61 



x + y + z and x^ + if + c-— xy -yz — zx, 

 the two real moduli which have previously been shewn to be- 

 long to the triplet {x, y, z.) — See page 53. 



Mr. Graves stated that he had obtained a multitude of 

 formulae concerning the functions A, m, and n, analogous to 

 the fundamental formulse'of trigonometry. Amongst the more 

 remarkable of these he pointed attention to one corresponding 

 to the well-known theorem of Moivre. 



By pursuing a similar course Me may frame a theory of 

 multiplets, admitting a like interpretation. In order to ac- 

 complish this we must assume a symbol k, such that k:"(1)=1, 

 whilst l,ic(l), K"(l), k'(1) • • • &c. are looked upon as units abso- 

 lutely differing in kind as much as unity differs from l/— 1. 

 The development of e"* into the form a + k/3 + k^7 + &c. will 

 give us a set of n functions, a, /3, 7 . . . • each depending upon 



one variable : and again, the expansion of e"* +'<''x+'''>/' -I- 



furnishes us with a series of n functions, a, b, r, &c , each 

 depending upon {n— 1) variables <p, x, '/'j &c. The multiplet 

 a _j_ K& -|- (c^c 4- &c. being now written in the form j«(a -f- kB -f- 

 K^r 4- &c.), which is equivalent to W2e'"?'4 "'x I "'./< l-&c. ...^ it 

 is evident that, if we call m the modulus, and 0, x> "^^ ^^- ^^^^ 

 amplitudes of the multiplet, we shall have the same theorems 

 concerning moduli and amplitudes that have been already 

 established in the case of the multiplication of couplets and 

 triplets. 



If, for instance, we form a quadruplet (tv, x, y, z) by the 

 aid of the symbol k, which is a pure imaginary fourth root of 

 positive unity, we shall find that the quantities [^w + x + y + z'], 

 [_(iv + y) — {x + z)^, and [_{w—yy^+ {x — s)^] are all moduli 

 of multiplication. The product of the three is equal to m*. 



Mr. Graves mentioned that his elder brother, Mr. John T. 

 Graves, had been the first to conceive the notion of employ- 

 ing the functions A, ju, and v in the interpretation of this 

 theory of triplets; but as they involve only one variable it is 

 not possible to bring a triplet in general into the form 



