284 Sir WrttiAM Rowan Hamitton’s Researches respecting Quaternions. 
g 
Thus, in particular, for any spherical triangle, of which the three sides may be 
briefly denoted thus, 
Sa aera: Loe 
RR 0 RRS RR O's (447) 
while the three corresponding vector units, directed to the positive poles of these 
three arcs, may be thus denoted, 
Sail epee RG Nh el (448) 
the following equation holds good, and may be employed, instead of (427), as a 
formula for spherical trigonometry : 
(cos. 6” + ¢” sin 6) (cos @ 4+-«sin @) (cos & + “sin é’) = 1. (449) 
Hence also may be derived this other and not less general equation, analogous to 
(431), and serving in a new way to express the result of the multiplication of any 
two numeral quaternions, in connexion with a spherical triangle : 
u(cos 6 + csin @) X p’ (cos 6’ + “sin 6’) = py (cos 0” — ¢’ sin@”’). (450) 
The sides of the triangle here considered are @, 6’, 6’, that is, they are the 
amplitudes of the two factors and of the product; and the angles respectively 
opposite to those three sides are the supplements of the mutual inclinations of the 
three pairs of vector units, /, (’; ¢’,«; 43 they are therefore, respectively, 
the inclinations of the two vector units “ and « to — ’, and the supplement of 
their inclination to each other. But, in the multiplication (450), « ¢, and — ¢’ 
are respectively the vector units of the multiplier, the multiplicand, and the pro- 
duct ; if then we agree to speak of the mutual inclination of the vector units of 
any two quaternions as being also the mutual inclination of those two quaternions 
themselves, we may enunciate the following Theorem, with which we shall conclude 
the account of this First Series of Researches :—Jf, with the amplitudes of any 
two quaternion factors, and of their product, as sides, a spherical triangle be 
constructed, the angle of this triangle, which 1s opposite to the side which repre- 
sents the amplitude of either factor, will be equal to the inclination of the 
remaining factor to the product ; and the angle opposite to that other side which 
represents the amplitude of the product, will be equal to the supplement of the 
inclination of the same two factors to each other. 
