1887. ] of the Quaternion Analysis. 165 
With regard to the question of sign, we have 
=—S8.s,5,s,=—S.s,8,8 
cme p= i CY 
S.s,5,8,=S.s,s,5,=S.s,8,8,=—S.s NSE SHE 
12°83 2°31 $8 
8°21 
Draw OL, OM, ON respectively perpendicular to s,, s,, s,. In the 
above investigation we have proceeded on the assumption that 
the three planes are so situated that a right-handed screw motion 
about OP, the direction of the motion of translation being from O 
towards P, will carry Z towards M, M towards NV, and NV towards 
LI. Thus if the order in which the three planes occur in the 
product correspond to a right-handed screw motion about OP, 
then the quantity we have equated to the scalar portion of the 
product is affected with a positive sign. If, however, the order of 
the planes correspond to a left-handed screw motion about OP, 
the said quantity is affected with a negative sign. 
Three other expressions for S. s,s,s, may be deduced from the 
formulae 
1 il 
S.S,8,8, = Ts, LY V3.5 VSS = Te 
Now the perpendiculars from the origin on V.s,s, and V.s,s, are 
respectively parallel to BP and CP, also TV.s,s,=sin B/p,p, and 
TV.s,s, = sin C/p,p,, and consequently we have 
LV. Vs,3,Va,= 7 TV.Vas,Vo, 
2 
TV.Vs,s,Vs,s,=sin B sin C sin a/p,"p,p,° 
Therefore 
sn BsinCsina sinCsinAsinb sin A sin Bsinc. 
P:PoPs P:P2Ps (OOO 
from which we deduce the well-known formula 
sss 
sin a/sin A = sin b/sin B =sin c/sin C. 
Another expression for S.s,s,s, is deducible from the formula 
8,787,’ + (S. 8,8,8,)” = 8," (Ss,s,)” + 8,” (Ss,s,)” 
+s, (Ss,s,)’ — 28s,s,Ss,s,S8s,s,. 
From this we deduce 
(S a 8,884) = 
ii 
ep ps {1 — cos’ A — cos’ B — cos’C'— 2 cos A cos Bcos C}, 
1if2rs3 
and combining the three sets of formule we obtain 
1 —cos’ A — cos’B — cos’ C — 2 cos A cos B cos C 
= sin’, sin’A * = sin’, sin’ B = sin’, sin’C 
= sin? B sin? sin’?a = sin’C sin? A sin?b = sin’ A sin? B sin’c. 
* See Article 11, 
