137 



both P and Q, become infinite, P : Q and P^ : Q* are both nothing, 

 both finite and equal, or both infinite; provided that the infinite 

 form is produced by substitution for x. If u=(v, w, ...), any rela- 

 tion which makes u v infinite either makes u w infinite, or is indepen- 

 dent of w. And if u v =<x> be produced by a relation containing v, 

 then u v dv + u w dw+...=0 and u vw dv + u vw dv+ ...=0 are relations of 



identical meaning. 



4. From the last it follows that U = const, is solved by making 

 any factor of d\J either or go . In dU=M(Pdx + Qdy), singular 

 solutions are obtained, as is known, from M = oo : it ought to be 

 asked whether M=0 does not give singular exceptions, that is, cases 

 in which U=const. arises otherwise than from P+Q,y' = 0. It is 

 found more convenient to treat these cases without actual separation 

 of the factor; that is, from dU=\J v dx + XJ y dy. 



5. In a former paper, the author insisted on the arbitrary func- 

 tions which enter the intermediate primitives : maintaining, for ex- 

 ample, that the primordinal of y"=0 is (j>(y',xy'—y)=0, for any 

 form of (p. Lagrange, he has since found, notices this extension, 

 and rejects it, because it leads to y' = a, xy'—y=b, as necessary 

 consequences of its ordinary solution. Mr. De Morgan maintains 

 his opinion, and observes that Lagrange's reason would make it 

 imperative to reject one of the two, y'=a, xy'—y = b, since either is 

 the necessary consequence of the other. 



6. In order to avoid the ambiguous use of the word singular, a 

 singular solution is defined as any one which, by the mode of obtain- 

 ing it, cannot have the ordinal number of constants : it is further 

 styled intraneous or extraneous, according as it is or is not a case of 

 the general solution. If y = \p(x, a) or a=A(x, y) give y'=^(x, y), 

 then dA=A p (y' — x)dx and ■^=—A x :A P are identical equations. 

 Every relation which satisfies A y =<x> is a solution, and a singular 

 solution ; except possibly, relations of the form #= const., which 

 must always be examined apart. Also, A y =cc is identical with 

 ip a =0. There can exist no solutions whatsoever except those which 

 are contained in A=const., A^, = oo , and (possibly) j?=const. 



Again, \v == (^ S' l Pu)x- Of this equation the author has found 

 neither notice nor use : supposing it to have ever been given, he 

 holds it most remarkable that it has not become common as the mode 

 of connecting the two well-known and widely used tests of singular 

 solution. It easily shows that x?^ 00 contains all extraneous solu- 

 tions, and all intraneous solutions which (as often happens) can be 

 also obtained by making a a function of x. It also easily gives a 

 conclusion arrived at by the author in his last paper, namely, that 

 when ^ =oo is satisfied and not 2/'=x» it follows that x»+X«X 1S 

 infinite. 



7. The author gives his own version of the demonstration of a 

 theorem of M. Cauchy, for distinguishing extraneous and intraneous 

 solutions. If y—P, P being a given function of x, satisfy y'=x(x> y), 

 that is, if P' and x(#> P) be identical, then y = P is an extraneous or 



