IN THE THEORY OF DIFFERENTIAL EQUATIONS. 521 



Whenever two solutions have a common point, Xy is infinite at that point, if y be finite. 

 Let y = P, y - P + Q be two solutions, and let P* be finite when Q = 0. The equations 



P- = x (a>, P), P-+QT = X («, P+Q) 

 are identically true ; whence 



Q' = x (*, P + Q) - x (•< ^) - X»("> P + OQ). Q, (0< 1), 

 if Xj, (#, -P) be finite. Hence (log Q)' is finite when Q ■ 0, which, Q being really a function 

 of x, cannot be : so that j^ (a?, P) is infinite. 



It has been commonly asserted that when Xj , = ec gives a solution, that solution is 

 extraneous. The well known proof (Lacroix, Vol. II. pp. 383 — 385) depends on a use of 

 expansions in the failing case of Taylor's Theorem. But the assertion itself is not merely 

 incorrect, but falsified by many ordinary examples. Frequent instances occur in which a case 

 of y = \jy (to, const.) has at each point contact with another case : and further, we have seen that 

 2^ = co may include ordinary solutions which are not even deducible from \^ = 0, that is, 

 solutions which are not singular even in the sense in which I have used the word. 



The following remark may suggest the means of inquiring further into this case. It 

 appears that when \}s a and (log yf/ a ) x are both made infinite by the same constant value of a, 

 there is a solution of y' = x which is wholly ordinary, contained in ^ = co , the relation which 

 has been asserted to give only extraneous solutions, and not even the rest of the cases which 

 come under \J/ a = 0. If this be the case, as from Xy = (log ^ o ) r it certainly is, how comes it 

 that instances have not appeared in the multitude of questions which have been exhibited in 

 elementary works ? The answer is, I think, of the same kind as must be made to the question 

 why the aptitude of Maclaurin's development to exhibit certain finite funtions in the form 

 + 0. to + .x 2 + ... has never been noticed until our own time. 



A function of a and a may become or co for a value of a independent of to in two 

 distinct ways. First, factorially, when the function f(x, a) can be exhibited as 



Fax \f{to,a):Fa}, 



in which Fa becomes or co and /(*, a) : Fa remains finite : as in e"— 1 which vanishes with 

 a, and is a x {{e ax - l) : o\, or Ox*. Secondly, non-f actor ially, where this transformation 

 cannot be made, as in e~ a ~ 2x , which vanishes with a, but cannot be reduced to a factorial form 

 in which the evanescence arises solely from that of a function of a alone. When <^ a becomes 

 or co factorially, the function of a disappears in ^ M :\^ a , and ^y ma y be finite : when non- 

 factorially, no such disappearance occurs, and \^««:^ a , which then becomes 0:0 or co:co , may 

 be finite or infinite, as it shall happen. 



7- M. Cauchy (Moigno, Vol. II. pp. 445 — 454) has given a criterion for distinguishing 

 between extraneous and intraneous singular solutions. This criterion I believe to be univer- 

 sally valid, and fully demonstrated : but as it seems to have attracted little or no attention in 

 this country, and as the demonstration may be much shortened without material alteration, I 

 subjoin my own account of it. 



The question whether y = P, a solution of y ~ % C*» #)» & anc ^ X (*» ^) ^ em g identical, 

 is intraneous or extraneous to y = \J» (to, a), the general solution, is clearly the same as the 



67—2 



