IN THE THEORY OF DIFFERENTIAL EQUATIONS. 519 



nor any biordinals except 



yy"- 2y" - ay'\ 2y' + xy"= by", 2xy* - Zyy '- xyy"= cy", 



nor any triordinal (in any case) except Zy'y'"- 3y" 2 = 0. If we ascend from this last, then, the 

 descending biordinals being a = A, b = B, c=>C, the h'rst primitive in ascent is <p (A, B, C) - 0, 

 where <p is arbitrary, and may contain any number of constants. It is obvious that <f) = 

 gives the triordinal common to dd = 0, dB =0, dC = 0, and can give no other, even in the 

 case of (pjA^ (p B B y „+ c C y ..= co , or (p v , nAtBiC = co . But the primitive of {A, B, C) = 0, 

 though it be xy = ax + by + c, is subject to cp (a, b, c) = 0. This is not a restrictive relation 

 between the values of a, b, c, since <t> may contain arbitrary constants besides a, b, c. 

 Thus, let the first primitive be A = mB or (y — mx)y" — Zmy — 2y' 2 =0 : the original is 

 xy = b (y + mx) + c, or xy ■ ax + by + c, where a = mb. These two forms are coextensive : 

 the only difference is, that according as a, b, c, or as m, b, c, are treated as independent con- 

 stants, the curves contained under 2y'y'"— 3y" 2 = are grouped in one or another of two 

 different ways. The primordinal is found by eliminating y" between <p (A, B, C) = 0, and 

 ^(^, B, C) = : and its original is xy = ax + by + c, subject to <p(a, b, c) = 0, ^( a » ^> c) = 

 and a similar remark. The original primitive is found by eliminating y and y" between 

 (p (A, B, C) = 0, x("^> ^> C) = 0, \js (A, B, C) = 0, which amount to A = const., B = const., 

 C = const., and give xy = ax + by + c. 



Lagrange, (Lemons, &c. p. 249 [1806]), after noting that, for instance, (y, xy' — y) = 

 gives y" = 0, does not admit it as a primitive, because it necessarily leads to y = a, xy — y =b. 

 This would be a sufficient reason for rejecting one of the last pair, since either is the necessary 

 consequence of the other. All the primordinal s of any biordinal can be expressed by means 

 of any two among them : in the case before us, simplicity would dictate y = a, xy 1 — y = b, 

 as fundamental forms. But more complicated biordinals have many primordinal forms of 

 equal simplicity: and further, in ascent we are generally obliged to take what we can come by, 

 without any means of telling whether or no we have got one of the forms we should have 

 preferred, if we had known the original primitive. Having 2y'y'" = 3y"*, and finding, by some 

 accident of the problem, that (y — l):y" 2 is an integrating factor, we are led to 



(x + y) y" + 2 {y - y' 2 ) = ay", 



which is not one of the forms selected by the common theory. 



6. I now proceed to some consideration of primordinal equations. By a singular 

 solution I mean any one, however obtained, which must, by the very method of solution, fail 

 to contain the ordinal number of arbitrary constants : it may be called intraneous or extraneous 

 according as it can or cannot be made a case of the ordinary primitive. 



Let y = x (' r ' y} ^ e tne P r i mor dinal deduced from y = >//(#, a), or its supposed equivalent 

 a = A (x, y). Hence dA = A y (y — ^) dx, and % = — A x :A y are both identically true. All 

 solutions of y = ^ not contained in A = const, are in A y = co ; except only, it may be, those 

 of the form x = const., which, though they satisfy y =x "*J ma ^ n S ^ otn s ^ es infinite, give 

 y'—X tne f° rm co — w . And, with the same exception, the converse is true, that A y = co 

 satisfies y = x' ^ or I* gives y = - A xy :A yy which (§ 3) agrees with - A x :A y , or ^, whenever 

 Vol. IX. Part IV. 67 



