240 MR. HARRIS ON SOME ELEMENTARY LAWS OF ELECTRICITY. 



position of this point within the surface, will depend on the distance between the nearest 



points of the spheres, and may be readily found by the expression z = (^ + ^^n —^ ^ 



a being- the distance between the nearest points of attraction, and r the radius. 



The points q q^ being thus determined for given distances between the spheres, the 

 whole force should vary between them, according to the general law (67.), and also 



as / 4. £ rV ^^^* ^^^ inversely as the distance between the nearest points, multiplied 



into the distance between the centres, as shown in sec. (TS.)*". 



72. These deductions accord very completely with experiment, so nearly, indeed. 



* Let CAD E B F be two hemispheres, attracting each other at distance A B. 

 Let AB = a, AM = a?, PM^y, AP = s, r = rad., and ir = 314159. 

 p. = absolute force exerted by a unit of force at a unit of distance. 

 P^ = Mm = (a + 2j?), 



Then unit of force at distance P » = ^ 



(a + 2 xY 



Now circumference whose radius = P M is = 2 tt y. 



And annulus, whose breadth = (is, = 27ryc?s; 



.*. Force exerted by an indefinitely small annulus at P on a coiTesponding annulus 



at;, = 2.y^.X^^-i^. 



But in circle y ds = r dx; 



.'. Force of annulus P on annulus p 

 Now 



TT fl 



r dx 



f 



(a + 2 xy 

 1 



l-K ixr dx 



C = — TT u r 



(a + 2 j?)2 ^ a + 2 J? 



, the corrected sum of which = total force from A to P. 

 + C. 



If X = 0, we have 



■K ikV — + C = 0, and C = tt /i r — 



a + 2x 



.'. — ir u r h C = TT u r I — Vwhenj:' = 0. 



'^ a + 2x \a a + 2x/ 



When X =i r, this expression becomes 2ir fx 



ri 



= the force upon the whole hemisphere. 



a (a + 2 r) 



Now area of hemisphere = 2 ir r^, and if q j" be the points in which we may suppose the whole force of 

 each hemisphere to be concentrated, we have, putting A g' = B 5 = z, 



2 7r/nr^ _ 2 7rjurg 1 _ 1 



(a + 2zy~ a{a + 2 r)' °^ (a + 2 s> "" « (a + 2r)' 



that is, (a + 2zY =ia{a+ 2 r). 



.-. a + 2 z = >/a (a 4- 2 r) = (a2 -I- 2 a r)*, 



(«2 + 2 a r)i — a 

 and z = 5 



When both hemispheres are equal, as we have supposed, and the distances variable, the attractive forces will 

 1 



vary as 



a (a + 2 r)' 



