134 Dr Glaisher, On expressions for the Theta Functions. | Nov. 14, 
Thus, for example, 
Pane af sinh 2 /at+sin 2 J apt 
Pp "Io cosh 2 J apt — cos 2,/arpt 
sin (272) dt 
” sinh 2 in27t . 
ay [ sinh LE yn) de 
Jo cosh 2at — cos 2art 
and also 
oO . 2 aa on . aaa 
Jt ae | sinh 2 /mpt—sin 2 /apt cos (2mf*) di 
0 
cosh 2.4/aut — cos 2 / mt 
aS ir sinh 27t — sin 27rt 
a 9 cosh 27t — cos Qart 
cos (2ut?) dt. 
§ 7. It may be remarked that these formulz for \/p are also 
of interest on account of their affording summations of the geo- 
metrical series of the second order as definite integrals, viz., we 
have 
| sinh 2 /ra t+sin2J/7at sin (Qa?) dt 
0 cosh 2,/7at—cos2/7at 
be ie sinh 27rt + sin 27t 
9 cosh 27rt —cos 2rt 
4 sin (2at’) dt 
Jat —J/a i” af sinh 2) rat — sin2,./7at 
Ja 0 cosh2./7at—cos2,/7at 
ids 3 sinh 2art — sin Qart 
9 cosh 27t — cos 2art 
cos (2Q7rt’) dt 
cos (2at”) dt. 
§ 8. In the previous paper in Vol. 11. of the Proceedings I 
gave the value of © (x) as calculated by each of the four methods, 
stating that there was a possibility of error. On comparing 
§§ 6 and 7 of that paper with the results of the present calculations 
I have found the following errata. In the two expressions for 
© (a) as definite integrals in § 6 (pp. 65, 66) at should be /at, and 
in the third line of p. 66 the term tia is omitted from the right- 
hand member of the equation. When the corrections are made 
the results in §§ 6 and 7 are in complete accordance with those 
contained in the present paper. 
