Cambridge Philosophical Society. 451 



which destroy the eiFect of its variation upon the form of differential 

 coefficients, it is called self-compensating. Thus <p{x, y, a) =0, 

 ^y(a', y, «)=:0, contain the self-compensating variable a. Similarly, 

 when ^{x,y, a, 6)=0 is accompanied by (p^da-\-<pi^db=0, a and b are 

 mutually compensative, and primordinally. The addition of 



<t>a(x\y) da + (i)b^x\y) db=:0 



makes a and b biordinally compensative. 



3. When a finite change in x makes an infinite change in y, it 

 makes an infinite change in y' : y, in y" : y\ &c. When either or 

 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 «,, infinite either makes ««, infinite, or is indepen- 

 dent oitv. And if ?<„=<» be produced by a relation containing v, 

 then ni,dv + Vu,div-\- ...=^0 and Ui,u,dv-\-u^u}dv-\- ...=-0 are relations of 

 identical meaning. 



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

 any factor of dU either or oo . In d\]=^M{?dx+Qidy), singular 

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

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

 in which U = const. arises otherwise than from P-f Qy' = 0. It is 

 found more convenient to treat these cases without actual separation 

 of the factor; that is, from d\]^\]^dx-\-\5 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" = is (p(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 tvvo, y'=a, xy' — y=6, 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 = y^{x, a) or a=A(x,y) give y'=x(.^>y)> 

 then dA = Ay(y' — x)^-^ ^^^ X^~^'^'- ^y ^^^ identical equations. 

 Every relation which satisfies A^=co is a solution, and a singular 

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

 must always be examined apart. Also, Ay=oo is identical with 

 i|/„:=0. There can exist no solutions whatsoever except those which 

 are contained in A=const., A^ = x , and (possibly) x=con£t. 



Again, x = ('"^S 4'«)j-- 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 Bhows that Xy^*^ contains all extraneous solu- 

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

 2 H2 



