POINTS OF THE INTEGRAL CALCULUS. [129] 



find it convenient to treat the groups of largest number in each line as containing the inde- 

 pendent equations, from which all the others are to be deduced. 



Section IV. 



On the condition of integr ability of (f> (#, y, y, y",...y in) )*. 



Let the successive differential coefficients of y with respect to x be denoted by p, q, r, s, t, 

 &c., using y, y", &c, when convenient. Let V be a function of x. y, p, q, &c, and let 

 Pr> V y , V p , &c. be its partial differential coefficients with respect to the several letters, taken 

 independently. Let the accent be the symbol of complete differentiation with respect to x, 

 so that we have in all cases 



r- V, + V y p +V p q+ V q r+V r s+ ... 



If V contain no letters beyond r, then s occurs only as seen in V r s ; and so on. 



Various proofs have been given of the theorem that Vdx is integrable per se, indepen- 

 dently of relation between y and x, when V y — V p + V" — ...vanishes identically. Euler 

 proved it by the calculus of variations. Lagrange gave a complicated proof involving the 

 same principles. Lexell gave so much more complicated a proof (according to Lagrange, 

 whose mention of it is all I know about it) that it is difficult to decide either upon its truth 

 or generality. Poisson gave another proof derived from the calculus of variations. M. 

 Bertrand gave two proofs,, one from the calculus of variations, another from the introduction 

 of definite integration with respect to a subsidiary variable. M. Jacques Binet extended the 

 second proof of M. Bertrand, and the resulting form of Poisson, so as to make them apply 

 to all cases, without any failure. (Poisson, Mem. Inst, 1829, p. 260; Bertrand, Jo. Ec. 

 Polytech. cah. 28; Moigno, Vol. n. p. 551.) Allusion is made to some investigations of M. 

 Sarrus, which I have not seen. 



The objection to all these proofs is that they are not fundamental : they introduce consi- 

 derations the farthest from elementary of any which can now be imagined, the principles of 

 the calculus of variations, and the use of expression by integration with respect to a variable 

 introduced for nothing but expression. This objection, however, is of no validity until a 

 simple and fundamental proof is given : and this I propose to do. 



Again, it is to be shewn, not only that the criterion is sufficient, but that it is necessary. 

 Some of the proofs of the latter point appear to me to fail entirely. They depend upon 

 the reduction of fVdx to an integrated portion together with an integral of the form 



" April 9, 1851. I picked up a memoir by M. Sarrus, which i takes no notice of it, except to observe that M. Sarrus does not 

 has the appearance of being part of the Journal de VEcole | use the calculus of variations. MM. Cauchy and Moigno pass 



Poly technique, and the signature has" Tome xi v. No. vii.ier 

 Janvier 1824." This memoir contains the proof here given, 

 in substance, though the equations on which the criterion is 

 founded are not demonstrated. It is singular that M. Bertrand 



it over altogether. But it must be observed that M. Sarrus 

 establishes only the necessity of the criterion, and does not 

 establish its sufficiency, except when the equations that give it 

 are granted with it. 



Vol. IX. Pakt II. 41 



