45 



wliich may accordingly be obtained by a combination of the 

 integrals (21) and (22) ; and the vector part of the quaternion 

 ty, of which the scalar part is thus = - h"^, may be expressed by 

 means of the formula : 



2F.iy=F^.m{iay=F.i'Z.maia', (27) 



which gives, by one of the transformations (17), 



F.iy = F.i-S..maS.ai; (28) 



so that we have, by (13) and (23), 



^.maV.adi = '2.mdaS.ai. (29) 



But also, by (21), because S. tda= 0, we have 



S . ma F. adt = - S . mda F. at + S . matda ; 



we ought, therefore, to find that 



S . m(da . at - at . da) = 0, 



or that 



= rs.m(F.ta.da); (30) 



which accordingly is true, by (13), and may serve as a verifi- 

 cation of the consistency of the foregoing calculations. 



IV. We propose now briefly to point out a few of the 

 geometrical consequences of the formulae in the foregoing sec- 

 tion, and thereby to deduce, in a new way, some of the known 

 properties of the rotation to which they relate ; and especially 

 to arrive anew at some of the theorems of Poinsot and Mac 

 Cullagh. And first, it is evident on inspection that the equa- 

 tion (22) expresses that the a.vis i of instantaneous rotation 

 is a semidia meter of a certain ellipsoid, fixed in the body, but 

 moveable with it ; and having this property, that if the con- 

 stant living force h^ be divided by the square of the length of 

 any such semidiameter i, the quotient is the moment of inertia 

 of the body with respect to that semidiameter as an axis: since 

 the general rules of this calculus, when applied to the formula 

 (22), give for this quotient the expression, 



'2..m{TV.aUif = - h^ t"^ = A^ Tt^ ; (31) 



