MR. W . H. li. RUSSEIAL ON THE THEORY OF DEFINITE INTEGRALS.

161 whence we lave I7r 2 2 dadvv(1 -v)- cos 0 RECOS0'cos (pv sin 0 cos 0+30-tan 0) 2 t (2 -3) ,4~p3s 24r+7r- It is to be particularly remarked, that. we may in many cases simplify the final results, which we obtain by means of these summations, by the use of the theorem 123n-1't1 nnn n Again, let (D-- 1)(D- i)(D-5)u-p(D-2)(D-4)sOu=O, and assume as the transformed equation (D -1) v- psv= 0. Then u=(D-2)(D-4)v 0= (D-2)(D-4)V, whence V=Ax2+Bx4, and v=CX(_x2)- )Sy+C2'X+C3'X whence u =CC(22x'-3pX22+3 +st+C2X'+ CX,x where the constants must be determined by comparison of this expression with the series to be summed. Thus we have 1'2*4 .3.4.5 p~x 8(p2X 24x 43 + 35p +3*.4 .5 .61.2+ 4X (V4.... . (VI.) 3614V ~ ~ ~ 3 Hence jivzS vzdvdz h(j2-3 +3)- Moreover we shall find 6f +2.5 2.3.5 .6 P2x21 30 o 0X 3x3 120 4.6 w4.5 .6 .7 1.2 +& ct5(p 4eX +4x)+ 4+ whence Vz4(1-v) sdvdz (p-2)'+ 1+) We shall also find X4{1+4P 23+45 p2x2 234 i 3+ 3 (VIII.) Hence Cv(I -v)_11vdv= W(p-2)s+ (p+2). These three last integrals can be obtained by ordinary integration. I have intro- duced them here partly for the sake of system, and partly because we shall require the series which they represent on other occasions.