Conducting Masses charged with Electricity. 185 



then 



and the potential energy of the system reckoned from the 

 equilibrium position is 



f=- Q 2 



^"-^IK 



If F„ a cos(j;>£ + e), we have for the motion under the 

 operation of both set of forces, 



8 



In actual liquids this instability, indicated by the negative 

 value of P', is opposed by stability due to the capillary force. 

 If T be the cohesive tension, the potential energy of cohesion 

 is given by 



P = ^TtCn-l)(n + 2)|jF^cr*. 



(pt + e), we have for the motii 

 )th set of forces, 



p -fcl)/(. + 2)T-_L% 2 }. 

 pal t 4tt a\ J 



IfT> . 3 , the spherical form is stable for all displace- 

 ments. When Q is great, the spherical form is unstable for 

 all values of n below a certain limit, the maximum instability 

 corresponding to a great, but still finite, value of n. Under 

 these circumstances the liquid is thrown out in fine jets, whose 

 fineness, however, has a limit. 



The case of a cylinder, subject to displacement in two dimen- 

 sions only, may be treated in like manner. 



The equation of the contour being in Fourier's series 



r = a(l + F 1 +...+F„ + ..), 

 we find as the expression for the potential energy of unit 

 length 



F=-| S(M _1)» 



Q being the quantity of electricity resident on length /. 

 The potential energy due to capillarity is 



P=i7mT 2(n 2 -l) &L, 



and for the vibration of type n under the operation of both 

 * See Proc. Roy. Soc. May 15, 1879. 



