410 



RICE 



ART. K 



It is an ellipsoid, and its position in the body is entirely independ- 

 ent of what axes of reference we choose. So the same surface 

 referred to the principal axes as axes of coordinates has the 

 equation 



By a theorem on quadric surfaces quoted in the Mathematical 

 Note, observing that 



^', T]' f ' correspond to x, y, z in the note, 



X', n', v' correspond to x' , y', z' in the note, 



ei, 62, 63, 64, 65, 66 correspond to a, 6, c, /, g, h in the note, 

 ei, €2, €3 correspond to a', b', c' in the note, 



we arrive at these three results: 



6263 + 6361 + 6162 



} (11) 



Now let the reader look at the equations (9) which give 6i, 62, etc., 

 in terms of the squares and products of the Crs coefficients, and 

 refer to the well-known rule for multiplying determinants 

 which will be found in any text of algebra. He will find that 

 the determinant in (11) is the square of the determinant 



(12) 



Thus the last of the equations in (11), on extracting the square 

 root, is equivalent to 



= rir2r3. 



(13) 



which is essentially equation [442], the third equation of (11) 



