273 



During- the absence of the Sub- Committee, the Chair 

 was taken, pro tern., 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.i3i3'/3" = 0; (1) 



in which 



/3 = V (v . aa'. V . a'" a'^, 1 



(i'=v(v.a'a"-v.a^a% (2) 



I3"=v{v.a"a"'.v.a''a), J 



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) of 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 aoRAMMARiTHM (by a 

 combination of the two Greek words ypafxfir] and apiO/iog, 

 which signify respectively a line and a number). In this 



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