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MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. From henc~ewe obtain, using the first of the two integrals and the series 2 2.3 p, 2.3.4 F3 6 6 l+-4p4+4 4.6 2 +&C.= M,(pi-2)zg+ f(p+2) -4.5.6 1.2.3 ir ddzdzcos2O z cos20cos(Pz sin 0cos +40- tan 0) 2J(p _2)s+ (p 2), U~ ~ r Or AZ Co z i and also gj ddzdzsec( S+ -c - z cos (c sin 0)cos ( Z sin 6+30) XC3(-)ZC3( )- The second integral will require in its applications, that we equate possible and impossibleparts, in other respects the results will be analogous to those we havejust obtained. There are one or two other methods of summation which I shall briefly notice. We see at once that 1+F+1 * 1P &c=- Now if (r) be any integer, = 2(- 1) -'d0 log, cos '2rio. r 7 I 2 Hence +2++2 + 23 T +&C.= dj~dOlogcosg,0s-.20 7r Whence ddlo "ji~d2) Whence J= ~~dO 1og:cos 0?-pCos2ecos (fAsin 2 0_ -2 0)=%E- . 2 The integral d0sin 0cos2rD2 - can be employed in the same way. 7r1 Again, jcosn 0cosn d=dd2 21 whence ,J XdKcos 0 sio=7r, 2 Hence using the series 1+1 22+1 3 24+&c.E- 2 2 22 we find ir ir ir 3d~dip cos7 ps 'cos'cos~cos(2G cos(h2 sin (20- .p) co'0 cos p+tan p-2 2 E-(S+)