TO THE THEORY OF MAGNETISM. 89 



this integral may be written 



/ A d- d-\ 



[ , , { dE r dE r dE r) 



dxdydz \ -j . -7- + -y- . + r- . -7- / , 

 J \ ax ax ay ay dz dz / 



which since SE= 0, and S - = 0, reduces itself by what is proved 



in Art. 3, to 



[da- fdE\ ., 7 , N [da- dE 



I -= } = (because dw = da) 7- ; 



j r \dwj ' ] r da 



the integral extending over the whole surface of the sphere dv, 

 of which da is an element ; r being the distance p', da, and 

 dw measured from the surface towards the interior of dv. Now 



I - -j- expresses the value of the potential function for a point 

 p, within the sphere, supposing its surface everywhere covered 

 with electricity whose density is -y- , and may very easily be 



\JLCb 



obtained by No. 13, Liv. 3, Mec. Celeste. In fact, using for a 

 moment the notation there employed, supposing the origin of the 

 polar co-ordinates at the centre of the sphere, we have 



E = E t 4- a (X t cos 6 + Y t sin cos vr + Z t sin 6 sin t*r) ; 

 E t being the value of E at the centre of the sphere. Hence 



- = X. cos + Y sin & cos isr + Z. sin sin -or, 

 da 



and as this is of the form U w (Vide Mec. Celeste, Liv. 3), we 

 immediately obtain 



7- = 47T/ \X cos & + Y sin 0' cos ' + Z sin & sin '}, 



r da 



where /, &, ' are the polar co-ordinates of p'. Or by restoring 

 x'j y' and z 



[da dE ( v / > \ , \r f > \ . rr r > M 



I j~ J 77 " l^-/ v* ~~ X /J T Jt , (y y,) + ^ v^ ~^/)l 



