112 ON MAGNETISM. 



dr . 



= sin u . sin X, 



dr 



-r = COS U. 



dz 



And thus 



,. dV dVl , dVl sinX dV . 



X= 7 -=+, -- eosw.cosX PT-' -- I- -7-. sum cos X, 

 ax du r aX r smu dr 



. dV dV 1 dVl cosX dV . 



Y=-j- =H ^-.-cosw.sinXH r .-. -: -- h -j- .smit.smX, 

 ay du r d\ r sin?* dr 



dV dV 1 , dV 



Z^- ^ .-smw + -r- .COSM. 

 2 aw r dr 



Now W= JTsinX Fcos X. Also the force in the 

 direction of radius of the .parallel passing through the 

 point of observation = J5fcosX+ Fsin X; from which, 

 combined with the force Z, 



N= Zsm u (JTcos X + Fsin X) cos w, 

 C Zcos u + ( X cos X + i^sin X) sin u. 

 Substituting the values of X, Y, Z t 



- 



JLf ' 7 ? 



r du 

 vr l dV 



W = -- : - .-7T-, 



r sin w X 



rfF 

 ~*- ; 



the same as the values found before. 



Every thing now depends on the function V: and 



