Sir W. R. Hamilton 07i Quaternions. 2^5 



i^ and /j^r, to be mutually rectangular.) that is, the arc R R' 

 = a quadrant, the two products (M.) and (N.) become +/p'/; 

 and thus, or by a process more direct, we might show that 

 if two imaginary units be mutually opposite (one being the 

 negative of the other), they are also mutually reciprocal. 



10. The equation (P.), which we shall find to be of use in 

 the division of quaternions^ may be proved in a more purely 

 algebraical way, or at least in one more abstracted from con- 

 siderations of directions in space, as follows. It will be found 

 that, in virtue of the definitions (A.), (B.), (C), every equation 

 of the form 



I. X A=» X , X 

 is true, if the three factors, «, x. A, whether equal or unequal 

 among themselves, be equal, each, to one or other of the three 

 imaginary units i,j, k; thus, for example, 



i.J k = {i .iz=i —Issk. k — )ij .k, 



J J i - U' -^= -«■= - 1 . «■= yj'i- 



Hence, whatever three quaternions may be denoted by Q, 

 Q'j Q"> we have the equation 



Q.Q'Q" = QQ'.Q"; (Q.) 



and in like manner, for any four quaternions, 



Q.Q'Q"Q"'=QQ'.Q"Q"'=QQ'Q".Q"', . (Q'.) 

 and so on for any number of factors ; the notation Q Q' Q" 

 being employed, in the formula (Q'.), to denote that one deter- 

 mined quaternion which, in virtue of the theorem (Q.), is ob- 

 tained, whether we first multiply Q" as a multiplicand by Q' 

 as a multiplier, and then multiply the product Q' Q" as a mul- 

 tiplicand by Q as a multiplier; or multiply first Q' by Q, and 

 then Q" by Q Q'. With the help of this principle we may 

 easily prove the equation (P.), by observing that 



^R ^'r' • ^*R' in = 2r • 2*R' m = — 4 = + 1 • 



11. The theorem expressed by the formulae (Q.), (Q'.), &c., 

 is of great importance in the calculus of quaternions^ as tend- 

 ing (so far as it goes) to assimilate this system of calculations 

 to that employed in ordinary algebra. In ordinary multipli- 

 cation we may distribute any factor into any number of parts, 

 real or imaginary, and collect the partial products ; and the 

 same process is allowed in operating on quaternions : quater- 

 nion-multiplication possesses therefore the distributive charac- 

 ter of multiplication commonly so called, or in symbols, 

 Q(Q'+Q")=:QQ' + QQ", (Q + Q')Q"= QQ"-^ Q'Q",&c. 

 But in ordinary algebra we have also 



QQ'=Q'Q; 



