468 MR. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 

 Hence i/ is determined by 



A - 2I + ( 2 e 



_ 

 A' - 2B> + C ~ c'' 



It may be noticed, though the fact is not directly connected with the present inves- 

 tigation, that the equation (50) is, if rendered a non-fractional equation, apparently 

 the most general quadrate-quadratic relation between s and x. But, as a matter of 

 fact, in order that the most general quadrate-quadratic relation of the form 



c ) + s (c^x 2 + 26^-f cj + (a.,x 2 + 2&,x +c- 2 ) = 

 may be expressible in the form (50), the condition 



Oy, 



must be satisfied. The proof of this is easy, as is likewise the verification that the 

 coefficients of the non-fractional equivalent of (50) satisfy the condition. 



Derivatives of Even Order. 



119. All the derivatives which have hitherto occurred have had the order of the 

 highest differential coefficient of the dependent variable entering into their expression 

 an odd integer, and the reason of this is that the dependent variable is the quotient of 

 two solutions of a linear differential equation having its right-hand member zero, so 

 that each solution contains implicitly in homogeneous form n arbitrary independent 

 constants, and the quotient of the two therefore implicitly contains 2/i 1 arbitrary 

 independent constants. Hence the differential equation satisfied by the quotient is of 

 order 2n 1. 



But, if we take the quotient of two solutions of the equation 



d'y 



(where x ls n t zero), these solutions are no longer linearly homogeneous in the n 

 implicit constants, and the quotient will therefore contain implicitly In independent 

 arbitrary constants. Hence the quotient- equations will be of even order; and like- 

 wibe the quotient-derivatives, if they exist. 



