522 Mr DE MORGAN, ON SOME POINTS 



question whether y = 0, a solution of y = x (x, P + y) — x (x, P), is intraneous or extraneous 

 to y + P = \lr (x, a), the general solution of the last. We need therefore only consider the 

 case in which y = solves y = x (*? 2/)' or m w hi c h X (*i °) van ' snes independently of x. 



It is first to be shown that when y = is extraneous, then / \x (x, y)\ *' dy begins finite, 

 that is, will be finite for some finite extent of value of /3. Since % = - A a :A , we have 



f Aydy or A (x, /3) - A (x, 0) = - / — — '-—- dy, (x constant). 



The general solution of y = x is a = A (x, y), and y = is extraneous, so that ^ (a?, 0) is 

 neither constant nor infinite for all values of x, but is a function varying with x. Hence, for 



some extent of value of /3, / A x -£~ l dy is finite. If then A, begin infinite at y = 0, it is clear 



that X A x y~ l dy (which begins finite) begins with elements infinitely great compared with those 

 f Ly'^dy : so that in this case Xx _1< ^ begins finite. But if A x begin finite, then, so long as 

 A x and x retain each one sign, 



/ — dy = A x (x, 0(3) / — , (0<1), or / — = , 



when ^4 (#, 0) is not = 0, this vanishes with /3, and begins finite : and A (x, 0) = 0, indepen- 

 dently of x, cannot be admitted, since it brings y = into the general solution. Hence f x~^y 

 begins finite whenever y = is an extraneous solution. 



The converse must now be shown : namely, that whenever f x~ x dy begins finite, x ( x > °) 

 vanishing independently of x, it follows that y = is extraneous. Let foX'^V =/( ,r 5 y), or 

 d y f {x, y) = dy--x (*« y)> independently of x : so that, m being any quantity, 



dyf{m,y) = dy; x (m, y). 



Since dy = v {x, y) . dx is made identical by y = \j/ (x, a), and since d/{/n, \// (<r, a)} is 

 identical with d\j/ (a, a) : x{ m > tyi** a )\' we nave 



d/ {m, J, (», a)} = * j-^A^i *», identicaUy. 



Now it is known that <p (x + h) — <px = <p' (x + Oh) . h, (0<l), provided that between 

 x + h and x there be no value which makes cj)'x infinite : and this though d)'x should be 

 infinite, <px being finite, at either extremity. 



Hence, 



> Y \«t> + Oh, \ls (x + Oh, a)\ . ,. 



for all values of x, h, and m, with such casual exceptions as may arise if the second side become 

 infinite between x and x +h, though these may be avoided by taking h sufficiently small. 

 Since f(m, 0) = for all values of tn, we see that if it were possible to assign a value to a 



