DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 73 



< Pu >t > o- div >a S-' 3 . ^w S^. 16 



= 4 ^ ( - *-} w 



O# 06' V *Y 0-S' ' * 



\ / / 



o* 08 f <y 



4 -<l> 



c- ^ / 

 7 ox ox 



Hence 



, _ d-w . d-w 4 o-<l> 2<1> B/' 



I P ", ., ~r 2< , 7 - ;r~s~; == -, = ~i . 



</?/- ax (I i] 7 ox dx ox ox 



or the value of I is the as-derivative of v. 



Similarly for the value of m. The solution is thus seen to be included in the form 

 given in 41 ; moreover, the value of iv obtained in the preceding investigation is a 

 solution of the equation there required to be satisfied, 



44. It has been seen ( 27) that the equations to which the method involving 

 derivatives of v, I, m, n, alone can be applied which method has been indicated as 

 the generalisation of AMPERE'S for the case of two independent variables belong to 

 a distinctly limited class. Moreover, even for equations in this class, it may happen 

 that an integrable combination of the subsidiary system cannot be constructed, or 

 that the quasi-general process sketched in 28 is not effective. Consequently it 

 becomes necessary to have some other method ; and this is provided by what has been 

 indicated as the generalisation of DARBOUX'S method for the case of two independent 

 variables. 



In the discussion, we shall consider only the case when it proves necessary to 

 construct the first derivatives of an equation F = ; but the explanations apply, 

 mutatis mutandis, to other cases when second derivatives of F = and derivatives 

 of higher orders should be constructed. Examples of the latter cases, when the 

 characteristic invariant is resoluble, have already been given in Section II. 



45. The original equation is 



F (a, 1), c, f, g, h, I, m, n, r, x, y, z) = 0. 

 The characteristic invariant is 



Ayr + H/K/ -f- Br/ - GI> - Yq + C = 0, 

 and it is supposed to be irreducible. The subsidiary equations are 



y I \<ly L dy / dx 



' ll '-L _ J-L] . H(^- - p'~\ + Bl^' -^Wft* 



5 dx J \dx dy J \dy 



VOL. CXCI. A. L 



