63 
of algebraical geometry. He has also arrived at simplifi- 
cations of this proof by introducing the ordinary imaginary 
Vv —1, treated by the ordinary rules; but his first investiga- 
tion, and the one which he prefers, has been founded on the 
rules of the calculus of quaternions, and consists in resolving 
by those rules the system of the two equations : 
ap” + pa _ap+pa_ ap”—p"a _ ap — pa | 
ap’ +p'a av+va’ ap’—p'a  av—va’ 
in which 
ata + jb+ke, 
v= tu+tyju + hw, 
p= intjy + ke, 
p’ = iz’ + jy! + kz’, 
p= ia” 4 Jy” +hz", 
i,j, k, being (as in former communications respecting quater- 
nions) three imaginary units connected by the nine non-linear 
relations 
oy k= —1; yok, jkai, i=]; jpi——h, j= —i, tha —j. 
The phrase ‘‘ equations of signification” is borrowed from 
Mr. De Morgan. Ifthe theorem be particularized, so as to 
correspond to that gentleman’s system of triplets, by making 
ozo we 0,56 => b=" = & 
then the equations of signification reduce themselves to 
=n, w= —li, C= — én, 
and the formula of multiplication resolves itself into the three 
relations 
aw” = we! + yz! + zy, 
y” = ay’ + yx’ — 22’, 
2" = az! + za/ — yy’. 
_ On the other hand, some of the results of the systems of the 
two Messrs. Graves may be reproduced by taking the same 
unit-line, u=1, v =w = 0, but employing that other axis 
for which a =b—=c. The equations of signification give 
then, more simply, 
