﻿a Terminated Straight Line. 359 



Hence, considering a prolate spheroid of revolution, c=b, the 

 attractions of the shell (a,b,b;e) will be proportional to those 

 of the shell ( VlF^h, \/¥-h, VW-h',e); or if, as usual, 

 b 2 = a 2 (l — e 2 ), then, if A increases and becomes ultimately equal 

 to b 2 , to those of the shell (ae, 0, 0; e) ; viz. this last is the por- 

 tion of the axis of x included between the limits x=—ae 3 

 x-=.-\-ae\ or say it is the terminated line x = ±ae; and I say 

 that the mass is distributed over this line uniformly. 



To see that this is so, observe in general that, in the spheroid 

 x 2 yZj±. 2 i 

 ~n ^ JM — = 1> ^ ne v °lum e included between the planes % = ct, 



/ b' 2 \ 

 x = *+ du, is = (y 2 + z <2 )du, =7r(6' 2 j^aP\dw, and thence, 



writing «'(l + e), fl'(l + e) for a\ b' } in the shell («', b\ b 1 ; e) 

 the volume included between the planes x = u, x = u + du is 

 = 7rb h2 .2e f dot; viz. this is independent of a, and simply propor- 

 tional to da. Hence, writing Z>' = 0, when the shell shrinks up 

 into a line, the mass must be distributed uniformly over the line. 

 It follows that for a line of uniform density the equipotential 

 surfaces are each of them a prolate spheroid of revolution having 

 the extremities of the line for its foci, and that, if we have a 

 shell bounded by any such surface and the consecutive similar 

 surface, with its mass equal to that of the line, then such shell 

 and the line will exert the 

 same attractions upon any 

 point P exterior to the 

 shell. The attractions of 

 the line are obtained most 

 easily by means of its po- 

 tential ; viz. taking S, H 

 for the extremities of the 

 line, and, as above, the s 6 h M 



origin at the middle point, and the axis of x in the direction of 

 the line, and writing 2ae for the length of the line, x, y, z for 

 the co ordinates of P, and r, s fo r the values of H P , S P (that is, 

 r= ^(x-ae) 2 + y 2 + z 2 , <?= \/ (x + aef + y 2 + z 2 ) , then the po- 

 tential is at once found to be 



-j T , x + ae + s . 



v = log — ; 



° x—ae-tr 

 and we can hereby verify that the equipotential surface is in 

 fact a spheroid of revolution having the foci S, H ; for, taking the 

 equation of such a spheroid to be 



** , y*+z* _., 



a 2 '*' a\l-e 2 )~~ ' 



2B2 



