DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 3 



Let any general differential equation of the second order be taken in the form 

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



When the proper value of v is substituted, this becomes an identity, so that, when 

 differentiated with regard to x, y, z, in succession, the results are identities. Hence, 



writing 



F, = 



8F 8F_ 



, 9F 



cv 



, 3F 3F 

 ; = -^ + -, n 



or 

 y 



8F 

 ''SI 



3F 



' di 



3F 



9F 3F 



^ + 9 



8F 



dm 



,.8F 



dm 



en 

 ,.BF 



8F 



have 



... + F,r- = o 



J 



By CAUCHY'S theorem, a solution of F = exists, determined by the values of r 

 and one of its derivatives, assigned for a relation between x, y, z. This implies that, 

 at all points on the surface represented by the relation, the values of v and, say, 8v/dx 

 are given ; and consequently the values of dv/dy and 80/82 are known at all points 

 on the surface. 



Taking now v, I, m, n, as known on the surface, and denoting by p, q, the derivatives 

 of 2 with regard to x, y, along the surface, we have 



dl = adx + hdy + gdz = (a + pg) dx -f- (h + qy) dy, 



dm = hdx + bdy -\-fdz = (h + pf) dx -\- (b + qf) dy, 



dn r= gdx -\- fdy + cdz = (g + pc) dx + (/+ ^c) dy, 



so that, as Z, m, n, are known everywhere on the surface, the quantities 



dl , dl 



are known along the surface. These equations require the relation 



B 2 



