232 The Astronomer Royal's Experiments on 



where k depends only on the intensity of the current, and where 

 P„ P 3 , P 5 are defined by the equation 



V 1— 2,27 COS V + X z 



If, therefore, X represents the resolved part, perpendicular to 

 the plane of the circle and towards it, of the force exerted by the 

 current on a unit of magnetism placed at Q, and if Y represents 

 the resolved part of that force parallel to the plane of the ciicle 

 and directed from its centre outwards, then 



Y dV ' dV 



A = jt, sin 6 j- cos 6. 



r . ad dr 



v dV n dV . h 



X = j^ cos 6 -f —r sm 6. 



r .do dr 



To calculate these quantities, we know that 



P, = cos^ 



P 2 =|(cos 3 (9--§cos0), 



P 3 = §& (cos 5 0- lg cos 3 6 + J| cos 6) . 



We shall only consider the case of those points for which r is 

 greater than a. Substituting these values in the expression 

 which in such instances holds for U, we have 



U^27r^|-g.^cos6'+^.^ r ^cos 3 (9- , -cos^ 



315 a 6 / ■. 10 a/I 15 A 

 "128T4 C0S ^¥ C0S ^63 C0S ^ 



From which, after some reduction, we obtain 

 ~ = ~l (-l + 3cos*6>)J + 1 • (9-90cos^+105cos^ 



- j|g (-75 + 1575 cos 2 61-4725 cos 4 + 3465 cos 6 0) ^ 

 + • • • • (I) 



+ jig(5S5cos6 l -3150cos 3 6' + 3465cos 5 6')^ 

 ~ }• • (2) 



