ON THE RECENT PROGRESS OF ANALYSIS. 250 



but in one of a later date (August 2nd) lie gave a formal 

 enunciation of his theorem, but without demonstration. The 

 two communications appear consecutively (Ast. Nacli. vr. p. 33). 



In No. 127 of the Nachrichten, VI. p. 133, M. Jacobi gave a 

 demonstration of his theorem. 



If we can so determine y in the terms of x as to satisfy the 

 differential equation 



di 1 dx 



-y) (i - xy v (i - &) (i - 



(M being constant), it is manifest that we shall have (^denoting 

 the elliptic integral of the first kind) F (kx) = MF (ty) , pro- 

 vided that y and x vanish together. The question therefore is, 

 how may the differential equation be satisfied, for it is clear 

 that by means of a solution of it we transform the elliptic in- 

 tegral F(kx) into another, viz. into F(\y). 



M. Jacobi shows that if y be equal to y, Z/and F being 



integral functions of x, the differential equation will be satisfied, 

 provided U and F fulfil two general conditions, the second of 

 which is found to be deducible from the first. He then makes 

 an assumption which is equivalent to assigning particular forms 

 to U and F, and thence shows, by a most ingenious method, 

 that these forms of 7 and F are such as to fulfil the first of the 

 required conditions, which, as has been said, implies the other. 

 He thus verifies, d, posteriori, the assumed value of the func- 

 tion y. 



In proving that the forms assigned for U and F have the 

 required property, it is necessary to pass from an expression of 

 the value of 1 y in terms of x to one of 1 \y in terms of 

 the same quantity. This is done by means of a remarkable 

 property of the functions U and F, namely, that if in both 



x be replaced by -j , -p- or y will (the constants being properly 



F 1 



adjusted) become j- T or . Therefore, in any form in which 



A< u ^~y 



the relation connecting y and x can be put, we may replace 

 x by j , provided we at the same time replace y by . 



This has been called the principle of double substitution, and 



172 



