ON THE RECENT PROGRESS OF ANALYSIS. 267 



are elliptic functions, and shows that this assumption satisfies 

 the differential equation already mentioned. It may be asked 

 what is gained by the introduction of elliptic functions into a 

 problem of which, as we have seen, particular cases (e.g. the 

 transformations of the third and fifth order) can be solved -by 

 algebraical considerations. The answer is, that the properties 

 of these functions enable us to transform the assumed relation 

 between y and x in a manner which would otherwise be im- 

 practicable. It is conceivable that any particular case might be 

 solved by mere algebra, l^ut it does not seem possible to dis- 

 cover in this way a general theorem for transformations of all 

 orders, and practically the labour of obtaining the formula 

 for the transformation of any high order would be intolerable. 



Having proved the theorem for transformation in nearly the 

 same manner as he had already done, M. Jacobi developes the 

 demonstration which, as we have said, Legendre hinted at in 

 No. 130 of Schumacher's Journal. 



He then proceeds to consider the various transformations of 

 any given order. We have seen that the modular equation for 

 those of the third order rises to the fourth degree, that is to say, 

 for a given value of the modulus of the original integral four 

 new moduli exist, corresponding to four new integrals, into 

 which the given one may be transformed. These four trans- 

 formations are all included in the general formula for the third 

 order; but it is to be remarked that in general only two of 

 the roots of the modular equation are real. Thus there are 

 two real transformations and no more. The same thing is true, 

 mutatis mutandis, of the transformations of any prime order (to 

 which M. Jacobi's attention is chiefly directed), that is to say, 

 there will be n + 1 transformations of the nth. order, n I of 

 which are imaginary. The two real transformations are called 

 the first and the second ; the second is sometimes called the 

 impossible transformation, because it presents itself in an ima- 

 ginary form*. Of the formulae connected with these two trans- 

 formations M. Jacobi gives copious tables. 



* Mr Bronwin, in the Cambridge Mathematical Journal and in the Pliil. Mag., 

 has made some objections to this transformation ; but from a correspondence which 

 I have recently had with him, I believe I am justified in stating that he does not 

 object either to M. Jacobi's result or to the logical correctness of his reasoning, 

 but only to the form in which the result is exhibited. 



