66 Mr Glaisher, On the theta function. [May 7, 



and 



l + 2q + 2q* + 2q' + &c. = 



= 2 A / - , V), — ^, cos {e)dt 



Y TTJo cosh /3t - cos l3t ^ ^ 



= 2 A/ - — 1 -^, ^. sm (e' dt, 



V TT J cosh /3i — cos /3i ^ ' 



where y8 = v'2 -^^ = ^2 log f - j . 



These results may be compared with those given in the 

 Quarterly Jouvjial. The latter were however obtained by the 

 direct method explained in the next section, and the formulae are 

 not identical. 



§ 7. We may a.lso apply the integrals (4) and (5) to the direct 

 summation of the series in (1), and it is thus found that 



-! + ©(.») = — cos f — r-^-j^^ KW. ^i 



^ ^ IT Jo \ TT J cosh 2 Rt — COS 2 Rt 



„ , , 4/f r- . 2KK'\ N 



@(x)= -^ sm f — , ^„ -rj^ dt, 



vr Jo V TT /cosh 2Ai - cos 2/ir« ' 



where 



M= sinh (A'- x) t cos {K + x)t + sinh (7^+ x) t cos {K- x) t 

 — cosh [K— x) t sin (K -\- x)t — cosh [K — x) t sin {K-\- x) t, 



and iy= a like expression, the only difference being that the signs 

 of all four terms are positive. Here x is supposed to lie between 

 A" and —K. 



§ 8. It will be noticed that none of the expressions for © (x) 

 put in evidence any of the properties of the function, and it does 

 not appear that it would be easy to deduce any theorems from the 

 integral expressions : but it is interesting to compare the different 

 integrals obtained by the four distinct methods. 



I should mention that as nearly all the results stated in this 

 abstract were obtained by rather lengthy analytical processes, there 

 is a possibility of error. All the work was throughout performed 

 twice independently, but at a very short interval of time, and 

 before the memoir is finally presented to the Society I intend to 

 recalculate de novo all the expressions. 



A commimication was also presented, by Mr J. W. Wareen, 

 On Curvilinear and Normal Coordinates (4th Exercise). This 

 will appear in the Transactions of the Society, and does not admit 

 of being given in abstract with advantage. 



