520 



Mr DE MORGAN, ON SOME POINTS 



A y = oo is produced by a substitution* for y ; that is, whenever it is produced by anything 

 except x = const. This last case, which satisfies y = x in the form co = co , requires special 

 consideration ; in what follows I suppose it excluded. 



Since y = >//(#, A) is identically true, we have 1 = \js a Ay, identically, \^ o meaning 

 \|/ (x, A). Hence \|/ a = contains all singular solutions except those in x = const. Again, 

 x (x, y) is identically y}, x (x, A) : whence X y = yj,^ = >J^:>// = (log ty a ) x . Of this equationf 

 I can find neither notice nor use : it may be directly deduced thus ; — 





.A y = 



'■'J!) 



J 



yl x 



A, 



y = - 



sty 



A^Ayy ^ ( *\ 



~*1 \~A~J~ 



Xr 



It follows that when \|/ o vanishes under a substitution for x, (log \^ a ) x becomes infinite with 

 log \}/ a ; and therefore also Xy But when \j, a vanishes under a constant value of a, \}/ a = oo 

 does not necessarily give (\}sj x = oo . Consequently, ^=00 contains all the extraneous 

 solutions, and may, and frequently will, contain intraneous singular solutions : in particular, it 

 must contain all those intraneous solutions which, besides being deducible from a specific 

 constant value of a, are also deducible from y = \j/ (x, a) by substituting for a a function of x. 

 So that Xj, = °° must give all the solutions, whether intraneous or extraneous, which have 

 contacts with ordinary solutions. 



The exception may be thus illustrated. Since \|/ tt = gives >//„= — \j/ aa . a x we have in 

 that case Xv~ ~ ^S^Ja' !- ^ ow y i r a = °> under any substitution for a, gives (logv// a ) a = 00: 

 but if a = const., or a x = 0, Xy takes the form co x 0, and is not necessarily infinite. 



Next, Xy or 0°g VOx mav b e infinite without \// = : as when yf/ a = 00 , or when yj/^ = 00 

 without \js a = eo . But whenever the result of ^ = co is not a solution of y =■ x> the relation 

 between a and x which turns y = \js (x, a) into that result gives \^„ = 00 : that is, an ordinary 

 solution can only meet Xy = 00 , when itself not a solution, in points of infinite curvature. 

 This was proved in my last paper, but the following is shorter. First, let \\/ a = 00 , yp being 

 finite : then, except when a = const, (a case here against hypothesis) we have \l/ x = co , and 

 therefore \j/ xx = 00 . Next, let \\r ax = 00 , and not \p a : then, unless x = const, (a case through- 

 out excluded), \|/ is not infinite ; so that ^^ being infinite, >//„ is infinite, except in the 

 extra-hypothetical case of a = const. Now \j/„, or y", is x* + X» • X' so t ^ lat a ^ t ' le re bations 

 resulting from ^ = 00 which make X* + XvX ni " te or zero, are necessarily solutions : as 

 otherwise proved in my last paper. 



* It must be remembered that the agreement of A : B and 

 A, : B x when A and B both vanish, or when either is infinite, 

 has only been proved when the singular state is produced, as it 

 were, by action upon z, that is, by substituting for 2 a function 

 of other letters. It is easy, when the state is produced by a 

 relation between z and other variables, to forget that the par- 

 ticular case in which z disappears from the relation may be, 

 and probably is, an exception. 



A caution of the same kind is necessary in another case. 

 The function and differential coefficient are so connected that 

 the second becomes infinite with the first, for a finite value of 

 he variable : but not when the function becomes infinite with- 



out action on the variable. 



+ The more complicated relations between Xj a "d tne 

 original primitive, mostly derived by Lagrange, have the dis- 

 advantage of connecting ^i„ and Xy with other functions. A 

 great number may be found in Lagrange's paper Sur les 

 Integrates particuliires (Berlin Memoirs, 1774, pp. 197-275) 

 and in the Lecons sur le Calcul des Fonctions. If, which is 

 more than I know, the equation in the text has ever been 

 given, it is curious indeed that it has not become common as 

 the mode of connecting the two best known and most widely 

 used tests of singular solution. 



