26 REPORT—1904. 
pendent constants as before ; the differential equations can be integrated, 
and (s)2,). . . (8,z,) are p of the 2p—2 moveable zeros of a function. 
In the third case, since ¢ is a quotient of two ¢-functions, ¢ has 
2p—2—p+1+1=p independent constants, 7.¢., it has p—1 arbitrary 
parameters, and thus, as Riemann states it, ‘the problem of determining 
p—1 of the 2p—2 quantities (s,2,) . . . (82)s%2,)-9) in such a manner that 
they shall be functions of the remaining p—1 and shall satisfy the 
p differential equations 
2p—2 
Stee or ae le 
1 
is completely solved if the 2»y—2 quantities are the moveable zeros of a 
»-function, and there is only one solution.’ Such pairs of quantities are 
said to be tied by the equation ¢=0.! And the solution of the differential 
equations, with the notation introduced above, is 
Qn -2 
2p—2 2p—2 2) 
k Ke (k = ‘ 
( Shae eh ie 2pk LY Pi A Soe wp) = days 2.) 
1 1 
where c, only depends upon the additive constants in w,, 7.e., upon the 
lower limits of the integrals ; or, in other words, the sum of the values 
which any one of the p linearly independent integrals of the first kind 
assumes at the 2»—2 moveable zeros of a -function is congruent to a 
constant which only depends upon the lower limits of the integrals. 
Part II. of Riemann’s ‘Theory of Abelian Functions’ is, as has been 
said, devoted to the consideration of the theta-functions, which he defines 
thus : 
+ co p p 2 Pp 
> G1 70,M,1, + 2D0,m 
S(.-m) =[ > 7 sc iia a sd 
where the summations in the exponents are with respect to p, p’, and 
that in the outer bracket with respect to m,...m,. The adoption of 
u,...u, the p linearly independent integrals of the first kind, in 
place of the general arguments v, . . . v, and of the moduli of periodicity 
of the w’s in place of the constants a,,,—an adoption which is duly 
justified by Riemann—makes, as he says, ‘log 3 a function of a single 
variable z, which when s, ~ resume their original values after an arbitrary 
continuous change in the position of 2, is changed by linear functions of 
the w’s.’* Thus 3 is a one-valued function of p arguments, but of a single 
point on the Riemann surface, which point is the upper limit of each 
of the p integrals which appear in the arguments of 9. The notation 
employed by Riemann has not been adopted by all following writers, for 
he does not use the symbol of integration with upper and lower limits 
associated with it ; he introduces instead a symbol of his own for the 
values of the integrals at the upper limits, and only mentions the lower 
limits in words. There is, for our purpose, a certain advantage in this 
notation, for it draws attention to the values of an integral w, ata certain 
set of points which form the different upper limits of the same integrand. 
Thus, if «,...«,, are the m points on the surface in which a rational 
function of s, 2 takes the same value, then, in Riemann’s notation, wu” is 
the value of w, (for w=1,...p) at the point ¢, (for v=1,.. . m), for 
' Loe. cit. § 16. 2 Loe. cit. § 17. 
