DEFINITE INTEGRAL OP THE VALUE OF AN EXPRKSSION. 107 



v 



r f f -' .//• + i- f ' J (xiv) 



^" co.s (;■ W — - , 



Let us assume that for any given positive integral value of;/ 

 _ if cos \rfl-(p+ l)j^ } _ "1 cos (rd-p '-) _ t ,„, (, » _'^. ^) 



we will show that this equation will also hold when ;> + 1 is written for p. For if we 

 integrate both sides of (xv) with respect to ;• we get 





pf e *' ^ dH = /• /" f fi ' 



cos I 



I +1 



,, _ ^' cos (r ^ — P i-^- ) . , - ^' cos (r ti -t n) 

 _ / f B „__ '^__ dO + I f f •' '-L^ dH + C 



H ' 



or 



' _ If ros (r H -tà-J n) ,, - ^' ros (r ft -^-^±-^- tt) 



(i'+l)/' f « — ^ '''^='- ./ ' '' 



ft ft 



dft 



,, -- cos (r ft — P tt) , -x 



+ i f i ^ - dft + C ^^"^ 



"^ ft" 



This equation is the same as (xv), with p + 1 written for p, except that it contains 

 the constant C. If, then, C = equation (xv) will hold with p + 1, written for p. 



In (xvi) put r = 0. Then, if p is odd, all the terms vanish except C, and the equa- 

 tion reduces to 



C= 



«- _ft 



If /> is even, let it equal 2q, and we shall have 



_ ^' 



(-1)"-^^ (2.y + 1) /'' ^ + ,dft=i-l}:h/ ^-^' • '^^ + 

 H ft 



or by (xii) 



^* ' ■ 1.3...(2ry + 1) ' n^ 2 " '*^ ' HH. .(2 ry — 1) v^ 2 "^ 



