1887.] Dr Glaisher, On expressions for the Theta Functions. 129 
Its habitat was Southport, Lancashire, where it was taken in 
the “slacks,” or swampy districts among the sandhills, 
The paper also dealt with the fertilization of the female in 
Araneina, and recorded an observation on this head with regard to 
the species Veriene rubens. 
Some points in the structure of the nets of orb-weavers were 
commented on, and an instance adduced illustrative of the 
possession of a certain amount of intelligence by spiders. 
Some typical British spiders were exhibited, together with a 
few specimens from Bermudas collected by Mr A. E. Shipley, 
Fellow of Christ’s College. 
(4) On expressions for the Theta Functions as Definite Inte- 
grals. (Second paper.) By J. W. L. GuaisHER, Sc. D., F.R.S8. 
§ 1. In Vol. 11. (1877) pp. 61—66 an account was given of 
four different methods of obtaining expressions for the Theta 
functions as definite integrals. I have recently worked out the 
expressions for the four Theta functions by the third and fourth 
of the methods described in that paper. 
Putting pa, p=, and w= pam A and denoting 
by 9 (u), 9,(u), 9,(u), @, oy the four Theta functions © ae ), H (wu), 
H (u+ kK), 8 (w+ B), we have 
i ae MWe Ce: 
w= ES ( ae: 
isan a Sin (=)"e ae 
eg s  5 | 
where J denotes n + 4. 
§ 2. By applying the third method, described in § 6 of the 
previous paper, to the summation of these series, I have found that 
2 \ sinh 2¢— sin 2 (¢ + x) 
Lae cosh 2¢+ cos 2 (t+) 
sinh 2¢ — sin 2(¢ — 3 é (=) di 
0 (u 
cosh 2¢ + cos 2(t— a) 
