394 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT 



Modification of the Coefficient Relations (III.). 

 19. By integration of equation (5), we have 



Xz' i(-D = constant; 



since the equations (iii) are homogeneous in the dimensions of X, this constant may 

 be taken of any arbitrary value other than zero, and so we may write 



(iv). 



With the form of equation (ii) adopted, this is the only equation which helps to 

 determine X and z ; and therefore we may consider z as arbitrary and, when an arbitrary 

 value is assigned, X is determinate. We therefore assume 



2 = a; + e/t, ........... (9) 



where c, an infinitesimal constant, is to be considered so small that squares and higher 

 powers may be neglected, and p. is an arbitrary non-constant function of a;*. From this 

 it at once follows that . 



z' = 1 + /, .......... (10) 



and, for values of k greater than unity, 



while from (10) and (iv) it follows that 



X=l-i(n-l) v ', .... . . . . (11) 



and 



d*-*\ . . 



__ = _ i(n _ 1) 



Hence 



W = X = 1 - i(n - 1) V '; Wj = 



and by (4), for values of r greater than unity, 



; W, = P r - i(n - 1)6 (l, 0, Po, . 



= P r -|(n-l)eT r 

 say. Also by (3) we have 



* The functions are shown by this process to be invariants only for an infinitesimal, but otherwise 

 perfectly general, transformation ; but the immediate purpose is to obtain the numerical coefficients and 

 not to prove the property of general invariance, which, otherwise known, could be derived by the 

 principle of cumulative variations. 



