304 ON THE RECENT PROGRESS OF ANALYSIS. 



is the fundamental period of the function. But if we had for 

 all the values of x 



f(x) =/{* + a] = 

 we should also have 



where m and n may be any integers, positive or negative. 

 Hence my. + nft may, provided a and ft are incommensurable, 

 which is implied in their being distinct periods, be made less 

 any assignable quantity, so that we may put 

 fx=f(x+e), 



where e is indefinitely small, and this manifestly is an inad- 

 missible result. Accordingly we see that one at least of the 

 periods of elliptic functions is necessarily imaginary. 



Again, similar reasoning shows that in a triply periodic func- 

 tion, that is in one in which we have 



f(x)=f{x+m(a.+ft^ l)+m f (oL+ft'>J l)+m"(a"+/8'V 1)} 

 for every value of x, m, m', m" being any integers, and in which 

 the three periods a + /3V 1, &c. are distinct, we can make 



/(*) =/(+) 



by assigning suitable values to m, m, m"; e being as before less 

 than any assignable quantity. Hence as this result is inadmissi- 

 ble, it follows that there is no such thing as a triply periodic 

 function. Whenever therefore a function appears to have three 

 periods they are in reality not distinct, and so a fortiori when it 

 appears to have more than three. But now we come to a diffi- 

 culty. For M. Jacobi proceeds to show that if we consider a 

 function of one variable inverse to the Abelian integral 



(a + ftx) dx 



X being of the sixth degree in x, this function has four dis- 

 tinct and irreducible periods. His conclusion is that we cannot 

 consider the amplitude of this integral as an analytical function of 

 the integral itself. In the present state of our knowledge, this 

 conclusion, though seemingly forced on us by the impossibility 

 of recognising the existence of a quadruply periodic function of 

 one variable, is not, I think, at all satisfactory. The functional 



