ELEMENTARY PRINCIPLES OF QUATERNIONS. 199 
because they contain not more than two vector factors, and they therefore are 
of the forms 
Aww, Or wp, Or Wry , 
where vz is a scalar, and X, », vectors. 
The eighth term will be 
aB x y, 
and the question is reduced to the demonstration that 
a8 x yanda x By 
are identical. 
The following demonstration may be, perhaps, the shortest. 
We decompose 8 and y into components parallel respectively to a triple 
rectangular system, of which the versors are p’, o’, 7’, and whose directions are 
the following— 
p parallel to a ; 
o perpendicular to a, and in the plane comprising a and £, led through a 
common origin ; 
7 perpendicular to p’ o”. 
Then we will have the expressions— 
Ge Alp 
(cee Bp’ + Bio’ 
y = Cp’+ C,o'+ Cy’. 
The six scalar coefficients A, B, &c., have determinate values, but we need 
not effect their determination. 
Then we have 
ax By = al B(Cp’ x p® + Cip’ x p'o’ + Cp" x p’z’) | 
+ B,(Cp’ x o’p! + C,p! x o? + C,p! x 07’) 
afs 4 y= A B(Cp” x p’ + Cp” x o” + Crom x T’) 
e B,(Cp’o’ x p’ + Cip’o’ x o + C.p’o" x et 
Now p’, o , 7 satisfy the relations set down in § 4 analogous to those between 
i,j, k. Therefore we have 
Ist, p’ x p> =p x (—1)=—p’ =p" x ’ 
2d, p’ x po’ = pr’ =— o’ =p xa" 
ad; "px pa =p(— 07) =— 7 =p KT 
Ap Xp — oT) o = 7p,—= po’ Xx p 
arhpX a a= — p = 1'o = eer 
' 6th, p’ x ot =p’ x pb =rxr == On ay 
VOL. XXVII. PART IL. Bd 
