The Rev. Brice Bronwin on some Definite Integrals. 497 



/dx . m cospx _ sinpx (sin.r) OT 1 /». / (sin.r) OT \ 



x n ( sin<r ) cos# — ^?cos.r x n pJ V < r n cos^/ , 



which is zero as long as the element of the integral is finite. 



it 

 Let us examine the points where it is infinite. Make x =— 



+ u, u infinitesimal. Then 



cospx _ cospu _ p cos pu^ 

 cos.r ~ u ~ pu 



/pdu cosp u _ Pd v cos v _ 

 p u J V 



taken from v= — oo to w= oo , the rest of the element remaining 

 constant. It will be the same at every other such point. The 



remainder «£ . therefore = 0. Hence 



w H-i — l^ll)jL_{A( W 2-2-2r- 1 --A 1 (m--^-4) w -- 1 + ...}( E - 



l P(w-l) 



As particular examples, 



/dx n cos2rx „ f*dx . cos2r# 



cos ^ 



/ 



^* • 9,.4.2 COs2r,3:, _ (~~ 1 )* ,r 



-r-(sin*) r + cosjr - 2 2r + 1 P(w-l)' 



(15.) 



jo a fraction greater than and less than 2. We may employ 



the method of summation here. 



2 pdx , . v sin z x . 

 Let \ z _ = / — (sin *)» . Since 



- . sin(z+l).r , sin(z— l):r _ « 2 +l , .*-l 



2s,n2 ' r = -io7F- + cos* ; 2 V« = V« *AM' 

 We find as before, z being odd, and 



_ x -^-l_ x ^+ 1 - A^ +1 -/ -/~ 2 + -V~ Z . 



A m,n —" A m,n ' *m,n ~ u m,n u m,n * w m,n' 



Putting for v z m „, &c. their values from (c), and continuing 

 the series till it terminates, adding and subtracting the addi- 

 tional terms, there results 



C 1 = i !i^{A(s + m)»- 1 -A 1 (*+m-2)»- 1 + ...} 



~ ^P(n-l) {A ' m -^- 2) ''"'- A ' ( '"~ g ~ 4) "'' + - } 

 Here n = m — (2 * — 1 ). As a particular example, n > 1, we 

 Phil. Mag. S. 3. No. \62.Suppl.Vo\,24>. 2K 



(F.) 



