1877.] theta functions as definite integrals. 65 



being as just defined, and 



Similarly, using the integral 



r°° t sin ct -. . „„ ,^, 



-2 — -r, cZi = i7re-«<' (5), 



Jo a' + i' ^ ^ ^' 



we find 



4 + 4 , / — -Ti + 4 , . , — vi + &c. 



a;^+ a* a;* + (a - tt)* a;* + (a + tt) 



and %{x)^sJ^-.[ [f{at,u)+f{at,-u)]^m{t')dt 



where /-(-^ ^^ _ «^^^ ^^/2 + ^^^ (^v^^ + ^^) 



and a, u are as before. These are the most simple and interesting 

 of the expressions for © {x) as definite integrals. 



The paper also contains the value of the series 



as a definite integral, obtained by means of (4) and (5). The re- 

 sults are stated in the British Association R-eport, Glasgow, 1876, 

 Transactions of Sections, pp. 15, 16. The integrals (4) and (5) were 

 used to sum the series l±q + q*±(f-\- &c. in my paper. On the 

 summation hy definite integrals of geometrical series of the second 

 and higher orders, (Quart. Math. Journ. t. xi. p. 328 — 343, 1871), 

 but I did not there obtain the value of @ {x). 



By putting a; = 0, and = — ^ in these formulse, we find 



l-2q + 2q' + 2q'-&o.^^—- 





sinh ^t + sin ^t . ^. , 

 cosh/3i+jcos/3i ^ ^ 



smh Bt— sin. ^t . ,,„, , 



- — ^-^ 7- sm (f)dt, 



cosh /Si + cos /Si ^ ' 



