Page:Philosophical Transactions - Volume 145.djvu/179

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160 MR. W. H. L . RUSSELL ON THE THEORY OF DEFINITE INT1EGRALS. Als f+ 8 3 +7 1081 6+&C. Also 7f 3 37 10.it2 3 - eX~ ~ ~ ~ ~ - 40 40x _ 3. . (III.) =5p~~pJX240 2COS. Ih -5 2x <(3e-2)3-3/ -)- (pi 2o s VS k + (px..j2)as2 SI -,- ( 2XI Whencew f jn dd COSG3COSOS(+ cos3 COSW 22 cO4Ji~L*cosd0cosp~sin (d+;)+ +~!- (tan O+tan; 3v3 V 33V V34/h~) whence 2C C, (3 Ipj2 2Z)s-l -3 _p2 si 2S --F Again, let the symboli ad equation be (D- 1)(D-2)(D-52u D-)suu- - 0- and let the transformnedequationbe (D - I) (D -2)v-Z2g2oV = (D-.1 )(D- 2)V, whence, u=(D- 3)v, 0=(D-3)V. Hence we find V-Cx' and V QX+ C2Xx~+C1X6s', whence u=- -2C~x+ Cpi2 22) )si+C,(-px2-2x)s-," we determineC1,C2,C, according to the series we haveto sum. Hence we find 2__ 2.3 2-v) _ 24x 6 6 xI+ + &C. "' (i'2x t 5~ ~22 +5 24 IZ/4 _ 1) ,4 (s/x'+ (IV)?2 2 2 22 By a similar,method we find- _ _ _ _ _ _ _ _ _ 120430( 2+ (+),V) x4{+ 2 347 12312s 2 22 JT&- WPj-_ 3xsv