Mr. J. Cockle on Differential Covariants. 227 



Then we have 



u 



~ Tr u a, 



Q ) = q+K 2 — +p-> 

 * l u r u 



Again, in^ and q substitute B, C, and F for b, c, and /respec- 

 tively, and call the results (P) and (Q). Then 



(P)=2K 2 B-(K 3 + K' 2 ) 



= p + 2K 2 ^, 

 (Q) = K 2 C-(K 3 +K' 2 )B 



Tr u" v! 

 2 * u x u 

 and, consequently, 



(P)=P, (Q) = Q. 

 Hence the expression 



(<\p, vilify 



is properly termed a differential covariant of the given differen- 

 tial cubic; for when the y in both is transformed by the same 

 factorial substitution, then the result obtained by transforming 

 that expression is the same as the result obtained by forming an 

 expression, 



in which P and Q respectively are the same functions of B, C, 

 and F that p and q are of b, c, and/. 



It will be remembered that a critical function remains unaltered 

 after factorial substitution (to which the division by u is an ac- 

 companiment), and that the differentials, with respect to the 

 independent variable oc, of critical functions are critical. 



That A causes this covariant to vanish is readily shown ; for 



{2K 2 +Ap)y' + (v + Aq)y, 



and 2K 2 + Ap--0, andp + /\q = 0. 



It may be convenient to exhibit the differential cubic and the 

 differential covariant above obtained in another shape. Let, then, 

 the cubic be written thus : 



dhi . d q y , dy , . 



Q2 



