[harkness] study of ELLIPTIC FUNCTIONS 95 



of the question whether a function of a single variable could have 

 more than two periods. Jacobi was influenced, on entering into this 

 field of research, by his belief that functions f (u; 2<w^ 2^^)^ for which 

 (o^/w is real but irrational, are non-existent; e.g. speaking of a doubly 

 periodic function X (u) and using the word index as equivalent to 

 period, he says "so wurde also die Funktion X (u) einen Index haben, 

 der kleiner als irgend ein'e gegebene Grosse ist und docli nicht ver- 

 schwindet. Das aber kann nicht sein." (Ueber die vierfach period- 

 ischen Funktionen zweier Variabeln. Crelle's Journal, 1834). 



It is easy to show that a one-valued analytic function f (u) cannot 

 admit an infinitely small period. For if u^ be an ordinary point, there 

 is a neighbourhood of Uq Avithin which f (u) takes the value f (u^) at no 

 point other than u,,; but if a period 2 fi exists such that u-|-2 fi lies 

 within this neighbourhood, f (Uq + 2 fi) = f (u^). The two statements 

 are in contradiction. 



It is also easy to show that an analytic function f (u) with a finite 

 number of values cannot admit an infinitely small period. 



There remains the possibility that there may be infinitely many- 

 valued analytic functions f (u) which admit an infinitely small period. 

 At the time that Gopel wrote his celebrated memoir Theoriae trans- 

 cendentium prim, ordinis adumbratio levis (published after his death 

 in Crelle, vol. 45, 1847), Jacobi was believed to hold that even in this 

 case such functions did not exist. Gopel took the opposite view. 



Triple periodicity. Jacobi proved in an entirely rigorous manner 

 that if we take a system of periods Î2 = Pi flj + P2 Î22 + P3 î^.î i^ 

 which the p's take all intégrai values, except Pi=P2=P3=o> then 

 whether fi^, ^2, Q^ are real or complex, the lower limit of | 2^2 | is 

 zero, (a) If two periods of the system, say pj fii+P2 fi2+P3 ^ 

 and pJ fii+P2 ÏÏ2+P3 • ^3 coincide, the lower limit o is obviously 

 attained; namely by (pj-pj) fii + (P2-P2) fi2 + (P3-pl) ^^3; (b) But 

 if this case does not arise, the lower limit is not attained, and the system 

 necessarily contains infinitely small periods. Jacobi showed that case 

 (a) means that f (u) is only apparently triply periodic ; in reality the 

 three periods fij, ^2, ^3 can be replaced by a pair J2i, ^22 (or possibly 

 even by a single period fi). That is, the system can equally well be 

 represented by 



Pi Ûl + P2 Î22, 



where Pj, P2 take all pairs of integral values, exclusive of the pair 

 Pi=P2 = o. Jacobi formulated his results in the following way; — ■ 

 "If an assigned function has three periods, these can either be com- 

 pounded out of two, or the function has an index which is loss than 

 any given quantity. Since this is absurd there is no triply periodic 

 function." It is of interest to recall the position taken up by Gopel 



