62 
The President stated that the remarkable researches re- 
specting algebraic triplets, made lately by Professor De 
Morgan and John T. Graves, Esq. in England, and here by 
the Rev. Charles Graves, had led him to perceive the follow- 
ing theorem : 
If the three symbols &, n, 2, or rather their squares and 
products, be supposed to satisfy the three following ‘* equa- 
_ tions of signification :” 
E (bn + cf) = a(n’ + 2), 
n (cf + af) =b (2? + &), 
 (a& + bn) = (6 + 0’); 
and if, by the help of these three equations, we eliminate any 
three of the six quadratic combinations &, n°, 2°, En, nZ, CE, 
from the development of the ‘* formula of multiplication,” 
(uf + vm + wZ) (0"E 4 y/’n + 22) 
= (af + yn + 20) (WE + yn + 22), 
and then treat the three remaining combinations of the same 
set (€?, &c.) as three entirely arbitrary and independent mul- 
tipliers: the three separate equations thus obtained between 
the fifteen real quantities abeuvwayca'y'2' vy" 2” will be 
such, that whether we project the four lines (w, v, w), (25 Y; 2), 
(x, y's 2’), (2”, y”, 2”), on the axis (a, b, c,) itself, or on the plane 
perpendicular to that axis, the four projections thus obtained 
will in each case form a proportion ; the proportionality of 
the projections on the axis being of the kind considered in 
ordinary algebra, and the proportionality of the projections 
on the plane perpendicular to the axis being of the kind con- 
sidered by Mr. Warren; that is to say, the lengths of these 
last projections are proportionals in the usual sense, and the 
rotation from the first to the second is equal to the rotation 
from the third to the fourth. 
Sir W. Hamilton has been able to prove this theorem by 
treating the three real equations between the fifteen rectan- 
gular co-ordinates a, b, c, &c. according to the known methods 
