472 MB. A. R. FORSYTH ON INVARIANTS, COVARTANTS, AND QUOTIENT- 



* 



s" , 2*' 



s'" , 3s" , 

 6* 4s'" 



S T , 5s* , 10s m , 10s" 

 s* , 6.s T , 1 5s* , 20.s m 



, 5s 1 , s , 

 , 15s", 6s 1 , 



the four rows next after the first two ; and so on. And the primitive of the 

 hyper-ntic derivative equation 



[(, z)}, = . . mv (54) 



is 



_ A 



s -^ 



B + BjZ + . . . + B^ Z"- 1 + C2 



(54'), 



where from the point of view of the derivative equation (54) the constants A, B, c 

 are arbitrary. 



Relation between the Derivatives of Even and of Odd Order. 



123. In the integration of the derivative equations the following connexion between 

 the two sets of derivatives is of interest. Let 



be the equation in the (n + l)tic derivative of odd order, and 



the equation in the hyper-wtic derivative of even order ; their primitives are of the 

 form 



AQ + AI& + ... 4- 



and 



B 



_ CQ + (V + . . . + C.-^- 1 



D 



