* 
i 
ON THE RECENT PROGRESS OF ANALYSIS. 77 
_ may, provided a and are incommensurable, which is implied in their being 
_ distinct periods, be made Jess than any assignable quantity, so that we may 
put 
; fxe=f(ete) 
where ¢ is indefinitely small, and this manifestly is an inadmissible 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 function, that is 
in one in which we have 
f (a) =f {x+m(a+ BV —1)+m! (a! +BY =1)4+-m"(a"+p'V—1)} 
for every value of x, m, m', m” being any integers, and in which the three 
periods a+ B/—1, &c. are distinct, we can make 
: S(2) =f (@ + £) 
by assigning suitable values to m, m!, m”; € being as before less than any 
assignable quantity. Hence as this result is inadmissible, 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 @ fortiori when it appears to have more than three. But now we come to 
a difficulty. For M. Jacobi proceeds to show that if we consider a function 
"(a+ Ph)de 
of one variable inverse to the Abelian integral Sharma X being of 
the sixth degree in 2, this function has four distinct 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 impossibi- 
lity of recognising the existence of a quadruply periodic function of one va- 
riable, is not, I think, at all satisfactory. The functional dependence, the 
existence of which we are obliged to deny, may be expressed by a differen- 
tial equation of the second order; and therefore it would seem that the 
commonly received opinion that every differential equation of two variables 
has a primitive, or expresses a functional relation between its variables, must 
_ be abandoned, unless some other mode of escaping from the difficulty is dis- 
covered. It is probable that some simple consideration, rather of a metaphy- 
_ sical than an analytical character, may hereafter enable us to form a con- 
E “sistent and satisfactory view of the question, and this I believe I may say is 
_ the opinion of M. Jacobi himself. The same difficulty meets us in all the 
Abelian integrals: as in the case of those of Legendre’s first class, namely 
_ where X is of the fifth or sixth degree, so also generally, the inverse function 
has more than its due number of periodicities. 
___Abel, in a short paper in the second volume of his works, p. 51, has in 
_ effect proved the multiple periodicity of the functions which are inverse to 
_ the integrals to which his theorem relates. The difficulty to which this gives 
_ rise did not strike him, or was perhaps reserved for another occasion. 
_ M. Jacobi next proves that his inverse functions of two variables are 
_ quadruply periodic, but that quadruple periodicity for functions of two va- 
Tiables is nowise inadmissible. 
__ A difficulty however seems to present itself, which is suggested by M. 
Eisenstein in Crelle’s Journal, viz. that if for each value of the amplitude 
L 
ae d 
_ the integral ¢ 2 or Wz (vide supra, p.75), has an infinity of magnitudes 
4 
real and imaginary, and the same is the case for ? y, it is by no means easy to 
AY a 
in 
