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(i) of multiplication of quaternions, and are obtained so 

 easily by developing and decomposing that formula, accord- 

 ing to the fundamental definitions (a) (b) (c). A new sort of 

 algorithm, or calculus, for spherical trigonometry, appears to 

 be thus given, or indicated. And by supposing the three 

 corners of the spherical triangle rr'r" to tend indefinitely 

 to close up in that one point which is the intersection of the 

 spheric surface with the positive semiaxis of x, each coordi- 

 nate a will tend to become — 1, and each )3 and y to vanish, 

 while the sum of the three angles will tend to become = tt ; 

 so that the following well known and important equation in 

 the usual calculus of imaginaries, as connected with plane 

 trigonometry, namely, 



(cos R + z sin r) (cos r' + 2 sin r') = cos (a 4- rO + « sin (r + r), 



(in which P = — 1), is found to result, as a limiting case, from 

 the more general formula (i). 



In the ordinary theory there are only two different square 

 roots of negative unity (+ i and — i), and they differ only 

 in their signs. In the present theory, in order that a qua- 

 ternion, w-\-ix+jj/ + kss, should have its square = — 1, 

 it is necessary and sufficient that we should have 



w = 0, x'' + y''-\'Z-'= + \; 



we are conducted, therefore, to the extended expression, 



\/ — 1 =i cos<j> -^-j sin (j> cos^ -f- k sin <j) sin ;//, (l) 



which may be called an imaginary unit, because its modulus 

 is = 1, and its square is negative unity. To distinguish one 

 such imaginary unit from another, we may adopt the nota- 

 tion, 



/k = ia +j(i + ky, which gives il — — 1, (l') 



R being still that point upon the spheric surface which has 

 a, /3, 7 (or COS0, sin ^ cos ^, sin <^ sinx//) for its rectangular 

 coordinates ; and then the formula of multiplication (i) be- 



