Sir W. R. Hamilton on Quaternions. 11 



shall be understood to involve four separate equations between 

 their respective constituents, namely, the four following, 



XV = W', X=- ,»', 1/ =s_7/', S = z'. 



It will then be natural to define that the addition or subtrac- 

 tion of quaternions is effected by the formula 



Q±Q' — 'vo±'w' + i{a;±a!') +j{y ±7/) +k{z±z'); 

 or, in words, by the rule, that the sums or differences of the 

 constituents of any two quaternions, are the constituents of the 

 sum or difference of those two quaternions themselves. It will 

 also be natural to define that the product Q Q', of the multi- 

 plication of Q as a multiplier into Q' as a multiplicand, is ca- 

 pable of being thus expressed : 



+ ixid + i'^x3(^ + ijxi/ + ikxzf 



+jy'^ -^jiyx' +/j/y +i^3/«' 



+ kz'nd + Jcizx' + kjzi/ + k'^zz' ; 

 but before we can reduce this product to an expression of the 

 quaternion form, such as 



Q Q' = Q" = -K)" + i x'< +iy' + k is", 

 it is necessary to fix on quaternion-expressions (or on real 

 values) for the nine squares or products, 



i% ij, ik,ji,j%Jk, ki, kj, F. 

 2. Considerations, which it might occupy too much space 

 to give an account of on the present occasion, have led the 

 writer to adopt the following system of values or expressions 

 for these nine squares or products : 



^•2 ==/ = ^2 _ _ 1 . ....... (A.) 



ij=k,jk9i,ki-j\ ...... (B.) 



ji——hkj——i,ik-—j\ .... (C.) 



though it must, at first sight, seem strange and almost unal- 

 lowable, to define that the product of two imaginary factors in 

 one order differs (in sign) from the product of the same fac- 

 tors in the opposite order (J/= '-ij)- It will, however, it 

 is hoped, be allowed, that in entering on the discussion of a 

 new system of imaginaries, it may be found necessary or con- 

 venient to surrender some of the expectations suggested by the 

 previous study of products of real quantities, or even of ex- 

 pressions of the form x + iy, in which 2'^= — 1. And whether 

 the choice of the system of definitional equations, (A.), (B.), 

 (C), has been a judicious, or at least a happy one, will pro- 

 bably be judged by the event, that is, by trying whether those 

 equations conduct to results of sufficient consistency and ele- 

 gance, 



