Sir W. R. Hamilton on Quartenions. IS 



4; is the longitude of R ; and let this point R be called the re- 

 ■presentative point of the quaternion Q. Let R' and R" be, in 

 like manner, the representative points of Q' and Q"; then the 

 equations (G.) express that i7i the spherical triangle R R' R", 

 formed hy the representative points of the tixio factors and the 

 product (in any multiplication of two quaternions), the angles 

 are respectively equal to the amplitudes of those t'wo factors^ and 

 the supplement of the amplitude of the product (to two right 

 angles) ; in such a manner that we have the three equations : 



R = S, R' = 9', R" = ,r-6" (H.) 



5. The system of the four very simple and easily remem- 

 bered equations (E.) and (H.), may be considered as equiva- 

 lent to the system of the four more complex equations (D.)j 

 and as containing within themselves a sufficient expression of 

 the rules of multiplication of quaternions ; with this exception, 

 that they leave undetermined the hemisphere to which the 

 point R" belongs, or the side of the arc R R' on which that 

 product-point R" falls, after the factor-points R and R', and 

 the amplitudes 9 and 6' have been assigned. In fact, the equa- 

 tions (E.) and (H.) have been obtained, not immediately from 

 the equations (D.), but from certain combinations of the last- 

 mentioned equations, which combinations would have been 

 unchanged, if the signs of the three functions, 



3/2;' — zj/', zc^ — x^^ xy^—ya^^ 

 had all been changed together. This latter change would 

 correspond to an alteration in the assumed conditions (B.) and 

 (C), such as would have consisted in assuming ij = — h, 

 ji=. +k, &c., that is, in taking the cyclical order kji (instead 

 oi ij k)y as that in which the product of any two imaginary 

 units (considered as multiplier and multiplicand) is equal to 

 the imaginary unit following, taken positively. With this 

 remark, it is not difficult to perceive that the product-poi^it R" 

 is always to be taken to the rights or always to the left of the 

 midtiplicand-point R', with respect to the multiplier -point R, 

 according as the semiaxis of + 2; is to the right or left of the 

 semiaxis of +3/, with respect to the semiaxis of + .r; or, in 

 other words, according as the positive direction of rotation in 

 longitude is to the right or to the left. This rule of rotation, 

 combined with the law of the modxdi and with the theorem of 

 the spherical triangle^ completes the transformed system of con- 

 ditions, connecting the product of any two quaternions with 

 the factors, and with their order. 



[To be continued.] 



