ON THE RECENT PROGRESS OF ANALYSIS. 255 



developed in the first volume of the Exercises de Calcul In- 

 tegral, which appeared in 1811. 



12. The theory of elliptic f auctions, as it is presented to 

 us by Legendre, may conveniently be considered under the 

 following heads : 



OL. The reduction of the general integral, 



Pdx 



Va + fix -f- 7Jj* 4- 8x 3 4- ex* 



in which P is rational to three standard forms, since known as 

 elliptic integrals of the first, second and third kinds*. 



This classification, though the reduction of the general in- 

 tegral had, as we have seen, been already considered by La- 

 grange, is I believe entirely due to Legendre. If we consider 

 how much it has facilitated all subsequent researches, we can 

 hardly over-rate the importance of the step thus made. It may 

 almost be said that Legendre, in thus showing us the primary 

 forms with which the theory of elliptic integrals is conversant, 

 created a new province of analysis : he certainly gave unity and 

 a definite form to the whole subject. 



For the three species of functions thus recognised Legendre 

 suggested the names of nome, epinome and paranome, the name 

 of the first being derived from the idea that it involves, so 

 to speak, the law on which the comparison of elliptic integrals 

 depends. But these names do not seem felicitous, nor have 

 they, I believe, been adopted. To this part of the subject an 

 important theorem relating to the reduction of elliptic integrals 

 of the third kind, whose parameters are imaginary, seems 

 naturally to belong. 



These three forms are 



f> /i-#x* ^ f" 

 -cV') ; Jo V ~T^ 2 :; J ( 



(i+no; 8 )y(i- 



Legendre always replaces x by sin <, so that the integrals become 



'* dd> 



/> d<f> /> . f 



I fn=i I Vi-c'sin 2 ^; 



J o Vi-c 2 sm 2 Jo v J 



(i + n sin 2 0) sji-c 2 sin 3 



The radical ^/i -c 2 sin 2 <f> is often denoted by A. 



The constant c is called the modulus; the second constant n (in the third kind) 

 is called the parameter. The modulus may always be supposed less than unity, 

 and if c = sin e, then c is the angle of the modulus. 



