ON THE RECENT PROGRESS OF ANALYSIS. 309 



In the twenty-fifth volume of Crelle's Journal M. Eichelot 

 resumed the subject of his former paper, and discussed it in a 

 very interesting memoir. This fundamental or principal result 

 may be said to be -a generalisation of M. Jacobi's. It is that in 

 the function 



fjb may have any value whatever. The resulting differential 

 equation, though rather more complicated than when, with M. 

 Jacobi, we suppose //, a root of fx 0, is still very readily in- 

 tegrable. We have thus an indefinite number of algebraical in- 

 tegrals, since the quantity /JL is arbitrary, but of course not more 

 than m 1 of them are independent. 



In the same volume of Crelle's Journal, p. 178, there is a 

 curious paper by Dr Hsedenkamp, in which the algebraical in- 

 tegrals of Jacobi's system are for the case of a polynomial of the 

 fifth degree under the radical deduced from geometrical con- 

 siderations. It is shown that in a system of curvilinear co- 

 ordinates (those of which MM. Lame* and Liouville have made 

 so much use), the equations of. the system are the differential 

 equations given by the Calculus of Variations for the shortest 

 line between two points. Consequently the finite equations of 

 a straight line are the integrals sought. This very ingenious 

 consideration is afterwards generalised. 



32. In the twelfth volume of Crelle's Journal, p. 181, M. 

 Eichelot has considered the Abelian integral of the first class. 

 The principal result at which he arrives is, that the only rational 

 transformation by means of which such an integral may be 

 changed into one of similar form is linear in both the variables 

 which it involves. By means of this substitution, he trans- 

 forms, under certain conditions, the integral in question into 

 a form analogous to the standard forms of elliptic integrals. 

 The subject of the division of hyper-elliptic integrals of each 

 class into three genera is also considered, and the same prin- 

 ciple of classification as Legendre's is made use of. The paper 

 concludes by pointing out an error which Legendre committed 

 in the application of his principle. Legendre had thought that 

 the formula of summation given by Abel's theorem for integrals 



f T e (?3' 



of the form I - 7 = could not involve a logarithmic function. 

 J Nx 



