with a Determination of the Limit beyond which it fails. 337 
that which arises from the introduction of another arbitrary factor into the last 
four terms of each octave ; for such new factor may be introduced without any 
infringement of the modular condition. Thus, 
(s? + bt? + cu’? + bev? + abew® + acx” + aby”? + az’ ) x 
(°° +60 + cw + bev® + abew? + acx®? + aby +az)= 
s+ bt? + cul? + bev’? + abew!”? + aca!” + aby!” + az!” 
in which a, 6, ¢, may be any values whatever. 
In the paper in which Mr. Graves first publicly announced the theorem for 
octaves (Phil. Mag. for April, 1845), a very interesting triplet formula is 
deduced by aid of the new theory of imaginaries. The formula adverted to is 
this, viz. : 
(ax +bay+cy’) (ax? +bx y,+ey') (ax, +ba,y,+ey;) = aaz,t+br,y,+ cy; 
An easy method of obtaining it, independently of that theory, has recently 
occurred to me; and as such independent verifications of the new doctrine, in 
the present stage of its progress, may not be superfluous, I may, perhaps, be 
permitted to offer it here, as a conclusion to this communication. 
It is already known (Barlow’s Theory of Numbers, p. 184), that 
(2° + bry + cy’) (a) + bx y, + cy) = % + bay) + Yer 
in which 
t= «x, — cyy, and y, = xy, + yx, + byy, 
Now, if instead of 0, c, we write a <, then it is plain, from the values of 2, y, 
here exhibited, that, in order to restore the integral character of the factors and 
of the resulting product, we shall have to write 
a(ax* + bry + cy’),(ax, + bry, + cy;) = ax, + bay, + Yo; 
similarly, 
a(ax, + bry + Cys) (AX: + bay, + Cys) = aaj + bay, + CY33 
consequently, 
a(ax’ + bay +cy’) (ax? +bary,+cy;) (ax, +ba,y, +cy;) = ax; +br,y,+ cy; 
Hence, the second member must be divisible by a*; therefore, each of the com- 
2 y 2 
