466 DR G. PLARR ON THE DETERMINATION OF 



each of the three sides of the solid angle. The position of the points of 

 tangence will of course be variable in each plane according to the orientation 

 given to the axes of the ellipsoid, but it is evident that on each of the planes 

 the positions of the point of contact will be unable to outpass certain limits so 

 long as the ellipsoid fulfils the condition of remaining tangent simultaneously 

 to the three planes : these limiting positions of the point of contact in one, as 

 for example, of the planes, will form a certain curve, and the proposed question 

 will be : the determination of that curve, the limiting curve as we shall call it in 

 the sequel. 



§1. 



Let a, /3, y designate the unit vectors in the direction respectively of the 

 three principal axes of the ellipsoid ; let a, b, c designate the lengths of these 

 axes ; then if co represents the central vector of a point on the surface of the 

 ellipsoid, and if we put 



aSam , BSBfo , ySyco 



or briefly 



aSaw 



the equation of the ellipsoid will be * 



Sg>0&) = 1 ; 



and if w designates the central vector of a point outside the surface, the equation 



So) o 0&) = 1 



will represent that of the tangent plane passing through the extremity of to. 



Let O designate the apex of the solid trirectangular angle, and i,j, k the 

 unit vectors in the direction of the three edges. We may assume that and 

 i,j, k remain fixed in position and direction. From this it follows that the 

 centre Ox of the ellipsoid and the direction of the trirectangular axes a, ft, 7 

 will be the variables of the question. 



We designate by 6 the vector OOj of the centre of the ellipsoid in one of its 

 variable positions, and by 



p, O", T, 



the vectors of the points of contact of the ellipsoid with the three planes 



respectively 



(j,k), (hi), (if) 



briefly designated, these vectors having their origin in O. From their defini- 

 tion they satisfy the conditions 



S/m = 0, S(r/'=0, St* = 0. 



