46 



where TF. a Ui denotes the length of the perpendicular let 

 fall, on the axis i, from the extremity of the vector a, that is, 

 from the point or element of the body of which the mass is m. 

 In the next place, the equation (26), wliich is of the first de- 

 gree in I, may be regarded as representing the tangent plane 

 to the ellipsoid (22), at the extremity of the semidiameter t ; 

 because this equation is satisfied by that semidiameter or vec- 

 tor I, when we attribute to it the same value (in length and 

 in direction) as before ; and because if we change this vector 

 I to any infinitely near vector i + Si, consistent with the equa- 

 tion (22) of the ellipsoid, this near value of the vector will 

 also be compatible with the equation (26) of the plane ; for 

 when the variation of the equation (22) is thus taken (by the 

 rules of the present calculus), and is combined with the equa- 

 tion (21), it agrees with the equation (26) in giving 



S.ydi = 0. (32) 



But the plane (26) is Jixed in space, on account of the con- 

 stant vector J and the constant scalar /«, which were intro- 

 duced by integration as above; consequently the ellipsoid (22) 

 rolls (without gliding) on the fixed plane (26), carrying with it 

 thebody in its motion, and having its centre fixed at the fixed 

 point of that body, or system, while t\ie semidiameter of con- 

 tact I represents, in length and in direction, the axis of the 

 momentary rotation. This is only a slightly varied form of 

 a theorem discovered by Poinsot, which is one of the most 

 beautiful of the results wherewith science has been enriched 

 by that geometer : for the ellipsoid (22), which has here present- 

 ed itself as a mode of constructing the integral equation which 

 expresses the law of living force of the system, and which might 

 for that reason be called the ellipsoid of living force, is easily 

 seen to be concentric with, and similar to, the central ellipsoid 

 of Poinsot, and to be similarly situated in the body. It may, 

 however, be regarded as a somewhat remarkable circumstance, 

 and one characteristic of the present method of calculation, 

 that it has not been necessary, in the foregoing process, to 



