OF ARTS AND SCIENCES. 355 



Expanding the numerator of equation [XVII.] by Taylor's theo- 

 rem, integrating between the limits J^, — |^, Jjy, — ^rj, and dividing 

 by the denominator, we obtain a value for G of the form 



^ = ^ + 2l[^d^ + '' df)+m)[^ rf^+'' ^4j+&c.[xx.] 



This is the general expression for the values of G. To find G^, G.^, 

 &c., we substitute the values of Q^, Q^, &c., respectively, taken from 

 equation [XVI. A]. Thus from a comparison of [XVI. A] and 

 [XVI. B] we see that Q^^ = cos a; from Figure 5, we see that 



_J2 



d 



(P' / X \ 2 xy"^ — x^ 



dt Vv^M^J "^ (^'"+lm " 



Substituting in equation [XX.], we find 



^0 — i r+24 z^ ^ 8 .*'' \ ' 

 or, since (? = 2 tt (7, 



G^/, Cj', &c. are found in the same way. 



[XXI. A.] 



Stt/x f 1 /2 25 35?A 5 /4x^-3y2x | 



+ 24S (3/2^(5/- 44.r2)+ 63xy (4x2-y2) j. 



