14 Derivation of one of the Equations of Laplace. 



C / ^\ r*''* ^" sin^ * + cos'2 dr. cos- ^ , cosV + co^tf. sinV"! 

 \Try\-r^ -r -T -r J' 



.^x p2sm.,W _co^ 



COS ^tt) + -——; . (cos* TT — 2 sin^S. sin^w) 1: 



' r- 3111 ^ ^ ' J ' 



(rfV \ cos i 

 ~dr) 'rTii^^' 

 j^ / dV \ p 2 sin v. cos v 2 sin ir. cos ir~j 



+ (^^ , rcos'9 + sin-fi.cos^fl-+ sin^fi.sin^Trl ; 



A — / /(f'^Vx ("sin-' ^ cos- ^. cos'-* , cos- ^. sin'' 9r~i 



\ df- / ' r'i ' 



((/'iV\ p sin'* * cos'* a- ~1 ^ / '^^^\ ' 



(/n'V ' L'''- «in'-^ ''^- s'H'- # J ' ' \d*- / ' rK sin'-* i y 

 . ^/ d-V \ p— sin^ . , cos^.cosir . . 



+ 2(-— --t). .cos Oh .sinfl. coS7^ + 

 cos t . sin * ... -1 



, Sin 9. sm X ; or, 0; 



(rf'-V \ n— sin* . • cos* . , 

 -.—>-- ) . — T—. . Sin 9 . cos TT -\ -.— - . sin 5 X 

 (/>-,cf* / Ijr. sin ^ r. sin S 



sin TT |; or, 0; 



(rf-V \ pcos^. cos* — sin* cos* cos A sin*"! 



dS.d'x ) ' L r ~ ' "rTsinT ''' r.sin^+ r J* 



or, O ; 

 Consequently,eqiiation (A), after multiplying by rS becomes 



— :~T-r ; which agrees with Laplace ; since the first two 

 terms are = r . (:^^) ; For ('—) is = V + r. (^) ; 



and, — : IS = •■— ; = ( — )+ — 



' dr t/' \dr / ' dr 



dC fL\ 

 ydW\,/dV\dr, \ d, / „ / ''V \ , / d"-V\ 



III. Dc- 



