284 ON THE RECENT PROGRESS OF ANALYSIS. 



function inverse to the integral I . , and <b c the 



JV(l-O (1-&V) 



corresponding function for the modulus c, then, on introducing 

 the inverse notation, the differential equation 



dy _ dx 



= a 



becomes of course d& add, with x = <$> C 6 and y = <$> k &. Hence 

 for a given increment a of 0, that of & is aa. 



Let us take the simplest case, and suppose y to be a rational 

 function of a?; then, as x or $ C remains unchanged when 6 in- 

 creases by a period of the function (j> c , y does so too ; that is (j) k O' 

 remains unchanged when & increases by a times a period of < c , 

 or in other words, a times a period of < c is necessarily one of < fc . 



Suppose now Jc and c to be both real and less than unity ; 

 then <j) k and (f> c have each a real period, here denoted by 2co k and 

 2co c respectively, and each an imaginary period vr k i and r c * re- 

 spectively, Tx k and TX C being both real. Let 6 receive first the 

 increment 2w c , and secondly the increment r, then, by what 

 has been said, 



m, ft, p, q being certain integers. But can these two equations 

 subsist simultaneously? Not generally, since if we eliminate 

 a and equate possible and impossible parts, we get two relations 

 among c0 c '& e a) k 'GT k , which are continuous functions of the two 

 quantities k and c. Hence both are determinate ; and if we 

 wish c to remain indeterminate, we must either make m and q 

 equal to zero, in which case a is impossible, or, making n and^? 

 equal to zero, assign a real value to it. When a is real we have 



(Oj, vr k 

 a = m q , 

 <o c * c 



and hence the remarkable conclusion, that 



0) k CO C 



- : :: q: m, 



^k c 



m and q being integers. 



-nr here is in M. Jacobi's notation 2JT, so that <f>6 = <f> (d+ imw F n-Gn), m and 

 n being any integers. 



