1872.] Mr. J. Stuart on Galvanomagnetic Attraction. 



69 



where L and M stand for the expressions on the right-hand side of (1) 

 and (2) respectively, and where ix depends on the strength of the current. 



To perform the integrations for the length of the bobbin in these 

 expressions, we have the formulae 



p=r . sin 0, 



&t? . sin 0= —r . dd : 



sin 2 

 and 



sm 



Making these substitutions for dx and r, the integrals with respect to ce 

 become integrals with respect to 0, which can be easily evaluated by a 

 continued application of the method of integration by parts, the limits 

 being from 0=a to 0=/3. If we then integrate the result thus obtained 

 with respect to a, from the limit b to the limit 6 + we finally obtain 



X b+ C— ¥ e , n V . / , /i * M 



\ — (cos p — cos a) + (cos 3 /3 — cos 3 a) } 



6p 2 



-A- 



39(cos 5 /3— cos 5 «)+ 15(cos 7 fl— cos 7 a)} 



7 , 5 75 



+ on ~ I ~ 9 ( cos P ~ cos a ) + 33(cos 3 3 - cos 3 a) 



b + c —b 1 



+ Qr . Q 6 — {— 7o(cos /3 — cos a) + 575(cos 3 /3— cos 3 a) 



+ 



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

 -1295(cos 9 /3- cos 9 a)+ 315(cos n /3- cos 11 a)} 



-= _^_{-S-(sin 3 /3-sin 3 «)} 



+ b + * : ~ ¥ { _12(sin 5 /3- sins «) + 15(sin 7 /3- sin 7 a)} 



+ — triS e 67 { + 120(sin 7 /3- sin 7 a)-420(sin 9 /3- sin 9 a) 

 8 9 op 



+ 315(sin u /3-sin u a)} 



+ 



These expressions for X and T will be converging for all points situated 

 at a greater distance than b + c from any point of the axis A B, 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 (6 + c). Eor such points, however, 



the expressions become illusory, assuming the form jjj. They may, how- 



