602 Prof. W. M. Hicks on the 



As this equation is independent of c, the shape of the trans- 

 lated mass remains unaltered, and its area simply varies as 

 a 2 , so long as the filaments are not so close that the shape of 

 the cross sections deviate considerably from circles. So long- 

 as they can be treated as such, the shape remains the same 

 however the energy alters. It may therefore be drawn once 

 for all. For this purpose the equation may be written in the 

 form suitable for logarithmic calculation with tables of 

 hyperbolic functions * 



y =-v/{ (coth #/4 — x) {x — tanh #/4) } . 



This cuts the plane of the filaments at a distance given by 



coth ci'/4 = x, 



x . x + 1 

 ° r 9 ==l0ge t ^l^ 



whence #=2*087288 =aa. 



It also cuts the axis of y &ty = / \/ > 6 . =/3«. 



The calculation is easily carried out by the aid of the 

 Smithsonian tables. The curve when drawn is seen to be 

 very close to a true ellipse. Indeed, if an ellipse be graphi- 

 cally constructed with the same axes it is difficult to dis- 

 tinguish any difference. For #=1, near which we should 

 expect a maximum deviation, the ordinates of the two are 

 1*5203 for the ellipse and 1*5257 for the boundary, corre- 

 sponding to a distance between the two curves of *0035a. 

 For practical purposes Ave may therefore take the area to be 

 it x 2*0872 x ^3a' 2 = 3'6 lira 2 , or that of a circle of radius 

 l-901a. 



The energies are 



El = 8^' 

 »i-£+2x^Wg(l + ^) f 



E 2 = 2x iAt%2 + i (mass of region 2) U 2 



* E.g. "Hyperbolic Functions." Smithsonian Mathematical Tables. 



