a homogeneous function of the ath gree of the variables x1, x,,...,, we have dua + xn dxn represent π3ua = a2ua, &c. and therefore, in accordance with the reasoning of Arts: and 4, an equation of which the second member expresses the plete, because the only, value of the first member when f is rational and integral, but a particular value when the firs member contains inverse factors. n-1 n-2 Hence, if we have any equation ƒ (π) u = X, where ƒ( of the form "+A‚π"1+ ̧π"...+A, and X is a series d homogeneous functions of the variables, suppose we get n X = X2+X2+... &c., u = {ƒ (π)} ̃1X +{ƒ (π)} ̃10 -1 = {ƒ (π)} ̃1Xa+{ƒ (π)} ̄1 X, ... + {ƒ (π)} ̄10 α = {ƒ (a)} ̄1 Xa+ {ƒ (b)} ̄1 X, ... + {ƒ (π)} ̄10, by (4) To find the value of the last term, we proceed, as in Art. to reduce it to a series of terms of the form A, (T-a i being the number of roots equal to a of the equation f(m)= Now it may, by an induction founded on successive appli tions of Lagrange's method for the solution of linear par differential equations of the first order, be shewn that α Ua, Va......Wa being arbitrary homogeneous functions X1, X2, X1 of the ath degree. n To this result we may give the symmetrical form (π — a) ̄10 = μ¿Ã¡ ̄1 + vaMi2...+Wa‚ L, M, &c. being logarithms of any homogeneous functi which are not of the degree 0. It remains to shew how it may be ascertained whether proposed partial differential equation can be reduced to form f()uX. d Let us resolve each symbol entering into π, into two, dx d dx and let represent as operating on x, only as entering dx; = = d d But as ", in (C), operates on the variables only as entering into π, which is a homogeneous function of those variables of the first degree, we may replace it by unity. We have therefore Tu (π- 1) TU. In the same way it may be shewn that 'u (r+ 1) (π − r + 2) ... Tu. And thus it is seen that any partial differential equation which is expressible in the form f()uX, on the hypothesis that operate on the variables only as entering into u, is reducible to the form (7) u X, independently of such restriction. This reduction having been effected, the solution can be found by means of (A) and (B), whenever the second member consists of one or more homogeneous functions of x, x2,... x. = d'u dx' dx' du du (∞ the +y + nu nx dx Here we have (2 — nπ' + n) u = x2 + y2+x3. Therefore {( − 1) − nπ + n} u = x2 + y2+x3, &c. whence u = {(π — n) (π − 1)} ̃ ̄1 {x2 + y2 + x3 } + {(π — n) (π − 1)} ̄1 0 10. We may, by simple transformations, reduce to the above case various other classes of equations differing from the above only as to the form of ; e.g. the class in which d d ... d -; +ad but, passing over such special forms, we shall consider the general equation f(π) u=X, where and each of the coefficients X, X,,...X, as well as X, may be any function whatever of the independent variables. And we design to shew, first, how it may be determined whether a given equation admits of reduction to the more general form above proposed; secondly, how, then, to integrate it. Suppose the given equation of the nth order; then the symbolical form in question, should the proposed reduction be possible, will be n-1 n-2 (π" +А ̧π”¬1+А ̧π”2 ... + A„) u = X ..............(4). Now the highest differential coefficients in the given equation will arise solely from the symbol π", and the terms in which they occur will enable us to determine the form of π. Thus, for two variables, we have |