1882.] certain Definite Integrals. 259 



■a h , a 6 , with (3, «y, B, /m, v, p; I have not observed that these equations 

 lead to an equation of condition between Oj, a 2 , . . . * 



This transformation will apply equally well to all integrals of the 

 form — 



f dx(fi(a Q + a^x + a 2 x 2 + a s x s -\-a 4 x 4 ') .... (222), 

 and f dx^>(a J ra 1 x + a 2 x 2 + a s x s + a ll x 4! +a 5 x 5 + a 6 x 6 ) . . (223). 



If we consider the integrals 



^cle sin (a + b0 + cez+ . . . aO n ) . . . . (224), 



and cos (a + £6> + c6> 2 + . . . /30») .... (225), 



where a. is not infinite, as it is in Fresnel's integrals, we must proceed 

 by expansion. 



Let Q = a + hd . . . e6 n , u=s'm G, v=cos 9, 



du dQ dv dQ 



de~~do' v ' d0~ ~ ~d9 ' ' 



d?u d 2 Q , / dQ\ 



do* de* u de^' 



Proceeding in this manner, and remembering that when 6=0, 

 ; sin0 = sin a, cos 6 = cos a, we are able to develope u and v with great 

 facility, and so obtain the integrals. 



The following method may be very generally applied. Let 



0(. e )=A o +A 1 (^)+A 2 fcY+ . . . 

 \x + a/ \x + a J 



where (a) is an arbitrary quantity. Then we shall have — 

 J )(x + a,y 



=wf * (iW!l!Wtf Y+ ... \ 



(Ao+Jk— W^YW ? = 5 Y+ ■■■) 



I " l x+a ~\x + a ! \x + a/ J 

 =2a{B .5=2 + afcY + a(5=£) 8 +... } .... (226), 



where B =A 1 , B 1 =A 1 + 2A , B 2 =A 3 + 2A 1 + 3A . . . 



The calculation of B , B x . . . is very easy. Form by addition the 



* Jan. 23. But Mr. Spottiswoode has discovered two conditions, which will, I 

 hope, be inserted in the next paper on the subject. 



