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 abc, 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, c, then the arc, or side of the triangle, 

 AB, will represent the amplitude of the multiplicand quater- 

 nion ; the arc or side ac will represent the amplitude of the 

 multiplier ; and the remaining arc 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 

 of the Philosophical Magazine for December, 1 844. 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 represetitative 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- 



