IN THE THEORY OF DIFFERENTIAL EQUATIONS. 525 



to prove that a= A satisfies y" = ^, so that we have nothing to do with the connexion between 

 y and y , except to note that a m A (x, y, z) produces dz - %(x, y,x).dx=>0 if we substitute 

 xdx for dy. I noted this in my last paper, as the means of showing that the primordinal 

 equation really involves an arbitrary function. 



I need not fully repeat the proof that dA = A^ (y" — ^) dx gives A y , = as containing ordi- 

 nary primordinal solutions of y" = ■%, and A y , = co as containing all the singular solutions. If 

 we take A y , m oo we have y" as obtained from A xy . + A m ,y + A y , y ,y" = 0. Now if A , = oo be 

 a relation which really involves y, and is not merely (x, y) = 0, this last equation gives the 

 same ratios of partial coefficients, and value of y", which occurs in A x + A y y + A ,y" = : that 

 is, A y , = oo gives y" = ^. The case of (x, y) = must be excluded, and considered apart. 

 We are not to infer that A y ,<= oo must, per se, reduce y" - ^ to 0, or dA:A , to an infinitely 

 small quantity of a higher order than dx. It is allowable to confound l:oo with in com- 

 paring infinitely small quantities with finite ones : but 1 : is really an infinite of an unattain- 

 able, or infinite order, just as oo itself is a magnitude of unattainable value. 



The biordinal y" = ^ is satisfied by the ordinary solution of A , = oo , but (as will 

 presently be shown) not necessarily by the singular solution : for the ordinary and singular 

 solutions of a primordinal usually give different forms and values to y". Since y = y\r(x, A, B), 

 y m. ^r x (x, A, B) are identical, we have 



Consequently, the singular primordinals are all such as satisfy either (and therefore both) 

 of the equations 



Hence ■^ a ^ bx —^ b ^ ax = is the criterion of detection, subject to further examination when 

 it is satisfied by \/r = 0, \// 6 = 0, which, combined with y = «|/, give a relation between x and 

 y independent of y. In every other case, the combination of y = \^, y'=\l, x , and the 

 criterion, gives the singular primordinal by elimination of a and b. If it should happen that 

 x does not appear in the criterion, which then takes the form (a, b) = 0, the singular solution 

 is (A, B) = 0, and is intraneous. 



Again, since y = \js, y = \^„ ^ = \J/„, are all made identical by substitution of A and B 

 for a and b, we have 



+m A s+ fc» ^- -l» & = { lo g OM'to" ^ J } «• 



Hence ^». = oo includes all the singular primordinals in which the criterion contains x, 

 but not necessarily those in which the intraneous form, (A, E) = 0, is brought out by absence 

 of x. But jj_, = oo includes cases in which ^ a ^ hI — ^Jv/'o* * s infinite ; and in all such cases, 

 those in {A, B) = excepted, we have \//„„=oo, or ^ x + XyV + XvX "* e • If ^=00, or 



