DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 479 



general solution of the equations (62). For this purpose, proceeding by the ordinary 

 method, we have to obtain a series of integrals of the subsidiary equations 



d_du_ du' du" du'" 



= : " (n- l)u~ 2(-2)tt'~ 3(n-3)* 



Now integrals of these are 



A = , 



B = (n - 1) Au" - (n - 2) u'\ 



so that by the theory of partial differential equations the most general solution of the 

 form-equation in (62) is 



<f> = function of u, (n 1) uu" (n 2) w' 8 , ... 



The number of independent integrals of the subsidiary equations necessary for the 

 construction of this most general solution is the same as the highest order of differ- 

 entiation that occurs ; each of the integrals when freed, by means of preceding integrals, 

 from all hut one of its arbitrary constants itself furnishes a solution of the form- 

 equation a conclusion from the ordinary theory of partial equations of this type. 

 With each new derivative of u of higher order supposed to occur in the concomitant, 

 there is a new subsidiary equation ; and consequently a single new integral is necessary, 

 which must of course include in its expression this new derivative. The earlier 

 investigations show how to derive such a function ; for, by taking the Jacobian of u 

 and the derived covariant involving what has hitherto been the derivative of highest 

 order, we obtain a function which involves the new derivative, is invariantive, and so 

 will furnish the new integral of the subsidiary equations. 



It thus appears that any identical covariant which involves at the highest the wth 

 derivative of u can be expressed in the form 



<M, U 2 , U s , . ...U.); 



and, as this result is true for all the values of m that can occur, we derive the conclusion 

 that the series of successive covariants already given is a complete series, that is, any 

 identical covariant can be expressed as an algebraical function of terms of the series. 

 132. These fundamental functions of the series which come after U 8 are not, how- 

 ever, in their simplest form ; they can be replaced by others, necessarily their 

 algebraical equivalent and involving the proper derivatives of u, but of lower order 

 in the variables. In fact, if the grade of the fundamental covariant be an even 

 integer and equal to 2r, the covariant may be taken in the form 



<k,=wu'>-a 1 V 2 '- 1 > + aju'V*- 2 '-. . . + (- l/a^u* + ...+(- 



