1G4: Wisconsin Acadenvj of Sciences, Aiis, and Letters. 



IS'ow by the definition of the Potential, it must be 

 (designating it by V) = T f ^ _£^^^_. 



i. e., the sum of all the quotients arising by dividing each 

 element of the entire sphere by its distance (x') from the 

 attracted point. 



Effecting this summation, we have 



When p is without the sphere, V = f '—^ — . 



« " within " " Y'=2 7rpr'-2Trp.x'. 



ISTow it has been pointed out that when the Potential Y 



has been found in any case, to get the attractions, we have 



only to diflerentiate Y once ; hence, when p is without, the 



.. .• 1 ^^ ddupr') . r^ ,, 



attraction along x = -.r— = — ^— i — ^ = 4-^0 q^ ^he 

 ° dx dx -^ X 



attraction is directhj as the maci of the sphere, and in- 



versehj as the square of the distance from the center ; the 



known law. 



When p is within the sphere, the attraction along x = 



dY d (2 rrp r^ - I Trp . x^) . , , " , , 



-V- = — ^^ — T '- = 2 Tip r' + 4 TTp . X, the value 



dx dx > d I ; 



of which will depend only on x (the distance from the 

 center), since the first term is constant. The attraction 

 within the sphere will therefore be directly as the distance 

 from the center ; which is the knovjn law in this case also. 

 These simple examples, it is hoped, will sufiiciently 

 illustrate the nature, mode of application, and usefulness 

 of the Potential Function when once an expression for 

 that function has been found. And we have carefully in- 

 dicated where full expositions can be found of the mode 

 of calculating this most important function in all the 

 cases that are likely to occur in the solution of physical 

 problems. 



