1873.] On the Motion of a Body about a Fixed Point. 233 



III. " Some new Theorems on the Motion of a Body about a Fixed 

 Point." By Edward John Routh, M.A v F.R.S. Received 

 December 21, 1872. 



Poinsot constructs the motion of a rigid body about a fixed point under 

 the action of no forces by means of an ellipsoid which has its centre at the 

 fixed point and rolls on a fixed plane. In this manner the relations of 

 the invariable line, and the instantaneous axis to each other and to the 

 other parts of the body, may be found by solid geometry. It is evident that 

 in many cases these relations are merely translations into the language 

 of solid geometry of certain properties of the spherical ellipse. In trying 

 to make use of the spherical ellipse in a general manner, I have been led 

 to some theorems which I think are interesting, and which seem to be new. 

 In what follows I shall restrict myself almost entirely to these results. 



1. Let a body be turning about a fixed point under the action of 

 no forces, and let the angular velocities about the principal axes at O be 

 Wj, u) 2 , w 3 . Also let A, B, C be the principal moments at the fixed point ; 

 then we know that if we put 



2 i 2 I 2 2 



W l + W 2 + W 3 =W 



Ao^ + B o 2 2 + C o, 3 2 =T, 



, 1 2 +bv+cv= g - 2 3 



T and G are constant throughout the motion, and io is the resultant 

 angular velocity. 



2. There are three straight lines whose motions we may consider : — 

 (1) the Instantaneous axis, whose direction cosines are ^, — 2 , ^ ; (2) 



the Invariable axis, whose direction cosines are ; (3) the 



(r Cr (x 



Eccentric axis, whose direction cosines are ^A^i VBo> 2 ^^ 3 . This 



Vt Vt ' Vt 



last axis is the eccentric line of the instantaneous axis with regard to the 

 momental ellipsoid at the fixed point, or, which is the same thing, the 

 eccentric line of the invariable axis with regard to the ellipsoid of gyra- 

 tion. Let a sphere of radius unity be described whose centre is O, fixed 

 in the body and therefore moving with it. Let these three axes cut the 

 sphere respectively in the points I, G, H. Let the principal axes at O 

 cut the sphere in A, B, C. It is our object to discuss the motions of 

 I, G, H relatively to A, B, C and in space. 



3. The equations to the cones described in the body by 01, OG, OH 

 respectively are 



A(AT - G> 2 + B(BT - G 2 )y 2 + C(CT - G 2 > 2 =0, 

 AT— G 3 2 , BT-G 2 2 , CT-G 2 2 A 



A B C 



(AT - GV + (BT - G 2 )?/ 2 + (CT - GP)z 2 = . 



VOL. XII. 



