20 EESEARCHES ON H. 



Hence from (9 § preced.) and (12), 



hhn J?{1 + cos.gY — ^ {I — cos. a)^ da 



X = 



r= 



z = 



4(2ae)^(^^)i A=^ A' 







*27t 



"Mian sin. a J. (1 + cos. a) + /(t (1 — cos. a) da 

 4 (2 aef ^V " ^ " ^' 







'■It 



h [{ab — hm) J. (1 + cos. a) + {ah + 67?i) (ti (1 — cos. d)] 



r(2^^1i70§ ■ ^ 







[J. (1 + COS. g) + (i[l — cos. cr)] d a 



Here also we find that Y vanishes at the limits. The values of X and Z, after 

 beino- developed, consist of two parts ; the one multiplied by cos. g, and integrable 

 by arcs of the circle and logarithms ; the other wholly dependent upon elliptical 

 functions. The £rst vanishes at the limits, and the second remains alone. The 

 values are as follow : 



X = 



'27t 



hln /{A^ — ^^){1 + COS? a) 2 [A^ + ^^) cos. a\ 



4(2ae)*(^fi)S 



( {A^ — l^^){l + cos:^a ) ^Z{A^ + ^^)cos.a \^ 







'27t 



_ Tc ( {fA^ + .f ^') (1 + COS.'' a) + Aii{f + f) dn.^ a 



^= I 4(2aef (J.^)iV A^ 



in the last of which we have made 



f^(ab — hn), f' = {ab + In), f + f == 2 ab. 

 Integrating between the limits ff =: 0, (T = 2 7t, or what is the same, taking the 

 complete functions as we have done already, we shall have 



ix— ^l!L(^z^) ("1 ^1 + 1 {F' — E')\ 

 ) (2aef(^,u)iW <? ^ 



^Oi C ' Oi c 



(14) 



\z. 



(2ae)^(^f.)r^-' •' ^'W <? ' h^(? 



which are the two components in the case of the imaginary roots of equation (2), 

 which will be an obvious case when the diiference of the axes of the ellipse is 

 great, and the length of the needle less than the transverse axis. 



