Sir Witi1am Rowan Hamunton’s Researches respecting Quaternions. 283 
rm, + ym; em, 103 
1 Ly Y 2 + 3 } (441) 
zm, + ym; + zm;= 0, 
represent the two cyclic planes of a certain other cone of the second degree, 
which has its vertex at the origin, and contains upon its surface the four points 
which are determined by the twelve following rectangular coordinates : 
md —m,C, mb — md, mc — mb; 
fe / / / ‘ ‘ 
cm, — dm, dm, — bm, bm, — em; 
(442) 
secre 
mim; — mms, mymi'— mim, mys’ — mimi! ; 
Ms M,— MM, M3M,— MM, Mm,—mM'm,. | 
It would have been easy to have given a little more symmetry to these last 
expressions, if we had not wished to present them in a form in which they might 
be easily combined with some that had been already investigated, for a different 
purpose, in this paper. 
50. If we denote by the symbol ?,,, that vector unit which is directed towards 
the positive pole of the arc rr’ (from the point rR fo the point Rr’ on the unit 
sphere), then the general formula (425) for the product of any two vector units, 
z, and 7,,, becomes 
ae 
Uy Iq = (COS + ty Sin) (7 — RR’); (443) 
and because the positive pole of the arc rr’ is the negative pole of the reversed 
arc R’R, so that in this reversal the change of sign may be conceived to fall upon 
the vector unit, 
te — — ta (444) 
while the arc itself may thus be regarded as not having changed its sign, but only 
its pole, we may also write, generally, in this notation, for the quotient of any 
two vector units, the expression 
. * . - . . ae 
Iggy! = — ty ty = (COs + ty, SiN). R’R. (445) 
Hence the associative principle of multiplication gives this other property of any 
spherical polygon, rr’r”..., which may be regarded as a sort of polar conjugate 
to the property (426), as depending on the consideration of the polar polygon, or 
polygon of poles, namely, the following : 
GR Ras Se Flare : a er 
(cos + ty, SIN) R'R. (COS + 7y7_/SIN) RVR’... (COS Fagin» SIN) RR" = J. (446) 
