172 ELECTROMETRY. 



and there were no extraneous action, the reciprocal action would be 

 the same as if the masses m and m' were respectively concentrated 

 in their centres, and we should have ^ 



mm 



If the balls, being of the same radius, were in contact at the 

 moment of electrification, the charges m and m' are equal, which 

 gives 



(3) < 2 =/^ 2 . 



For equal distances / and /' this expression becomes 



m 2 = 4C/( A + a) sin - tan - . 



2 2 



When the radii are unequal, we may obtain the ratio of the 

 charges m and m' by means of tables calculated by M. Plana.* 



If the balls were at a great distance, the potential V of each of 

 them would simply depend on the charge it possesses ; but this 

 hypothesis is generally inadequate. 



A first approximation is obtained by supposing that the external 

 action of each mass is the same as if it were concentrated at the 

 centre. 



We have thus (177) in the case of two balls of equal radius r, 

 and the distance of which d is represented by cr, 



d+r c+ i 

 / r V v2 



( A \ / V 2 I I = r^i 



\d+r) (^+i) 2 



_w 2 _;;/ 2 i 

 = ~d* = T" 2 ^ ' 



These formulae are insufficient when the minimum distance of 

 the balls does not greatly exceed their diameter, and the tables 

 given by Sir W. Thomson f stop at the value c=4, which corresponds 



* PLANA. Mem. de VAcad. de Turin (2), Vol. vn., p. 71. 1845. See 

 MASCART. Traite de fElectricite Statique, Vol. I., p. 281. 

 t SIR W. THOMSON. Reprint of Papers, pp. 96 and 97. 



