288 
a+Pi= cs, a—'/P = cay ¢ = De; p= FA, Gl) 
so that a is the centre of the ellipsoid, & a variable point on 
its surface, c the fixed centre of an auxiliary sphere, of which 
the surface passes through the fixed point a, and also through 
the auxiliary and variable point p, while B is another fixed 
point, we obtain the equation: 
EA= + U.DA+T.DB; (42) 
which gives 
(ca)! = FU.DA.T. DB, (43) 
and shows, therefore, that the proximity (ua)—' of a variable 
point 8, on the surface of an ellipsoid, to the centre a of that 
ellipsoid, is represented in direction by a variable chord Da 
of a fixed sphere, of which one extremity a is fixed, while 
the magnitude of the same proximity, or the degree of near- 
ness (increasing as © approaches to the centre a, and dimi- 
nishing as it recedes), is represented by the distance DB of 
the other extremity D of the same chord va from another fixed 
point B, which may be supposed to be external to the sphere. 
This use of the word ‘‘ proximity,” which appears to be a 
very convenient one, is borrowed from Sir John Herschel : 
the construction for the ellipsoid is perhaps new, and may be 
also thus enunciated :—From a fixed point a on the surface 
of a sphere, draw a variable chord pa; let D’ be the second 
point of intersection of the spheric surface with the secant DB, 
drawn to the variable extremity p of this chord from a fixed 
external point B; take the radius vector Ea equal in length 
to p’B, and in direction either coincident with, or opposite to, 
the chord pa; the locus of the point £, thus constructed, will 
‘be an ellipsoid, which will pass through the point 8. This 
fixed point B (one of four known points upon the principal 
ellipse) may, perhaps, be fitly called a povx, and the line BE 
a polar chord, of the ellipsoid; and in the construction just 
stated, the two variable points p, p’ may be said to be conju- 
gate guide-points, at the extremities of coinitial and comu- 
