with a Determination of the Limit beyond which it fails. 339 
the letters 2, m, n, 0, to be combined with my three letters, 2,7, k, into one common ima- 
ginary or symbolic system. Thus, as I had already (in October and November, 1843) 
communicated to him and to the Academy the fundamental equations of quaternions, 
namely, 
(a) 
S 
I 
= 
S 
a 
1 
= 
=~ 
= 
I 
> 
ae 
which may be concisely summed up in the formula 
Sap Sle aie= ale (b) 
so he proposed to found a theory of octaves on the following equations, 
Pepe Pale Sy ai = Sil, 
@ =jk =lm=on = —kj = —ml= —no, 
J =ki =n =mo=—ik = —nl = —om, 
k=y =lo=nm=—ji =—ol =— mn, 
: | 
=miz=nj =ok =—im=—jn = —kho, (c) 
m=tl =q =kn =—li =—jo =—nk, 
n =jl =i0 =mk=—]j =—oi = —km, 
0 =ni =jmakhl = —in = —mj=—lkh; | 
which he communicated to me, in a letter dated January 4, 1844, and which may be 
concisely expressed by the single but continued equation, 
C= Pa/e=(Pair Sr =o= = | j 
=ijk=ilm =ion=jln=jmo = klo=knm. J (d) 
«© In other words, as I had introduced the consideration of an imaginary triad, or 
ternary cycle of square roots of negative unity, namely, ik, in which each is = the pro- 
duct of the two that follow it in the cyclical succession, ihiyh, if those two factors be taken 
in their order (‘= jk, &c.), but is equal to the negative of that product, if the order of the 
two factors be reversed (t= — hj, &c.); so J. T. Graves extended this view to the consi- 
deration of seven such triads, that is tosay, my triad and six new ones formed on the same 
type, namely, 
ijk, ilm, ion, jln, jmo, klo, knm. (e) 
‘* And as I had shown that, with the equations (a) or (b), the product of two quater- 
nions is a quaternion, 
(w+ ix +jy + hz) (w + ia’ + jy + kz’) 
= w + ia’ + yy’ + hz’, (f) 
