390 MR. A. R. FORSYTE ON INVARfANTS, COVARIANTS, AND QUOTIEXT- 



13. Tlie first invariant. In the particular cases of n = 3 and n = 4, Biuosciii lias 

 shown (/. c. 5) that there is a function of the coefficients such that 



remarking that the invariantive forms remain the same for differential equations of 

 higher order ; and HALPHEN has, for the general equation, obtained this invariant by 

 another process. Before passing to more general investigations it is easy to see that 

 this result follows from equations (5)', (6)', (7)' ; and the deduction of it requires 

 modifications of those equations which are subsequently of great use. We have, 

 from (4), 



W = X , W 1 = X'; 

 and from (3) 



p l( n _1\'-Z Z " P z'-l 



\J >n -\ IV* L l z > ^ 1, 1- 



while, generally, 



n z' 



^m, m z 



so that (5)' now is 



= 2'X' + i (w - 1) Xz" . . . * . v . , (5), 



an integrated form of which will be subsequently taken. Writing with BRIOSCHI 



we have 



2" =2'Z, 



2 1U = Z' (Z' + Z 2 ), 



X' = -i(n-l)XZ, 



X" = -i(-l)X{Z'-i(H-i)Z 2 }, 

 X ui = - ^ (n - 1)X,{Z" - f (n - 1)ZZ' + i(n - 1) 2 Z 3 }. 

 Again, 



C., ._ = A (n - 2) 2'-* {4'^ + 3 (n - 3) 2" 8 } 



= h ( n - 2 ) 2 '"~ 2 { 4Z/ + ( 3 - 5 ) Z2 } ; 



C..!, _ 2 = i (n - 2) 2 '"- 3 2" = 1 (n - 2) 2'-*Z, 

 by means of which (6)' changes to 



Xz' 2 Q 2 = - 3 \- (n - 2) (4Z' + (3 - 5) Z 2 } + i (n - 2) ZX' + i (X" + P 2 X), 

 which, on substitution for X" and X', reduces to 



2Z' - Z 2 = - (P, - Q/ 2 ) ._.,.,., . (<p) _. (G). 



