342 



that we can pass to a confocal surface, by changing ^ and B to 

 ri ,, and tQ respectively, where t is a scalar. 

 " Again we have, identically, 



^-^rF=/'>+P3; (4) 



if for conciseness we write 



P, = (^-6l)-iS.(„-6)V; (5) 



P2=Y.(^-e)-^V.(r, + e)p. (6) 



But p, is the perpendicular from the centre a of the ellipsoid 

 on the plane of a circular section, through the extremity of 

 any vector or semidlameter p; and p, may be shewn (by a 

 process similar to that which I used to express Mac Cullagh's 

 mode of generation) to be a radius of that circular section, 

 multiplied by the scalar coefficient S . (,, - 0)- 1 (^ + 0), which 

 is equal to 



03 _ ^2 Trt'-TO' ac 



If, then, from the foot of the perpendicular let fall, as above, 

 on the plane of a circular section, we draw a right line in that 

 plane, which bears to the radius of that section the constant 

 ratio of the rectangle (ac) under the two extreme semiaxes to 

 the square (b'^) of the mean semiaxis of the ellipsoid, the equa- 

 tion (2) expresses that the line so drawn will terminate on a 

 spheric surface, which has its centre at the centre of the ellip- 

 soid, and has its radius =|; this last being the value of the 

 second member of that equation (2). And, in fact, it is not 

 difficult to prove geometrically that this construction conducts 

 to this spheric locus, namely, to the sphere concentric with 

 the ellipsoid, which touches at once the four umbilicar tangent 

 planes." 



The Rev. Charles Graves communicated the following 



