OF THE INDEPENDENT VARIABLE, 



185 



If we employ equations (1) of Art. 206, we must put 

 the relations between x, y, and z, in the form 



4.1 f dr x 



therefore -7- =--77-5 : 



dx A/iar+V 



x 



. a , 



= - = Sin U COS 0, 



dr y 



-j- = - = sin sm <p, 



dy r 



dr z a 



-j- = - = cos V t 

 dz r 



d6 z 



dx 

 dO 



x _ cos 6 cos <f> 



~~~ 



y _ cos 6 sin 

 ~ 



dz 



d^ 

 dx 



sn <> 



r sin 6 ' 



X 



COS 



dy ~ x 2 + y 2 ~ r sin ' 



therefore 



^ = smecos(j>~ + 



cos ^ cos <A 



sm 



sin 



du n . , du cos 8 sin <> du cos d> du 



-y- = sm 0sm<f>^-+ - -72 H = fl-jr 



dy ^ dr r do r sin 6 d<p 



du _ fidu sin d du 

 dz dr r dd 



which will be found consistent with (1). 



(2), 



