260 ON THE RECENT PROGRESS OF ANALYSIS. 



by means of it we pass from the expression of 1 y to that of 



1 , and thence obtain that of 1 A?/. It is to be ob- 



\y' 



served that this principle is used merely to prove a certain pro- 

 perty of the functions U and V. Of course, as the change of x 



into = implies that of v into in the finite relation between 

 Jcx \y 



these quantities, the same thing will be true in the differential 

 equation by which they are connected, a remark which may 

 very easily be verified. But, on the other hand, it by no means 

 follows that because it is true in the differential equation there- 

 fore any assumed finite relation between y and x having this 

 property is the integral required. The property in question 

 therefore does not enable us to verify any assumed value 



of y. 



This remark has reference to a communication from Le- 

 gendre which appears in No. 130 of Schumacher's NacJirichten^ 

 VI. p. 201. In it he gives an account of M. Jacobi's researches, 

 and an outline of the demonstration of which we have been 

 speaking. I find it impossible to avoid the conclusion that this 

 great mathematician mistook the character of the demonstration 

 in question, and that to him it appeared to be in effect a mere 

 verification of the assumed value of y by means of the principle 

 of double substitution. He remarks that the direct substitution 

 of the value of y in the differential equation is impracticable, 

 but that M. Jacobi had avoided this substitution by means of 

 1 une proprie'te particuliere de cette equation qui doit 6tre com- 

 mune aux integrates qui la repre'sentent.' This property is the 

 principle of double substitution; and after showing that it is 

 true of the differential equation, the writer proceeds thus : 

 * Ce principe une fois pose*, rien n'est plus facile que de verifier 



1' equation trouvee y = -^, car par la double substitution on 



obtient la m6me valeur de y a un coefficient pr&s qui doit etre 

 gal a 1' unite ;' and, after a remark to our present purpose im- 

 material, concludes, ' Ainsi se trouve demontre'e ge'ne'ralement 



liquation y = -painsi que, etc.' 



As we have seen, such a verification would be wholly in- 

 conclusive, nor is the essential point of M. Jacobi's reasoning, 



