TRANSACTIONS OV THE SECTIONS. 



15 



On a Series Summation leadinr/ to an Expression for the Tlicta I< unction as a 

 Definite Integral. By J. W. L. Glaishee, M.A., F.B.S. 



By means of the formula giving the resolution of l+x" into its linear factors, 

 and of the equation 



sin (a-x) sin (a+x) ^ f ,_ ^ 1 f n ___?!_ 1 J i ^_ \ 



sin'a 1 «•-' J 1 (a-Tvy J \ {a+n)' J 



it can be shown that 



^+^'1 ii+(;^35)^'J {^"^(VH^^P^j r+c^^^Wj I {^2hyny 



= 2~" (cosec y ) '" P, P3 Pj . . . . P„_i if n be even , 

 and =2~'Yco8ec y)'" (cosh — -cos ^jPj P3 P, . . . . P„-2 if «be uneven, 



1%= ^ cosh I — sin — 1— cos — a— a- cos— 1 J. 

 1 \ b 2n} b\ 2nl J 



X ^ cosh ( sin-—\- cos-- «+^cos— - I; 



\ \ b 2n) b\ 2nl J 



and cosh .r, sinh x are written as usual for the hyperbolic cosine and sine of ,r, viz. 

 cosh x=h((f-\-e~''), sinh z = h{e'' — e"'). 



where 



Taking the logarithm and differentiating, we have 



1 1 



. =..-if 1 



1 



^.=''_l_(a_6)--^'''^ x''"'+(a+6f' a-'"+(«-26,)^ 

 .r 4-(«+26) J 



and =""r " 

 o 



. , 27r,r 

 smh— j- 

 



, 27r,r iJ:rr« 



cosh — ; cos -=- 



6 b 



;j— , +Q1 + Q3 • • • • +Qu-2 > if « be 



uneven ; 



where 



Q,r = 



cosh (-J^sin^^J -cos -^(«-.r cosg;^) 

 . rjT . , I2itX . 7-tt\ rtr . 27r/ j-ttn 



cosh (_8m2^)-cos-y(«+xco8 2^j 

 Using the integral, 



o r*.r"-'sin(cV), 



