in a Letter to R. T. Graves, Esq. \Q3 



We have, then, this first law for the multiplication of two qua- 

 ternions together, that the modulus of the product is equal to 

 the product of the moduli of the factors. 



Division of quaternions is easy. The equations (B.) of 

 multiplication give 



ra'= (a^ + ^,2 _^ c^ + rf^)-' {a a" + 5 &" + c c" + rf dJ') ; 

 J h'={a^ + h'^^c' + d'')-'{-ha" + aW^dc''-ca<)', 

 ^^•'' ) d^{a''--\-U' + c' + d'')~'\-c a'< -^ a c" + bd<'-dV<)\ 

 ^d'={a'' + h''-\-c^ + d^)-'[-da'^ + ad!^^cW-bd'). 



The modulus of the quotient is the quotient of the moduli. 

 A quaternion divided by itself gives for quotient (1,0, 0,0) = 1. 



Addition and subtraction require no notice. 



Making 



,y. V ra=ja,cos g, i=|tx. sin g cos cj), c=/x sin q sin<^ cos \J/, 



^ ''' \ d—i^ sin q sin <fi sin \i/, 



I would call q the amplitude of the quaternion ; <^ its colati- 



tude; and ^ its longitude. The 7nodidiis \s [i. Representing 



the three coefficients 6, c,^Z of the imaginary triplet ib+jc-{-kd, 



by the rectangular coordinates of a point in space ; jw-sin q, as 



being the radius vector of this point, might be called the radius, 



or perhaps the length, of the quaternion. We may speak of 



the inclination of one quaternion to another, and its cosine is 



x» • . • w /.( ix bV + cd + dd' 



cos <b cos ii>'+ sm 4 sm <±' cos (\I/'— \I/) = — , -. ; — ,. 



^ ' fJbfjJ smq sm q' 



If with the amplitudes q, q', p" of any two factors and their 

 product, as sides, we construct a spherical triangle, the angle 

 opposite to the amplitude of the product will be the supple- 

 ment of the inclination of the factors to each other; and the 

 angle opposite to the amplitude of either factor will be the in- 

 clination of the other factor to the product*. 



This theorem of the spherical triangle, combined with the 

 law of the moduli, requires only besides the following rule of 

 rotation, to decide on which side of the plane of the two fac- 

 tor-lines the product-line is found^ in order to complete the 

 geometrical construction of the equations (B.) of multiplica- 

 tion of quaternions. In whichever direction, to the right or 

 to the left, the positive semiaxis of/ must rotate round that of 

 k in order to approach to that of^'; in the same direction must 

 the multiplicand line turn round the multiplier-line in order 

 to approach the product-line (or the plane containing that 

 product-line and the multiplier-line). 



If the factor-lines coincide with each other in direction, 



* An improved form of this theorem of the spherical triangle was com- 

 municated by the writer to the Royal Irish Academy in November 1843, 

 and has been published in the Philosophical Magazine for July 1844. 



