61 
e+y+zand a +y?+2°--vy—y2z—24, 
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 
formule concerning the functions A, mM, and N, analogous to 
the fundamental formulz 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 we may frame a theory of y 
multiplets, admitting a like interpretation. In order to ac- 
complish this we must assume a symbol x, such that «"(1)=1, 
whilst 1,«(1), «°(1), «°(1)... &c. are looked upon as units abso- 
lutely differing in kind as much as unity differs from / — 1. 
The development of e? into the form a + KB + xy + &e. will 
give usa set of n functions, a, 3, y-..- each depending upon 
one variable @: and again, the expansion of ex textvt..-- 
furnishes us with a series of m functions, a, B, r, &c, each 
depending upon (n—1) variables 9, x, ~, &c. The multiplet 
a+b +x«°c + &c. being now written in the form m(a+ «B+ 
er + &c.), which is equivalent to meetx| yb &e..-.) it 
is evident that, if we call m the modulus, and ¢, x, ~, &c. the 
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 (w, x, y, z) by the 
aid of the symbol «, which is a pure imaginary fourth root of 
positive unity, we shall find that the quantities [w+a+y+2], 
[(w+y)—(#+2)], and [(w—y)? + (w—z)*] 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, », and » 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 
m(AX +m + ev). 
