768 



PROFESSOR STLVESTEE ON THE MOTION OF A RIGID BODY 



becomes 

 //cos 



S--I^(c-b) cos/3cosy=0, '-^^-l(1- J) cosyco8«=0, 

 ^f^-L(B-x)cosacos^=0 



(3) 



The above equations suffice to express the relations of the angles which the invariable 

 line in space makes with fixed lines in the moving body to one another and to the 

 time : to complete the solution it will be sufficient to express in terms of the time, or of 

 any quantity dependent on the time, the position of any of the planes drawn through a 

 principal axis and the invariable line. 



The letters X, Y, Z, 1 retaining their previous signification, let ZZ' represent the 

 infinitesimal angular displacement of Z due 

 to the rotation <y, about X in the time dt. 



Then 



But 



or 



ZIZ'=ZZ'^^^^^^'= ZZ'^^^^- 



sinlZ' sinlZ 



cos NX cos IX cos a 



cos NY cos lY cos /3' 



cosNX=- 



COS a 



and 



'/(C08<»)«+(cOS/3)* 



sin TZ=\/l — (cos y)''=\/(cos a)'' + (cos ]3)* 



(cos «)* 



dt 



Hence ZIZ'=L 



(C0sa)* + (C08^)« 



relation 



CU; : a>2 : Wj : : A cos a : B cos /3 : C cos y 



shows that the invariable line coincides in direction with the pedal to the radius vector drawn in the direction 

 of the instantaneous axis. 



2. Consequently the length of such pedal being 



(cos cif (cos (if (cob y)' 

 ~X~ + ~B~+~C~ 



which is constant, a plane drawn at that constant length perpendicular to the invariable line touches the ellip- 

 soid in every position into which it turns, and therefore the ellipsoid with its centre fixed roUs on such plane. 

 This proves the identity of the two motions qua space. 



3. The moment of inertia in respect to the instantaneous axis being represented by the inverse squared length 

 of the radius vector of the ellipsoid in the direction of that axis, the square root of the vis viva (a constant) is 

 proportional to the angular velocity divided by the radius vector drawn to the point of contact, so that the 

 former is proportional to the latter ; this completes the representation by expressing through means of the 

 dlipsoid the relation of the motion of the associated free body to time, or at all events it gives the law from which 

 that relation may be extracted. 



The above contains the whole sum, pith, and substance of Poinsoi's ellipsoidal mode of representation. 



