Sir W. R. Hamilton on Qtmternions. 243 



constituents of a product, or by the definitions (A.), (B.), (C), 

 we have 



There are, therefore, in this theory, hifinitely many different 

 square roots of negative one, which have all one common mo- 

 dulus = 1, and one common amplitude = — , being all of the 



form 



i^ — 1 = « cos ^ + Jsin (^icostf/ + ^sin 4> sin i^, . (L.) 



but which admit of all varieties of directional co-ordinates, that 

 is to say, co-latitude and longitude, since ($> and 4/ are arbitrary; 

 and we may call them all imaginary units, as well as the three 

 original imaginaries, i,j, k, from which they are derived. To 

 distinguish one such root or unit from another, we may de- 

 note the second member of the formula (L.) by ?V^, or more 

 concisely by i^, if R denote (as before) that particular point of 

 the spheric surface (described about the origin as centre with 

 a radius equal to unity) which has its co-latitude = 4>, and its 

 longitude =\I/. We shall then have 



i^ = ia. +7/3 + ky, i\--\, . . . (U.) , 

 in which 



a = cos ($), /3 = sin <^ cos vj/, y = sin ^ sin ^, 

 u, |8, y being still the rectangular co-ordinates of R, referred 

 to the centre as their origin. The formula (I.) will thus be- 

 come, for any spherical triangle, 



(cos R + in sin R) (cos R' + i^/ sin R') = — cos R" + /R//sin R". (F.) 



8. To separate the real and imaginary parts of this last 

 formula, it is only necessary to effect a similar separation for 

 the product of the two imaginary units which enter into the 

 first member. By changing the angles R and R' to right 

 angles, without changing the points R and R' upon the sphere, 

 the imaginary units /r and i^i undergo no change, but the 

 angle R" becomes equal to the arc R R', and the point R" 

 comes to coincide with the positive pole of that arc, that is, 

 with the pole to which the least rotation from R' round R is 

 positive. Denoting then this pole by P", we have the equa- 

 tion 



^R^R' = ~ cos R R' + ?p//sin R R', . . . (iVI.) 



which is included in the formula (I'.), and reciprocally serves 

 to transform it; for it shows that while the comparison of the 

 real parts reproduces the known equation 



cos R cos R'-sin R sin R'cos R R'= -cos R", . (K'.) 

 R2 



