Sir WittiAmM Rowan Hamirton’s Researches respecting Quaternions. 279 
45. Multiplying both members of this formula (427) into cos xr” — ¢,,sin Rr”, 
we put it under the less symmetric but sometimes more convenient form, 
(cos R + 7, sin R) (cos R’ +7%,,sin R’) = —cosr” +7%,,sinr”. (428) 
Developing the first member of this last equation, and substituting, for the product 
of the two vector units, its value (425), we find that it resolves itself into the 
two following formule : 
cosR cos k’ — cos RR’ sin R sin R’ = — cos Rr” ; (429) 
7, iN R cos R’ + 2,,sinR’ cosR + 7, sin R sin R’ sin RR’ = 2,,sinR”. (430) 
Of these two equations, the first agrees with the known expression for the cosine 
of a side rr’ of a spherical triangle rr’r”’, regarded as a function of the three 
angles R, R’, RX”; and the second expresses a theorem, which can easily be verified 
by known methods, namely, that if a force = sin x” be directed from the centre 
of the sphere to the point Rr”, that is, to one corner of any such spherical triangle 
rrr”, this force is statically equivalent to the system of three other forces, one 
directed to R, and equal to sink cos R’; another directed to r’, and equal to sin r’ 
cos Rk; and the third equal to sinr sink’ sinrr’, and directed towards that pole 
P” of the arc rr’, which lies at the same side of this arc as does the corner r”. 
46. In this, or in other ways, we may be led to establish, as a consequence 
from the principles which have been already stated, the following general formula 
for the multiplication of any two numeral quaternions : 
qXq7 = p(cosk +7%,sinR) X p’ (cosr’ + 7,sinr’) 
= pp! {cos (w—R”) + 2,, sin (w7—R”)} ; (431) 
and to interpret it as being equivalent to the system of the three following rules 
or theorems. First, that (as was seen in the twenty-seventh article), the modulus 
uw” of the product is equal to the product py’ of the moduli of the factors. 
Second, that if a spherical triangle rx’ Rr” be constructed with the representative 
points of the factors and product for its three corners, the angles of this triangle 
will be respectively equal to the amplitudes of the two factors, and to the supple- 
ment of the amplitude of the product ; the amplitude r of the multiplier quaternion 
q, for example, being equal to the spherical angle at the corner r of the triangle 
just described. And third, that the rotation round the product point, x”, from 
VOL. XXI. 2P 
