288 



a + j3 = CB, a— /3=CA, (T = DC, jO = EA, (41) 



«o that A is the centre of the ellipsoid, e 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 d, while b is another fixed 

 point, we obtain the equation: 



EA = ± U . DA -^ T . DB ; (42) 



which gives 



(ea)-^ = ^Z U . DA . T . DB, (43) 



and shows, therefore, that the proximity (ea)-^ of a variable 

 point E, 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 e 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 T)\ 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 da ; let d' be the second 

 point of intersection of the spheric surface with the secant db, 

 drawn to the variable extremity d of this chord from a fixed 

 external point b ; take the radius vector ea equal in length 

 to d'b, and in direction either coincident with, or opposite to, 

 the chord da; the locus of the point e, thus constructed, will 

 be an ellipsoid, which will pass through the point b. This 

 fixed point b (one of four known points upon the principal 

 ellipse) may, perhaps, be fitly called a pole, and the line be 

 2l polar chord, of the ellipsoid; and in the construction just 

 stated, the two variable points d, d' may be said to be conju- 

 gate guide-points, at the extremities of coinitial and conju- 



