DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 481 



This is an invariant, and eo is a solution cf the equation (as will be verified immediately 

 in connexion with a cognate case) ; and it involves the (2r -j- l)th derivation oft*. 



As the function next in succession beyond ^> sr+l we take <(*& + & which has already 

 been found. It is not difficult to see that 



3n - 4r 5 . 



differs from u*<f> 2r+z by a resoluble function, 



Replacing now the quantities U^, U 8 , ... by the functions <f>, we can enunciate the 

 result of 131 in the form : 



Every identical covariant, which is a function of and its derivatives alone, 

 can be expressed as an algebraical function of u, U 2 , U 8 , $ 4 , ^5, ... 

 134. Example II. The derived invariants which are functions 0/"8 3 . 

 The form-equation for these invariants / is 



* - 1 



and in order to obtain the most general solution of this equation it is necessary to 

 obtain a proper number of integrals of the associated subsidiary equations 



~ 1.66, 2.70', 3.88T, 



Integrals of these involving derivatives of B 3 in successive orders are 



A = 6 3 , 



B = A6' 8 - * e'., 



C = A ! B" 3 - 4A6 3 6" 3 + ^ e' 3 , 

 D = A6* 3 - G6 S 6" 3 + ^ eV, 



When we proceed to construct the general solution of the form-equation by modifying 

 these integrals so that each may contain only a single constant, the right-hand sides 

 are the successive invariants derived from 8 3 , or are algebraically equivalent to them ; 

 and thus the required general value of / is 



= function of 6 3 , 



j, .,, 



The derived invariants, which arise in successive formation after B^j, are not in their 

 simplest forms ; they can be reduced in a manner similar to that adopted for the 



MDCCCLXX XVIII. A. 3 Q 



