Xxxiii 
duct of the moduli of the two factors is equal to the modulus 
of the product. It is clear also, that if we construct a spherical 
triangle anc, of which the three corners, or the radii drawn to 
them from the centre of the sphere, represent the directions of 
the three lines a, b, ¢, then the are, or side of the triangle, 
aB, will represent the amplitude of the multiplicand quater- 
nion; the are or side Bc will represent the amplitude of the 
multiplier; and the remaining are or side ac the amplitude of 
the product, so that the spherical triangle will be constructed 
with these three amplitudes for its three sides. . And we see 
that in the triangle thus constructed, the spherical angles at 
A and c, which are respectively opposite to the amplitudes of 
the multiplier and multiplicand, are equal to the respective 
inclinations of the axes of the multiplicand and multiplier to 
the axis of the product of the quaternions ; and: that the 
remaining spherical angle at B, which is opposite to the am- 
plitude of the product, is equal to the supplement of the 
inclination of the axes of the factors to each other: a form 
almost the same with that under which the fundamental 
connexion of quaternions with spherical trigonometry was 
stated by Sir William Hamilton, in his first letter on the 
subject, to John T. Graves, Esq., which was written in Oc- 
tober, 1843, and has been printed in the Supplementary number 
ofthe Philosophical Magazine for December, 1844. The other 
form of the same fundamental connexion, which was commu- 
~ nicated to the Academy in November, 1843, may be deduced 
from the foregoing, by the consideration of that polar or sup- 
plementary triangle, of which the corners mark the directions 
of the axes of the factors and the product, and were then 
called the representative points of the three quaternions com- 
pared. If the order of the factors be changed, the (positive) 
axis of the product falls to the other side of the plane of the 
axes of the factors, being always so situated that the rotation 
round the axis of the multiplier from the axis of the multipli- 
_ cand to that of the product is positive ; multiplication of qua- 
