262 O^V THE RECENT PROGRESS OF ANALYSIS. 



demonstration of this theorem received * le dernier degre de 

 rigueur*.' 



14. In 1829 M. Jacobi's great work on elliptic functions, 

 the Fundamenta Nova Theories Functionum Ellipticarum, was 

 published at Koenigsberg. It contains his researches not merely 

 on the theory of transformation, but also with respect to other 

 parts of the subject. But the great problem of transformation 

 is the fundamental idea of the whole work ; the other parts are 

 subordinate to it, or at least derived from it. The subject is 

 treated with great fulness of illustration and in a manner not 

 unlike that of Euler. 



M. Jacobi begins by considering the possibility of trans- 

 forming the general transcendent whose differential coefficient 

 is unity divided by the square root of a polynomial of the fourth 

 degree. Subsequently, having shown that this transcendent 

 may be transformed by introducing a new variable y equal to 

 the quotient of two integral functions of a?, and also that the 

 general transcendent may be reduced to one of the form 



dy 



he proceeds to consider the latter in detail. 



The first step of this reasoning, viz. the possibility of the 

 transformation, depends on a comparison of the number of the 

 disposable quantities in the assumed value of y with that of the 

 conditions required, in order that the quantity under the radical 

 in the transformed expression may be equal to the square of an 

 integral function of x multiplied by four unequal linear factors. 

 It is shown that the number of disposable quantities exceeds 

 by three that of the required conditions. But, as Poisson has 

 remarked in the report already mentioned (Mem. de Vlnstitut. 

 x. p. 87), and as M. Jacobi himself intimates, this does not 

 amount to an absolute a priori proof of the possibility of the 

 transformation ; non constat but that some of these conditions 

 may be incompatible. 



Granting however the possibility of putting the quantity 

 under the radical in the required form, it is shown, as in 

 Schumacher's Journal, that this condition is not only necessary 



* Verhulst, Traite Elemmtaire des Fonctions EUiptiques. 



