49 



it gives, therefore, by (38), 



(»V^ - 7i"2> = (crH n'')7 : (40) 



that is, by (37) and (36), 



(n"^ - fi^n'^y = n'^S . tnaS . ay + S . mm'V. aa'S . aa'y. (41) 

 Such being, then, the solution of this linear equation (21) or 

 (39), the sought equation of Mac Cullagh's ellipsoid becomes, 

 by (26), 



{ri^n^ - 7i''2)A2 = w^S . m{S . ayf+'S. . mm'(S . aayf ; (42) 

 and we see that the following inequality must hold good : 



-ji'^^>0. (43) 



.2„/2 



n 71 



If then a new and constant scalar g be determined by the con- 

 dition, 



(nV--n'V*'+^V = 0, (44) 



(where y^ is still equal to the same constant and negative 

 scalar as before), we may represent the internal conical path, 

 or relative locus, of the vector 7 in the body, by the equa- 

 tion : 



= gY + n^S .m{S. ayy + S . 7il7n'{S . aa'y)-. (45) 



We see then, by this analysis, that the straight line y which is 

 drawn through the fixed centre of rotation, perpendicular to the 

 plane of areas, describes within the body another cone of the se- 

 cond degree : while the extremity of the same vector 7, which 

 is a fixed point in space, describes, by its relative motion, a sphe- 

 rical conic in the body, namely, the curve of intersection of the 

 cone (45) and the sphere (44) : which agrees with Mac Cul- 

 lagh's discoveries. Again, the normal to the cone (45), which 

 corresponds to the side 7, has the direction of the vector de- 

 termined by the following expression : 



and this new vector is always situated in the plane of 

 areas, and is the side of contact of that plane with another 

 cone of the second degree in the body, which is reciprocal to the 



VOL. IV. E 



