266 ON THE RECENT PROGRESS OF ANALYSIS. 



M. Jacob! lias shown that no function* can be triply periodic, and 

 therefore these inverse functions possess the most general kind 

 possible of periodicity, a property which gives them great 

 analytical importance. 



Following M. Jacobi, we shall henceforth give the name of 

 elliptic functions to those which are analogous to circular func- 

 tions. It is on this account better to call Legenclre's functions 

 elliptic integrals than, as he has done, elliptic functions (vide 

 ante, p. 253). 



By the new notation we are led to consider a great variety 

 of formulae analogous to those of ordinary trigonometry. The 

 sine or cosine of the amplitude of the sum of two quantities may 

 be expressed in terms of the sines and cosines of the amplitudes 

 of each, &c.f ; and we have only to make the modulus equal to 

 zero to pass from what has sometimes, though not with much 

 propriety, been called elliptic trigonometry to the common 

 properties of circular functions. 



" M. Jacobi gives a table of formulas relating to the new 

 elliptic functions, and proceeds to apply their properties to the 

 problem of transformation. It was in this manner that he had 

 treated the problem in the Nachrichten. As in his earlier essay, 

 he assumes y equal to a rational function of #, whose coefficients 



* i. e. no function of one variable. 

 *f* The fundamental formulae are 



sin am u cos am #A am v + sin am v cos am uA am u 



7 *-->" . 

 i - k 2 sin 2 am u sin" 5 am v 



cos am u cos am v - sin am u sin am vA am wA am v 

 cos am (u + v)= - 19 . .. -r-n - ; 



i - k 2 sm j am u Bin 2 am v 



A am -uA am v - k 2 sin am u sin am v cos am u cos am v 

 A am (u + v) ,.. . r- ^- - 



i - k 2 sm 2 am u am 2 am v 



being the modulus, and A am u A/I -fc 2 sin 2 am u. If 



where & 2 -f &' 2 = i, then it may be shown that 



sin am (u + ^K) = sin am u, 

 and 



sin am (u + iK' \/ i) = sin am u, 



so that 4# is the real and iK' V -^ tne imaginary period of sin am u. Hence it 

 is obvious that we shall have generally 



sin am (M + 4m+ inK' /J - i.) = sin am u, 

 m and n being any integers. 



