472 On certain Definite Integrals, [Jan. 25, 



tude of what are usually called binomial integrals. It will be seen 

 that these formulae include a vast number of such integrals as 



f dx - x - . . (239). f d *'*\ . . (240). 



f dx . a 2 * 



J s/a + lx* + cx 4 < + . . . + ex^^l-x* ' ^ ; * 



f dx . x 2n 



J^ / a^-bx i + cx i + . .. + ea; 2 ~*</l— a?' 2 



If we have an integral of the form 



where P and Q are functions of 0, we have by applying the symbol 



0— to the integral 



die 



/£?00(P) 6 p *Q=0(a)a*& (243). 



As an example of this 



f^0{0(cos 3 <9 e 3 '"V cos ^^ (244), 

 Jo 



from whence 



Ji cftte* 6088 e C0S3 e cos cos3 ® s * n ^ cos3 cos cosB ^ s * n 3#) 

 o ( 1 + 2/t cos 3 6> cos 30 cos 6 # 



7 ef . . . (245). 



In the same way we may 'deduce general formulae from integrals 

 (86), (87), (116), (118), (122), (129), (130), of the present series. 



The following integrals were obtained by the use of reciprocal 

 functions : — 



^(Zflcos^^cos (asin0+r0)= . . . . (246), 



FiPtostf*. CQS ^-"5» s (r-l)*^ (27) 

 Jo l-2*cos0 + a 2 2*- V ; 



r* ^ tt(i+^-) 



J o ( 1 - 2x cos r6 + a 2 ) (1 - 2/3 cos s0 + /3 2 ) (1 - a 2 ) (1 - /3 2 ) (1 - 



(248), 



where r and 5 are prime numbers. 



j n d0{e"* ei (p (cos 0e«) + 6«~ 5 '0(cos e -*')} = 2tt0(1) . . (249). 



