306 ON THE RECENT PROGRESS OF ANALYSIS. 



elliptic functions, viz. the reduction and transformation of the 

 integrals themselves, and the theory of the inverse functions. 



But before considering these I shall give some account of what 

 has been done in fulfilment of the suggestion made by M. Jacobi 

 at the close of the Considerations Generates. Mathematicians 

 have succeeded in effecting the integration of the system of 

 differential equations to the consideration of which we are led by 

 Abel's theorem, and which is commonly designated by German 

 mathematicians as the " Jacobische system;" its existence and 

 its integrability having been first pointed out by M. Jacobi. 



In Crelle's Journal (xxm. 354), M. Eichelot, after modifying 

 the form in which Lagrange's celebrated integration of the dif- 

 ferential equation of elliptic integrals is generally presented, ex- 

 tended a similar method to the system of two differential 

 equations which occurs when we consider the Abelian transcen- 

 dents of the first class. He thus obtains one algebraical integral 

 of the system. In the case of Lagrange's equation one integral 

 is all we want; but in that which M. Eichelot here discusses 

 we require two. Now if in the former case we replace each of 

 the variables by its reciprocal, we obtain a new differential equa- 

 tion of the same form as the original one, and integrable there- 

 fore in the same manner; and if in its integral we again replace 

 each new variable by its reciprocal, that is by the original 

 variable, we thus, as it is not difficult to see, get the integral of 

 the original equation in a different form. That the two forms 

 are in effect coincident may be verified a posteriori. But the 

 same substitutions being made in M. Eichelot' s equations, which 

 are of course those we have already mentioned at p. 302, the 

 first of them becomes similar in form to the second, and vice 

 versa the second to the first. Thus the system remains similar 

 to itself; and if in the algebraical integral we obtain of it we 

 again replace the new variables by their reciprocals, we fall on 

 a new algebraical integral of the original system ; this integral 

 being, which is remarkable, independent of that previously got. 

 Thus the system of two equations is completely integrated. 

 Extending his method to the general system of any number of 

 equations, M. Eichelot obtains for each two integrals, but of 

 course these are not all that we want. At the conclusion of his 

 memoir M. Eichelot derives from Abel's theorem the algebraical 

 integrals of the "Jacobische system." 



