109 



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

 constants, and no more, in its complete primitive. 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 which the admitted primitives of 

 the second order are 



V 1 = const., U 2 = const., U 3 = const., 



he maintains that the general primitive of the second order is 



/(U x , U 2 , U,)=0, 



where f 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 differential 

 equation of the rath order to a partial differential equation of the 

 first order with ra independent variables. 



4. On the criterion of integr ability of cp(x, y, y f , y",...). — If 

 we denote the differential coefficients of y bjp, q, r, s, &c, it is well 

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

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

 to relation between y and x, is 



V, J -VJ+V q "-Vj"+... = 0, 



the accent denoting complete differentiation with respect to x. This 

 has usually been established, either by the calculus of variations, or 

 by a process of elaborate expression of the actual result in terms of 

 definite integration with respect to a subsidiary variable. Mr. De 

 Morgan, after some remarks upon the manner in which certain 

 proofs of the necessity of the criterion fail, gives a very simple ele- 

 mentary proof founded upon the following theorem. If U be any 

 function of x, y, p, &c, — as far say as s, for an instance, — then 



(u') a =u;, (uo^iy, (u'),=u;+u,. 



(U'),=U f ' + U„ (U'),=U/ + U 9 , (U') 3 =U S ' + U„ (U') 4 =U\. 



Mr. De Morgan takes it to have been hitherto unnoticed that the 

 formulas V p — V g ' + V r " — ..., V g — V/+ ..., so much used in this sub- 

 ject, are, when V is integrable, nothing but the differential coefficients 

 oifVdx, with respect to y, p, &c. 



[But since the paper was communicated, Mr. De Morgan has found 

 the above theorem, and its consequences, in a memoir by M. Sarrus, 

 apparently belonging to the Journal de VEcole Polytechnique, and 

 printed in 1824. No notice is taken of this method by MM. Ber- 

 trand, Binet, or Moigno, who have written on the subject since 

 M. Sarrus.] ^ 



