234 The Astronomer Royal's Experiments on 



these expressions, we have the formula? 

 p — r. sin#, 

 8x . sin 0= — r . $6 ; 



.*. bx = r . -y 

 sm 2 # 



and p 



sin 6 



Making these substitutions for 8x and r, the integrals with re- 

 spect to x become integrals with respect to 6, which can be easily 

 evaluated by a continued application of the method of integra- 

 tion by parts, the limits being from 6^=u to 6= ft. If we then 

 integrate the result thus obtained with respect to a, from the 

 limit b to the limit b + c, we finally obtain 



— = ^ ~ { — (cos ft— cos a) + (cos 3 ft— cos 3 a) j- 



t -5 75 



6 + c ~ 6 * _ 9 / cog ^ __ cog x + 3 3 ( CO s 3 /3- cos 3 a N 



— 39(cos 5 /3— cos 5 a) + 1 5 (cos 7 ft— cos 7 a) J- 



b + c -b 7 c 75 (cos ^ -cos a) + 575 (cos 3 ft- cos 3 a) 

 896/ < V y A 



— 1590(cos 5 /3-cos 5 a)-{-2070(cos 7 /3— cos 7 a) 



-1295(cos 9 /3-cos 9 a)H- 315 (cos^-cos 11 *)}- 

 + ......, 



I = b + c ~^ 3 j + (sin 3 /3- sin 3 a) j- 



5 



-fi± c _^A 5 |- 12 (sin 5 ft- sin 5 a) + 15 (sin 7 /3-- sin 7 *)} 



7 7 , 7 



+ b + c ~ 1 { + 120(sin 7 /3- sin 7 a) -420 (sin 9 /3- sin 9 a) 



+ 315 (sin 11 ft- sin 11 «)} 



+ 



These expressions for X and Y will be converging for all points 

 situated at a greater distance than b + c from any point of the 

 axis AB, inasmuch as they are composed by adding together 

 corresponding terms of series which are then all convergent. 

 Among other points, these expressions hold for such as are 

 situated on the axis external to the bobbin, and not nearer A or 

 B than by the distance (b + c). For such points, however, the 



