24-4- Sir W. R. Hamilton o» Qiiaternions. 



the comparison of the imaginary parts conducts to the follow- 

 ing symbolic expression for the theorem of the 6th article : 

 /r sin R cos R' + iR'sin R'cos R + /p/'sinRsinR'sinRR' = z^/zsinR". (K".) 



As a verification we may remark, that when the triangle (and 

 with it the arc R R') tends to vanish, the two last equations 

 tend to concur in giving the property of the plane triangle, 

 R+R'+R" = 7r. 



9. The expression (M.) for the -product of any two imagi- 

 nary units^ which admits of many applications, may be imme- 

 diately deduced from the fundamental definitions (A.), (B.), 

 (C), respecting the squares and products of the three original 

 imaginary units, 2,j, Ic^ by putting it under the form 



(2«+y/3 + /t:7)(2a'+i/3' + ^y') = 



- (««' + /3/3' + yi) + 2(/3y' -7/3') +i(ya' -ay') + ^(a^' -/3a'); (M'.) 



and it is evident, either from this last form or from considera- 

 tions of rotation such as those already explained, that if the 

 order of any two pure imaginary factors be changed, the real 

 part of the product remains unaltered, but the imaginary part 

 changes sign, in such a manner that the equation (M.) may be 

 combined with this other analogous equation, 



^R' ^R = ~ cos R R' — 2p// sin R R'. . , . (N.) 

 In fact, we may consider — ?'p// as =/p//, if P/' be the point 

 diametrically opposite to P", and consequently the positive 

 pole of the reversed arc R'R (in the sense already determined), 

 though it is the negative -pole of the arc R R' taken in its 

 former direction. 



And since in general the product of any two quaternions, 

 which differ only in the signs of their imaginary parts, is real 

 and equal to the square of the modulus, or in symbols, 



jt*(cos9 + ?Rsin fi) X ]«< (cos5 — z^sinfi) = ]«,% . . (O.) 



we see that the product of the two different products, (M.) and 

 (N,), obtained by multiplying any two imaginary units toge- 

 ther in different orders, is real and equal to unity, in such a 

 manner that we may write 



«r2r' •2r'«"r = 1; (P.) 



and the two quaternions, represented by the two products 

 2RfR/ and ?R'?R, may be said to be reciprocals of each other. 

 For example, it follows immediately from the fundamental de- 

 finitions (A.), (B.), (C), that 



ij .ji^ikx —k——Jc^=sl', 



the products ij and ji are therefore reciprocals, in the sense 

 just now explained. By supposing the two imaginary factors, 



