JoLY — The AssoGiative Algebra applicable to Rypers^jace. 117 



55. The motion of a rigid body has been treated with almost suffi- 

 cient fullness in Art. 42, In that article it was proved, with even 

 greater generality than here requisite, that c may be chosen in the 

 equation (t = (t^+ FjOe, so as to render o- perpendicular to all the 

 planes of rotation of O in an odd space, and zero in an even space. 

 All that remains is to consider how e may be expressed in terms of 

 o-Q and O ; or, in other words, to solve for p an equation of the type 

 tJT = V-P,p, or again, to invert the linear function 



<^p = F^fip = zs. 



Stated in the last form, p = cji'^zs is a definite vector when the 

 equation <^a = is impossible ; when it is possible, the solution is 

 indeterminate, but of the form p = <f>~^zj + ^o-- 



In the first place, I shall give a solution of the problem depending 

 •on the reduction of O to the canonical form 



O = «i2«i4 + a-ziizH + &c. 

 in 2m units {i). Let 



€ = Bill + $2,^2 + . . . + ^2m+1^2»i+l> 



-in which 2m + I units occur ; then 



FiOe = 012 (- e2h + eiiz) + au{- ej^ + e^ii) + &c. ; 

 and in this 4^+1 does not occur; consequently, «2m+i cannot occur in 



(T - O-fl. If 



O'o = *1^1 + ^2^2 + . • . + *2m+1^2m+l) 



the simplest legitimate value for o- is a-- S2,„+i4m+i- Endeavouring 

 to satisfy a- = 0-^+ FjOe under these conditions, it is obvious that 



1 1 „ 



^2 = ^ij ^1 = H 52, &c. ; 



•or that (?2 = — Sii{(rQ - o-), $1 = Siz^cr^ - o-), &c. ; 



or again, that 



e = - 'XhS'h€ = — {iiSt'z (oTq - 0-) - hSii^OTQ - o-)} + &c. + e^mnhm^i 



U12 



" FiS ■— . {O-Q - Cr) + e2»,+l«2m+l. 



