ELEMENTARY PRINCIPLES OF QUATERNIONS, 193 
becomes Uc, namely, : 
Ue) = aaa feds he] — Uc, 
where we put 
ay = b,¢ ore: Cyd 
b, = Ga — aye 
Cy = Ayd = b,a . 
The general result we have arrived at is this. The products pw and ap are 
expressed by a quaternion, whose scalar is for both Spa, namely, 
Spa = (— Tp Ta cos ozs) 
=— (ax + by + cz), 
and whose vectors are + V(pa) and V(sp) =— V(pz) respectively. 
Namely, 
Vpo = TpToa sin cs Ur 
(bz — cy) + j(ca — az) + kay — ba) . 
I| 
It follows that 
pa = Spa + Vpo 
_ Also, it follows that ° 
| A eS 
U(pa) = — COS pa + Uz sin po 
~~ se 
U(ep) =— cos po — Ursin pa. 
| And these expressions are consequences of the relations 
ea oy et | 
Jk=—hj=t, M=—-th=j, G=-—ji=k 
(Up) =— 1 
Up Uc =— Uc Up = Ur. 
The expression of the scalar shows that when the factors are at right angles, 
|namely, when o@ coincides in direction with that of Uc, then the angle on 
being = 90°, the scalar is = zero, and the product pa or po reduces itself to its 

The expression of the vector shows that it is of the same direction as the 
perpendicular to the plane of both factors drawn through a common origin, and 
having for its positive part the part which we have defined by the rule laid 
down at the instance of the determination of the value of Q. 
| The condition that g =+ 1 being supposed to be fulfilled, the direction of 
