DIFFERENTIAL EQUATIONS OF THE n"' OrDER. -19 



Subtracting the latter of these equations from the former, we get 



{>=n-l 



^v K' - ^: a:"' + s ^r a;:;, ^„.-„ ... - s Ar ^r -»-.,. = o...(39). 



C=l 



Changing in tlie first sum q to q — 1, the diftereuce between the 

 two sums in this equation will become 



S" [^T Af - D^'J Af^ 



""-p. Q ' 



which is =0 by equ. (28). The equations (39), therefore, become iden- 

 tical with (37). Thus 



When there are n distinct first integrals to (12), they will satisfy the 

 conditions (33), ivhich must be satisfied, in order that these integrals may 

 render the equations (15) integrahle. 



§■ 6. 

 The remaining integrations and deriving of the primitive. 



We suppose the equation (20) completely solved. 



1) Assuming that this equation has not equal roots, if we can inte- 

 grate the equations (21) and (23) for every one of these roots, we get n 

 distinct first integrals, which give such values of the ditferential coëftlcients 

 of z of the {n — 1)'' order, as will render the equations 



d:„^2-i,< = ^„-i-i.idx + z„_^_,^,^, dy (i = 0, 1, 2, ... (n — 2)) . . . (40) 



integrable. These equations being integrated, we take out the values of 

 the dift'erential coefficients of z of the (n — 2)'"' order and substitute in 

 the equations 



t^-,,-3-,-..- = -V2-,-, , d^v + r„_3_, ,+1 dy [i = 0, 1, 2, . . . , (n — 3)] ... (41), 



after the integration of which we get expressions for the differential coeffi- 

 cients of z of the («, — 3)'''' order and so on. Going on in this way , we 

 will finally get ^j o ^^^'^ ~o,i ^^ functions of x, y and z. Substituting 

 their expressions in the equation 



d~ = ^1,0 da; -\- Zq I dy ^ 



