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



y = coefficient of p in [x(z + p) - x(z)} r 



(eh-fg)p 



/ _ i y- r 



ffl - 1! 



m _r!r_l! 



+ 



so that, writing 

 we have 

 Hence 



m!m-l! ^_ r 

 ' r 1 ! m r! 



115. The method of reduction of the determinant transformed by the substitution 

 of this last relation is conveniently indicated by the reduction of the cubic derivative. 

 Denoting ds/dz, d 2 s/dz*, ... as before by s 1 , .s", . . . and dsjdx, d 2 s/da?, . . . by s ta % . . ., 

 we have 



[s,z] 3 = 



30s, 



- 12P<f>s m + 36^^% 



+ 60^} , 



Multiply the second and third columns by Xj and X 2 respectively and add to the first, 

 choosing Xj and X 2 so that s u and s, no longer occur in the first constituent of that 

 column ; it will be found that s u and s t have disappeared from the other constituents. 

 The value of ^ is 2<f>, of Xj is 2< 2 . Multiply the third column by X 3 and add to the 

 second, choosing X 3 so that s { no longer occurs in the first constituent of the new 

 second column ; it will be found that, for the value of 2<f> of Xg, s, has disappeared 

 altogether from the second column ; and we have 



[*] = 



30s, 



6 s T - 



