266 Mr. Nicholson on Legendre's Functions and the 



Whittaker, l Modern Analysis/ Chap, vii., or a similar 

 work. 



The simplest case is the series 



cos ~ — cos — - + cos —■ 



L 'L 'J, 



which we shall not dwell upon. 

 The expansion produced is 



^-q =P„(/*)-P 1 (/i'>+P 2 ( / a)... + (-V'P„Oa) + ... (6) 

 2 cos g 



where 6 is not zero or it, but may have any intermediate 

 value. Therefore, by the integral properties of Legendre's 

 functions, the integral on left being finite at both limits, 





(/0<*A* = (-)»_ 2 



6 v ' 2n + l 



cos. 



or 



J o P„(cos0)sin^=(-)'<^— i- 

 The next case is the result: 



IT 



- = sin z + 1 sin Sz -f 1 sin 5r + 



holding from z = to 7r, both exclusive. 

 Therefore between $ = and 2-7T, 



rin(2» + l)* 



^ o 2n + l 



Hence 



sin(2rc + l)^ 



(7) 



■ - 



1 C T 

 2 .-'^f"/ _i 2 — — — 1 



2/t + l it *" o 2m+1J ^/^(costf — cos<£) 



-Pn(/,) = 2^ f- '_* 



7T 



-a 



n v # 



■9 ,y/cos' | - 



cos- 2 



the excluded values of (j> being outside the range of 



integration . 



