932 
PROFESSOR A. CAYLEY ON THE SINGLE 
, »- , f [dmcln 
taken over the area included between the two rectangles. We have (m, n)=vi-{-—.n, 
= m J r i9n suppose, where (a being' negative) 6= —is positive : the integral is 
[U 
J J (m 
dmdn f 7 If 1 
dm. —I 
J J (to + idn) 
i6\jn-\-i6n) _ v 
= Udm(— 
1 17 J \m — i dv m -J- idv 
1 m—idv 
= - Jon-■ 
id & m + idv ’ 
or 
finally between the proper limits the value is 
2 f., /fit — idv 
=rfi lo s 
fJL+idv 
— idv'\ 
+ idv') 
, dv 
-1 
where the logarithms are log (/x— i6v)= log \//A-j-m— i tan 1 —, &c., and the tan 
t 
denotes always an arc between the limits — \tt, +lpr. Hence if A =co, , = 0, the 
2 . . . 2ir 27r 2 7r 3 
value is -(—Of— 0f+iiri+i7ri)=-7r,=-; or K=4-—. Hence finally 
'iu v a a 
[p^-G=co]-l[p^z., = 0]=exp( ). 
64. We have a,=log q, negative ; hence taking r such that log q log t=tt' 2 , we have 
ci' =log' r, also negative; and r, like cq, is positive and less than 1. We consider the 
theta-functions which depend on r in the same manner that the original functions did 
on q, say these are 
A(w, r) = A (0, r) Till 
A 
m+ 
-a > 
n—. 
TT l J 
B (u, r) = B (0, r) IHI<J 1 ^ 
C(u, r) = C (0, r) nn 
— a 
m + n — ; 
TTl 
—a 
vx + n —. 
TTl 
D (u, r) = D'(0, r)umi\ 1 + 
, cf 
to + n — ; 
TTl 
