52 



1. If two couplets be multiplied together, the modulus of 

 the product will be equal to the product of the moduli of the 



factors. 



2. The amplitude of the product will be equal to the sum 

 of the amplitudes of the factors. 



One of the most important analytical properties of the 

 couplet x-\- \/ —\ .y consists in this, that the equation 



X -{■ \/ — \.y = Q 



is equivalent to two, viz. x ■=. Q and ?/ = 0. 



As regards the geometric interpretation of the foregoing 

 results, it is sufficient to observe that the symbol \/ — 1 has 

 been explained as denoting rotation through a right angle ; 

 whilst the couplet «+ V — 1. has been taken to represent 

 both the length and the direction of the right line drawn from 

 the origin to the point whose rectangular coordinates are x and 

 y : the length of this right line is obviously r ; and it is in- 

 clined to the axis of x at an angle equal to B. 



The problem now proposed by Mr. Graves is to assign two 

 distributive symbols, i and k, of such a nature that (1) the sum 

 or product of two triplets, x-\-iy -\-kz and a;, + lyi + k^i, shall 

 be itself a triplet of the same form : that (2) there shall be 

 theorems concerning the moduli and amplitudes of triplets, 

 similar to those already enunciated for couplets : that (3) the 

 equation a; + /?/ + ks = shall be equivalent to the three, 

 X zz 0, y — 0, z =. : and that (4) the symbols i and k shall 

 admit of a geometric interpretation analogous to that which has 

 been provided for the symbol \/ —\. 



The preceding conditions will be complied with, if we as- 

 sume I and K to be distributive symbols of operation, which, 

 when combined, are subject to the following laws : 



iK{a) zz a : Ki{a) =z a : K^(a) = i(a) : i^(a) = K{a). 



We must, at the same time, agree to regard i(l) and k(I) as 

 units absolutely ditfering in kind from each other and from 



