48 



and thus would shew that the vector y is (in the body) a va- 

 riable semidiameter of an ellipsoid reciprocal to that ellipsoid 

 (22) of which the vector l has been seen to be a semidiameter ; 

 and that these two vectors y and t are corresponding semidia- 

 meters of these two ellipsoids. The tangent plane to the new 

 ellipsoid, at the extremity of the semidiameter -y (which ex- 

 tremity is fixed in space, but moveable within the body), is 

 perpendicular to the axis t of instantaneous rotation, and in- 

 tercepts upon that axis a portion (measured from the centre) 

 which has its length expressed by h"^ Ti^, and which is, there- 

 fore, inversely proportional to the momentary and angular 

 velocity (denoted here by Ti), as it was found by Mac Cullagh 

 to be. To find the equation of this reciprocal ellipsoid we have 

 only to deduce, by the processes of this calculus, from the 

 linear equation (21), an expression for the vector -y in terms 

 of the vector i, and then to substitute this expression in the 

 equation (26). Making, for abridgment, 



n' = -S .ma^; n'2 = -S.W7n'(F. aa')-; 1 



n"^ = + ^.mm'm"{S.aa'a"f ; i ^^^^ 



so that n, n', n", are real or scalar quantities, because the 

 square of a vector is negative ; and introducing a characte- 

 ristic of operation a-, defined by the symbolic equation, 



o- = S . rtiaS .a, or at = S . inaS . ai ; (36) 



it is not diflUcult to show, first, that 



((t' + nV + /O t = - S . mm' V. aa' S. aa'i ; (37) 



and then that the symbol a is a root of the symbolic and cubic 

 equation, 



(7' + nV + «'2(7 + n''2=0; (38) 



in the sense that the operation denoted by the first mem- 

 ber of this symbolic equation (38) reduces every vector i, on 

 which it is performed, to zero. But the linear equation (21) 

 may be thus written : 



(«T+«^)« = y; (39) 



