47 



make any use of the three axes of inertia, nor even to assume 

 any knowledge of the existence of those important axes ; nor to 

 make any other reference to any axes of co-orditiates whatsoever. 

 The result of the calculation might be expressed by saying 

 that " the ellipsoid of living force rolls on a plane parallel to 

 the plane of areas ;" and nothing farther, at this stage, might 

 be supposed known respecting that ellipsoid (22), or respect- 

 ing any other ellipsoid, than that it is a closed surface repre- 

 sented by an equation of the second degree. With respect to 

 the path of the axis of momentary rotation i, within the body, 

 it is evident, from the equations (21), (22), that this path, 

 or locus, is a cone of the second degree, which has for its equa- 

 tion the following : 



7' S . m{ V. laf = - A-(S . ma V. laf ; (33) 



where the symbol y^ by one of the fundamental principles of 

 the present calculus, is a certain negative scalar, namely, the 

 negative of the square of the number which expresses the 

 length of the vector y, and which (in the present question) 

 is constant by the law of the areas. Thus, according to ano- 

 ther of Poinsot's modes of presenting to the mind a sensible 

 image of the motion of the body, that motion of rotation may 

 be conceived as the rolling of a cone, namely, of this cone 

 (33), which is fixed in the body, but moveable therewith, on 

 a certain other cone, which is the fixed locus in space of the 

 instantaneous axis t. 



V. But we might also inquire, what is the relative locus, 

 or what is the path tvithin the body, of the vector j, which 

 has, by the law of areas, a fixed direction, as well as d^ fixed 

 length in space: and thus we should be led to reproduce some 

 of the theorems discovered by Mac Cullagh, in connexion 

 with this celebrated problem of the rotation of a solid body. 

 The equations (26) and (32) would give this other formula, 

 >S.£g7 = 0; (34) 



