I 



Cambridge Philosophical Society. 567 



and so on. If, then, <p(^, y, y', 3/". y '",..) = be a given diflferential 

 equation, and if it be found that 



^fy.XY'-Y, X, A. -Il\ = o 

 \ Y" Y"3' 7 



can be solved ; it is seen that the original equation can be solved by 

 eliminating X between ^=Y' and y=XY'— Y. 



'I'he general method of which this is a particular case, is as follows. 

 Let /(a:, y, X, Y) = have its diflFerential equations of the first order 

 formed on two suppositions : first, that X and Y are constant ; se- 

 condly, that X and y are constant. Let these diflferential equations 

 be 



X=ib{x, y, y') x=(p(X, Y, Y') 



Y=*(t, y, y') y=y(X, Y. Y'). 



These equations may be used instead of the first two pairs of cor- 

 relatives in the preceding example : and each diflferential coefficient 

 of Y IS expressible by means of the same and lower diflferential 

 coefficients of y ; and vice versd. To get convertible forms, as in 

 the instance above, /(.i-, y, X, Y) must be chosen so that x and y are 

 simultaneously interchangeable with X and Y. 



Mr.De Morgan gives a similar extension of the method as applied 

 to partial diflferential equations. 



3. On the constants of a primitive equation. — It is usually left to 

 be collected from induction that the equation of the wth order has n 

 constants, and no more, in its complete piimitive. Mr. De Morgan 

 proposes an d. priori proof of this point, on which, as in all such 

 cases, it would be presumptuous to decide until it has been tho- 

 roughly examined. 



He further proposes an extension of the meaning of the term so- 

 lution, in the case of all the primitives intermediate between the 

 differential equation and the original primitive. Thus, supposing 

 an equation of the third order, of M'hich the admitted primitives of 

 the second order are 



Ui=const., U2=const., U3=const , 

 he maintains that the general primitive of the second order is 



/(U., U,. U3)=0, 

 where / is any function whatsoever : and, starting from this last 

 equation, he determines a general primitive of the first order in a 

 similar way. 



This view is supported by the reduction of a common diflferential 

 equation of the «th order to a partial differential equation of the 

 first order with n independent variables. 



4. On the criterion of integrability of <p(x, y, y', y",...). — If 

 we denote the diflferential coefficients of y hyp, q, r, s, &.C., it is well 

 known that the condition which is both necessary and sufficient, in 

 order that 'V=<ji(x, y,p, q,...) may be integrable without reference 

 to relation between y and x, is 



v,-v;-fv;'-v;"+...=o, 



the accent denoting complete diflfcrentiation with respect to x. This 



