388 MR. A. R. FORSYTE ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



In this equation it is necessary to change the independent variable from z to x, in 

 order to compare it with (ii). We have* 



_ ^. 



",-! ! dr" 

 where 



( Jm 



A. Mt , = Limit, when p = 0, of - [<j> (x -f- p) <f> (x) }' 

 and 



(2). 



It at once follows that, writing m ! C w> , for A t , and denoting differential coefficients 

 sometimes by dashes (chiefly when there are powers higher than unity) and sometimes 

 bj Roman numeral indices, we have 



C Mi , = ==* = coefficient of p m in {<f> (x + p) - $ (x)}' 



ItV i 



i.e., in (p^ + ^p^ + i.p^ +...)' ...... (3). 



Substituting now in the semi-transformed equation, we immediately find the 

 coefficient of d'u/dz' -f- s ! to be 



ro! d"-*X 





2ln-2l Z -2.' -3.' ' ' ' sin -s -21 



r = n-> n \ ft = n-r-, 



' 



- r - 



^ 



r r if, 



with the symbolical interpretations P = 1, Pj = 0. But the present form of the 

 equation must be effectively the same as (ii.), and the coefficients of corresponding 

 derivatives of u must therefore be proportional to one another. In the transformed 

 equation the coefficient of frujdz" - n 1 is (here s = n, so that r = and t = are 

 the only values for terms in the summation) A^X, or the coefficient of d"u/dz* is 

 A_,X/n ! ; that is, it is Xz'*. Hence we have 



TO 



* ScHLailiLCH, ' Vorlesungen iiber einzelne Theile der hoheren Analysis' (3 e Auflage, 1879), p. 5. 



