278 Sir Wi11am Rowan Hamitton’s Researches respecting Quaternions. 
Connexion of Quaternions with Spherical Geometry. 
44. Let pr, Rk’, R’,..R"~" be any points upon the surface of the unit 
sphere, so that they may be generally regarded as the corners of a spherical 
polygon upon that surface ; and let them be regarded also as the determining or 
representative points (in the sense of the forty-second article) of the same num- 
ber of vector units, 7, 7,,&c. Then the associative property of multiplication 
will give, on the one hand, the equation 
MWe Dag tems WanlOgnn socio Caos nie == (id) (421) 
because 
QS BS, =.= — 15 (422) 
and, on the other hand, on substituting the expressions for these vector units, 
involving their respective direction-cosines and the three fundamental units, 
i, j, k, which expressions are of the forms 
i, =ta+7B+hy, ty aia +98 + ky's.. (423) 
we shall have, for the product of the two first, by the fundamental relations (8), 
the expression 
ig ty = (ia + j8 + hey) (ia! +56! + hy’) 
= — (aa! + 68’ +77) + (By — vB’) Hire’ — ary’) + B(ap!— Ba’), (424) 
that is, 
Lig Igy = — COSRR’ +2,, SiN RR, (425) 
if rk’ denote the arc of rotation in a great circle, round a positive pole Pp”, from 
the point r to the point R’ upon the sphere, with other similar transformations for 
the other binary products. By combining these two principles, (421), (425), 
it is not difficult to see that, for any spherical polygon, regarded as having its 
corners R, R’,..at the positive poles of the sides of another polygon, the following 
formula holds good : 
(cosr + 7,sinR)(cosR’ +2,sink’).. (cosn™~) + 2,60- sink”) =(—1)"; (426) 
in which the symbols r, r’,.. under the characteristics cos and sin, denote the 
(suitably measured) successive angles at the corners R, Rr’. . . In particular, for the 
1if/ 
case of a spherical triangle, rx’r’, the formula (426) gives this less general for- 
mula, which, however, may be considered as including spherical trigonometry : 
(cos rk +7,sin R) (cos R’ +7,sin R’) (cos R” + i,sinR”)=—1. | (427) 
