OF A CURVE AND ITS APPLICATION. 155 



cos <h A <b 



.: .—II - —r=±= (Bd> + Cd> 3 + J>0 5 + &c.) 



VV- s s/m'-cj) 2 \/»» ! -f 



+ s/rrf-q? (B + 3 C<p + 5 D(p* + &c.) 



Hence cos = A + m 5 5 - (B<p* + C<p*+ Ddf + &c.) 



+ 3 Crf(/> ! + 5Z>m 2 4 + 7Em*(f) 6 + &c. 



- (50* + SC0 4 + 5Z>0 6 + &c.) 



2 d>* 6 



But cos = 1 - -2— + J- £■ . 



r 1.2 1.2.3.4 1.2.3.4.5.6 



Hence equating coefficients; 

 A + m*B=l, 2J5-3Cm 2 = , 4C-5Z>m 1! = , 6D-7Em ! = 



1.2 1.2.3.4 1.2.3.4.5.6 



1 lEm* 



Hence D = — - + — -— : 



1.2.3.4.5.6.6 6 



1 5Dm? 1 m» 5.7. Em* 



"~ 1.2.3.4.4 4 1.2.3.4.4 1. 2.3.4 s . 6 s 4.6™ 



1 3Cm 2 _ 1 »re 2 m 4 3. 5. 7 Em? 



" 1 .2.2 2 " 1.8.. « ~ 1 . 2 8 . 4 2 + 1.2*. 4 s . 6* 2.4.6 = 



to 2 m 4 m 6 



Taking the integral from = to (p = m, we have it = J. 



51. In the same manner 



r sin (bdd> w, . 



J / g _ ^ 2 " a sin-1 ~ + V «* 2 - 2 (6 + c0 2 + d0 4 + e0 6 + &c.) 



sin a (h 



—/ —— - / - / (6 + c0 2 +d!0 4 +&c.) 



\/V-0 2 v/m 2 -0 2 ^/rri'-d? 



+ \/»» 2 - 2 (2c0 + 4d0 3 + 6e0 5 + &c.) 

 Hence sin ■» a - (60 + c0 3 + d0 5 + &c.) 



+ 2cm* + 4dra 2 3 +6em 2 5 + &c. 

 - (2c0 3 + 4d0 5 + &c.) 



But sin = - — + 2. &c. 



r ^ 1.2.3 1.2.3.4.5 



Hence equating coefficients 



a = 0; 6 - 2cm 2 = 1 ; 3c-4d.wj 2 = ; 5d - 6em s «= ; &c. 



1.2.3 1 .2.3.4.5 



1 6em" 



Hence a = + 



1.2. 3.4.5.5 5 



20—2 



