468 DR G. PLARR ON THE DETERMINATION OF 



it will be necessary to assume that the values of u, v, w will continually be 

 positive in consequence of our admission that the ellipsoid be contained in the 

 one octant of which the edges are in the directions + i, +j, + k . 

 We have thus 



A consideration of what takes place in the two other tangent planes will give 

 corresponding values of v and of w. 

 We thus have the three expressions 



w 2 = SJe<f>- 1 k 



with the' restriction that the determination of the square roots of these expres- 

 sions be the positive one exclusively. 



By the value of N and of analogous values for the cases of <r x and r t , we 

 have now 



ft ■* «*"**' 



1 17 



1 w ^ 

 These values are to be introduced into 



p=6+Pi, <r— &c. 

 We may at once introduce also the following notations : — 



x 1 = Sj<f>~ 1 Jc, x = Sj<pk , 



1 = S^- 1 j, z = Si<j>j . 



§ IIL 



Let us now calculate the variations of the above vectors when the directions 

 of a, fi, y vary. It will be sufficient for our purpose to consider the variation 

 of p only. 



We represent the unit vectors a, ft, y by 



a=piq 

 = pjq 

 .y=pkq, 



