64 



Proceedings of tlie Lie i/al Irish Academy. 



From the foregoing Table 1, we can write clown the coefficient of the 

 compounded rotator by simply collecting terms, and can, by a simple 

 calculation, find the angle and axis planes of the compound rotation. 



' Table 2. — Showing Multiplication of Eotator into a Vector. 







X 



i 



y 

 J 



k 



w 

 h 



ei<p\ 



1 



i 



J 



k 



h 



e<piy 



Ji 



J 



— i 



-h 



k 



ei<py' 



wji 



J 



— i 



h 



-k 



e<pia 



14 



-h 



k 



-J 



i 



dicpa 



wkj 



h 



k 



-J 



— i 



Ocpi^ 



ik 



-k 



-h 



i 



J 



ei<p0 



wik 



-k 



h 



i 



-J 



dcpyy' 



uji .Ji 



— i 



-J 



k 



h 



d(paa 



oikj . hj 



i 



-J 



-k 



h 



dcp^H' 



cuik . iJc 



— i 



j 



-k 



h 



6 (pay 



wji . kj 



k 



h 



i 



J 



6(pay 



wkj .ji 



k 



-h 



i 



-J 



epa'^ 



wkj . ik 



J 



i 



h 



k 



d(po.ff 



wik . kj 



J 



i 



- h 



-k 



dcp^'y 



wik .ji 



h 



k 



J 



i 



Ocp^y' 



wji . ik 



-h 



k 



J 



— i 



01 is written for cos 0, and for sin 0. 



Table 2 shows the effect of multiplying the quaternion 

 (^1 + Otti) (<^i + ^tt'i) into a vector. 



The direction cosines of a transformation can be found from Table 2 

 by collecting the terms : xi+yj+ %h + ivh becomes 

 V[^ { ^i</)i - ^</. (7/ - aa' + /S/3') } + y { - e</)i7 - ^ic^y + ^(A (a'/3 + a/3') } 

 + 3{ %/3 + ^i<^^' + ^^ (ay' + a'y) }+«<;{ ^<^ia - ^it^a' + ^(^ (yS'y - /5y') ) ] 

 ^Jlx[64>,y + e,4>y'+e4>{o!li + a^')} + y{e,^, - 6cl>iyy'+aaf-/3f3')} 



+ z{- e^,a - 0,4>a'+ d4> {fS'y + /3y') \+w\ Ocji^/S - 6y<i>[i' + ^0 (ay'- a'y) } ] 

 + ^ [a; { - %^ - ^i<^yS'+ ^</) (ya'+ a'y) } + y { ^c^itt + ^i</)a'+ ^<^ (^y + ^'y) } 



+ z{ei</)i - ^</)(aa' + /8/i' - yy') J + io[e,cf>,y - e<^y' + e^{a! ^ - a/3') j] 



+ h [x{-e<fi,a + ^i</)a'+ Ocji (/8'y - |8y') } + y {- 6'<^i/3 + dict>(S'+ ^<^ (ay'- a'y) } 



+ z{- 64^,y + e,^y' + e</.(a'/8 - a^') } + w\e,<i>, + 6<i>{yy' + aa' + /3^')}]. 



