524 Mr DE MORGAN, ON SOME POINTS 



It is very important, in this subject, to distinguish between the means of detecting singular 

 solutions, and that of establishing their general properties. For instance, the primary being 

 <p (a?, y, a) = 0, we cannot be certain of all the singular solutions until (besides the case of 

 x = const.) we have completely examined (p a -(p v = 0, both in <p a =0 and <p y = od . Never- 

 theless, we may be sure of the general properties of those solutions by reasoning either on 

 d> a = or on d) y = co: for <p(x, y, a) = can be examined with perfect generality either in the 

 form y - \^ (x, a) = 0, or A (x, y) - a = 0. Consequently, $„'■$„ = may be distinguished, 

 under the name of the criterion of detection, from <p a = or d) y = OS , the general property. 



9. I shall now treat biordinal equations fully, and triordinals to such an extent as will 

 show how the general theory commences. An TC-ordinal equation has 2" distinct kinds of 

 primitives possible a priori, there being the choice of ordinary or extraneous singular at each 

 step of the ascent. Generally, the singular solution of a singular solution is not a solution 

 of the final equation. 



The common theory contains a defect which ought to have been striking. If 

 <h(x, y, a,b) =0 give y" = %(* 5 y, y')i it is clear that if a and b be made compensatory to the 

 second order, that is, if d a>b <p = 0, d ab d> j{!/ = 0, then y" m ^ will remain true : and also that 

 a singular primordinal is produced, satisfying y" = ^, and having no arbitrary constant. It is 

 assumed that all the singular primordinals are thus obtained. This is true, but to the 

 inference we have no more right than to the deduction of A = 0, B = 0, from A + B = 0. 



Let y = y\t{po, a, b) be the primitive, which, combined with y = ^/,, gives a = A(w, y, y), 

 b = B(x, y, y). The primordinal of y"= ^(a?, y, y) is f(A, B) = 0, / being arbitrary. Any 

 given curve y = lira; may be made to solve some one form of the primordinal, namely, that 

 obtained by eliminating x between A = A (x, -mx, is ' x) and B = B(x, tstx, sr'x). But as 

 y = srx, taken at hazard, will not satisfy y" = ^, we may presume that it is a singular solution 

 of the primordinal. And this may be verified as follows. Let/ (.4, B) = be the result of 

 the elimination; then dB:dA = -f A :f B . But, as we shall see, dB:dA = B y ,:A y ,; whence 

 /a^ + /b^ = 0. If f (A,B) = give y' = K (x,y), we have K,— (f A A, + f B B ll ):(f A A ll .+f B B,); 

 whence not only y- srx, but any solution of f(A, B) = 0, seems to give K y = co . The truth 

 is that all solutions which make variables of A and B, are singular : for, f(A, B) = being 

 B = /u.A, the ordinary primitive is y = ^(x, a, fxd), which gives A = a, B = fxa. 



We must not suppose that we are here dealing with a peculiar or restricted class of equa- 

 tions. Every primordinal is in an infinite number of ways reducible to f(A, B) = 0, where A 

 and Bare made constants by the ordinary primitive. Thus y' 2 = l is coextensive with 

 y 2 + a? 2 — Zxyy' = (yy — xf, the ordinary primitive of which makes both sides constant. It is 

 obtained by throwing ± y = x + a into the form y~ = a? + 2ax + b, where b = a 2 . 



Given y = -arx, required a key to the primordinal equations which have y = <srw for a sin- 

 gular solution. Take any equation y = \p (%, a, b), determine a = A (x, y, y), b = B (x, y, y), 

 eliminate x between a ■ A (x, <srx, tst'x) and b = B (x, nrx, nr'x), and write A (x, y, y) and 

 B (x, y, y) for a and 6 in the result. 



We know that a = A and b ■ B give but one biordinal equation y" = ^ (x, y, y), where 

 X - ~ (-4* + ^ V y)'-^ V ' - ~ ifi» + ByVY-B,,- We need not appeal to the original primitive 



