1887.] Theta Functions as Definite Integrals. 133 
§ 5. In the formule of the last section we may substitute 
2 
in the right-hand members a for w, and multiply by oe if ab 
ie 
the same time we interchange on the left-hand side Jkp and 
Jk'p. This transformation corresponds to the change of modulus 
2 
from & to k’, for by this change » is converted into = and the 
Me al Ts MO : 
multiplier “— is equal to ““,. We thus find 
ere i Jp 
sinh 2¢+ sin 2t ne / 2 
Jp = a cosh 2¢ — cos 2t* inthe > ) dt 
—. 4 f> sinh 2¢—sin 2t Qut? | 
=1+ 7 [, cosh 2t — cos 24° os (Se a 
— 8 f° sinhécost+coshtsint . /2ut’ 
kip ie i} cosh 2¢ — cos 2¢ ( ) ale 
8 (° sinh tcost¢— cosh ¢ sin é 2ate 
age | cosh 2¢ — cos 2¢ ( ae 
= A fie sinh! 2¢— sin 26 /2pt 
Jip ==. See a a 
a4 sinh 2¢ + sin 2¢ ey) Af 
cosh 2¢ + cos 2¢ ( R ‘ 
Jkk'p* = 
8 (” (Cc+Ss) sinh 2¢— (Cc — Ss) sin 2¢ ee (an a 
TJ (cosh 2¢ + cos 2t)” No ) 
aS 0 (Cc — Ss) sinh 2¢ + (Ce + Ss) sin 2¢ AM = dt 
‘ (cosh 2¢ + cos 2t)’ ( 
§ 6. The factor “es by which the integrals in § 4 were 
multiplied may be removed by transforming them by the sub- 
Bre a 1 . 
stitution t=,/7pr, and the factor _ may be removed from the in- 
tegrals in the last section by putting ¢= 7. 
* We may obtain the formule in this section directly by giving to x its limiting 
yalues in § 3 and transforming the integrals by the substitution mt=2r. 
