DIFFERENTIAL EQUATIONS OF THE SECOND ORDER, 65 



We can at once infer the result of 35 as to 'the solutions which, being of the 

 suggested form, explicitly involve u and 17 only ; but other solutions in a finite form 

 do not suggest themselves. 



38. It need only be remarked that, with the slightest change of notation, all the 

 preceding results can be associated also with the equation 



* I ""T n ~~ ^ ^KTT ' 



,v~ cy* ct- 



where c is a constant. 



Application to V 2 y + /cr = 0. 



39. Consider next, but more briefly, the equation 



where K is a constant ; in our notation, it may be written 



+ l> + c + K-V = 0. 

 The characteristic invariant equation is 



the same as for V~v = 0. Substituting for a and b their values as given in 1, it is 

 seen that, on account of the characteristic equation, the term in c disappears from 



the result ; and we have 



T dl dn din dn 9 o 



~" d,c * dx dj " d]i 



The subsidiary system thus is 



1 = 0, J = "1 



dm dn <U__ (In 



d;<; dj/ dy dv 



dv , dv_ _ . ( 



the third of which is an analytical consequence of the fourth and the fifth. As i 

 the preceding example, the general solution of I = is 



z = u + xp (u) + yq (u), 

 where p and q are functions subject to the relation 



and in the subsidiary system the differentiations with regard to x and y are partial 



VOL. CXCI. A. K 



