45 



which may accordingly be obtained by a combination of the 

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

 ly, of which the scalar part is thus = - Jt^, may be expressed by 

 means of the formula : 



2F.iy=V2.m{iay=F.i'2.maia; (27) 



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



V.iy^F.i^.maS.ai; (28) 



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



S . ma V. adi = S . mda S . ai. (29) 



But also, by (21), because ^S*. ida= 0, we have 



S . ma V. adt = - S . mda V. aii- 'Si . maida ; 



we ought, therefore, to find that 



S . m{6a .ai-ai. da) = 0, 



or that 



= rS.m(F.m.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 axis i of instantaneous rotation 

 is a semidiameter 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 ivith 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, 



S.m(TF.«t70' = -A2 4-2 = ^2 2--^; (31) 



