273 
During the absence of the Sub-Committee, the Chair 
was taken, pro tem., by George Petrie, Esq., V. P., when 
Sir William R. Hamilton read a paper on the expression 
and proof of Pascal’s theorem by means of quaternions; and 
on some other connected subjects. 
This proof of the theorem of Pascal depends on the fol- 
lowing form of the general equation of cones of the second 
degree : 
s. BB/B" = 0; (1) 
in which 
Bom wm (vaad's va ala”), 
B’ = v(v.a‘a’.v.a’a’), (2) 
Bre ¥ (Va av. a’ a), 
a, a’, a’, a’, a’, a", being any six homoconic vectors, and 
s, V, being characteristics of the operations of taking sepa- 
rately the scalar and vector parts of a quaternion. 
In all these geometrical applications of quaternions, it is 
to be remembered that the product of two opposite vectors is 
a positive number, namely, the product of the numbers ex- 
pressing the lengths of the two factors; and that the product 
of two rectangular vectors is a third vector rectangular to 
both, and such that the rotation round it, from the multiplier 
to the multiplicand, is positive. These conceptions, or defi- 
nitions, of geometrical multiplication, are essential in the 
theory of quaternions, and are hitherto (so far as Sir William 
_ Hamilton knows) peculiar to it. If they be adopted, they 
oblige us to regard the product (or the quotient) ef two in- 
clined vectors (neither parallel nor perpendicular to each 
other), as being partly a number and partly a line ; on which 
account a quaternion, generally, as being always, in its 
_ geometrical aspect, a product (or quotient) of two lines, may 
perhaps not improperly be also called aGRaMMARITHM (by a 
combination of the two Greek words: ypauprh and apAude, 
which signify respectively a line and a number). In this 
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