280 Sir Wirt1am Rowan Hamitron’s Researches respecting Quaternions. 
the multiplier point, which is here denoted by R, ¢o the multiplicand point, denoted 
here by r’, is positive ; or, in other words, this rotation is in the same direction 
(towards the right hand, or towards the left), as the rotation round the positive 
semiaxis of z or of k(= 7), from that of x or of 2, to that of y or of y. The 
same third rule may also be expressed by saying that the rotation of a great semi- 
circle rownd the multiplier point r, from the multiplicand point r’, towards the 
product point x”, is positive; whereas the rotation to the same product point, 
from the multiplier point, rownd the multiplicand point, is, on the contrary, 
negative. (Compare the remarks in Note A, printed at the end of the present 
series. ) 
47. The associative character of multiplication shows that if we assume any 
three quaternions q, 7’, g’’, and derive two others ¢,, 7,, from them, by the equations 
W=% TT =U (432) 
we shall have also the equations 
99" = 99, = 9" (433) 
q’ being a third derived quaternion, namely, the ternary product gq q. Let 
‘’ be the six representative points of these six quaternions, on the 
same spheric surface as before; then, by the general construction of a product 
assigned in the foregoing article, we shall have the following expressions for the 
six amplitudes of the same six quaternions : 
/ ur 
Ut TN TBAB 
ea) = my sey) tare fo 
Sy By, Se pRl ss Si Be =7— RRR; , 
hs ! Le, s ONT Be / ye 
= 2, 2 i BURRS) WO RL, Bn Ss = RP (434) 
eo’ = R,, rR” r= mil! mY mp g/— a—R RR —7r—R Re! z, ; 
x’ RR, being the spherical angle at k, measured from rr’ to RR, and similarly in 
other cases. But these equations between the spherical angles of the figure are 
precisely those which are requisite, in order that the two points r, and r,, should 
be the two foci of a spherical conic inscribed in the spherical quadrilateral 
rR’ RR”, or touched by the four great circles of which the arcs rR’, Rk’ R"; 
r’ Rr’, RR, are parts; this geometrical relation between the six representative 
points RR’ R”R,R,R” of the six quaternions 4, 9’, 9”, 99', 7, 997'9", which 
may conveniently be thus denoted, 
BB in) REBAR, (435) 
