Electrostatic and Magnetic Fields. 475 



centre on the tangent at E is p, the area is ^pdsd/t/x, and so 

 the charge is ^crpdsdK/K. 



Now through P let an ellipse confocal with the given 

 ellipse LAK be described. A point A on the latter curve 

 corresponds to P on the confocal, and a point E' on the 

 confocal corresponds to E on the given ellipse. Join E to 

 P and A to E'. These lines have the same length, r. Let 

 p be the length of the perpendicular from the centre on 

 the tangent at E', and denote the angle between the line 

 AE' and that perpendicular. Let also p ', Q be the corre- 

 sponding quantities for the point P and the line EP. [Care 

 of course is to be taken that the lines are reckoned in the 

 directions indicated by the letters, and that the perpen- 

 diculars are regarded as drawn both inward or both out- 

 ward, so that there is no ambiguity as to the signs of the 

 cosines.] In a former paper (Phil. Mag. April 1907) I have 

 proved for two confocal ellipsoids the geometrical theorem 

 (not, apparently, previously known) 



p sec0=p o ! sec # , (4} 



where p\ 6 and yV, O refer to pairs of corresponding points 

 on the ellipsoids, and have applied it to the complete and 

 instantaneous evaluation of the integral for the force pro- 

 duced at an internal or external point by an elliptic homoeoid, 

 and hence to the solution of the problem of the attraction of 

 a solid ellipsoid of uniform or of varying density. 



The geometrical theorem as stated above asserts that the 

 product p sec 6 is invariant over the confocal ellipsoid : 

 exactly the same theorem holds of course for the given 

 ellipsoid. Moreover, since an ellipsoid is one of its own 

 confocal?, the theorem holds also for any two points P, Q on 

 a given ellipsoid, the perpendiculars from the centre on the 

 tangent planes at P, Q, and the chord joining these points. 

 The theorem holds also for any confocal surfaces of the 

 second degree. 



In the present case let us take on the confocal ellipse 

 through P ? the element of arc ds ', the points on which 

 correspond to the points on ds ; then since the equation of 

 the confocal is 



:'^ + t ^-=k, (5> 



ar + u tr-t-u 

 we have, as can easily be proved, 



pds= rr-z —r—, ip' ds' '. . . . (6) 



