[138] PROF. DE MORGAN, ON SOME POINTS OF THE INTEGRAL CALCULUS. 

 When we grant the absolute identity of 



^» (<*> y, y) + ^ y («. y, y) ■ y + >/y {x, y, y') . f (x, y, y), 



and (p{x, y, y , >// (x, y, y')\ 



we grant all that is necessary to prove that y" = <p (x, y, y, y") follows from y" = \J, (x, y, y'), 



and no more. 



When we grant that q = \js (x, y, p) satisfies 



d 1 , dq dq 



j- + — p + — q - <p (*, y, p, q) 

 dx dy dp 



in which x, y, p, are independent variables, it can be shown that we have granted the preceding 

 identity, and no more. Consequently, every solution of the partial differential equation gives 

 an integral of 



y -<p(x,y,y,y), 

 and every integral can be thus obtained. 



It can further be shewn that any solution of (l) is ordinary or singular, according as 

 the solution of (2) from which it arises is ordinary or singular. 



July 3, 1851. 



ERRATUM. 



At the beginning of Section 1, for <t>(x+y \J -I), read <p (x, y,a + b *J-\) 



