288 Sir Witiram Rowan Hamirton’s Researches respecting Quaternions. 
is easily seen, on reverting to first principles, to be the choice of the cyclical order 77h, 
rather than 7/j, or the choice of the upper rather than the lower signs in the assumptions 
ij=—jizth, jho—hj=+i, hiz—th= 4). (26) 
This gives a clue, which may be thus pursued. Let 
x” = te cos $,s y = re sin 6, cos ww, at = ni sin 6,” sin W's | (27) 
“u" 
vy = r," COS by Oy = a sin by cos his zi," = Tax sin by, sin his 
then, by (12) and (16), and by the meaning which we have assigned to r”, we have 
yr? — a fk pw? = wy? 7/2, (28) 
«* By (9), 7,” is perpendicular to the plane of 7’; and therefore, by (10), 7” is in that 
plane, being, in fact, the projection of 7’ thereupon, This projection is entirely fixed by 
the construction already given ; and it remains only to determine the direction of the 
perpendicular, 7,,’, as distinguished from the opposite of that direction. Anda rule which 
shall fix the sign of any one of the coordinates, x,”, y,”, ,;’, will be sufficient for this 
purpose. It will be sufficient, therefore, to study any one of the equations (8), for 
instance the first, namely, 
pF = a! on! 
aw," =yz —2y’, 
and to draw from it such a rule. 
‘¢ Substituting for y, 2, y’, 2’, their values (16), we find 
x,” = py’ sin O sin 6’ sin ¢ sin ’ sin(l’/—y) ; (29) 
so that (the other factors having been already supposed positive) 7,” has the same sign as 
the sine of the excess of the longitude y/ of 7’ over the longitude y of r. But these longi- 
tudes are determined by the rotation of the plane of ar round the positive semiaxis of «, 
from the position of ay towards the position of wz, or from the positive semiaxis of y 
towards that of z; which direction of rotation is here to be considered as the positive one. 
Consequently, x,” is positive or negative, according as the least rotation round +2, from 
r to 7’, is itself positive or negative ; in each case, therefore, the rotation round a”, and, 
consequently, round 7-,”, or finally round 7”, from r to 7’, is positive. The rotation round 
the product line, from the multiplier to the multiplicand, is constantly right-handed or 
constantly left-handed, according as the rotation round + 2 from +7 to + 4 is itself right- 
handed or left-handed. Hence, also, to express the same rule otherwise, the rotation 
round the multiplier, from the multiplicand to the product, is (in the same sense) constantly 
positive. In short, the cyclical order is multiplier, multiplicand, product ; just as, and 
precisely because, we took the order jk for that in which the rotation round any one, 
from the next to the one after it, should be accounted positive, and chose that 7 should 
be =k, not -—k. The law of the moduli, the theorem of the spherical triangle, and 
the rule of rotation, suffice to determine entirely the product of any two quaternions. 
