IS* Sir W. Rowan Hamilton on Quaternions. 



and consequently 



0-=— (o-— p)p((j--p)-'=— XpX-», . . (239.) 



if X be the directed chord tr—p itself, or any portion or pro- 

 longation thereof, or any vector parallel thereto. If then p, pj, 

 Pv •'• Pm be any series or succession of unit-vectors, while X^, 

 X2, ... X„ are any vectors respectively coincident with, or 

 parallel to, the successive and rectilinear chords of the unit- 

 sphere, connecting the successive points where the vectors 

 p • . Pn terminate; and if we introduce the quaternions, 



§'i = Xi; §'2=Vi; 9'3= Va^ ; &c., . . (240.) 



we shall have the expressions, 



pi = -9iPQ\~'-> pi=+Q^9^~'i ps^-qmr''* &c. (24.1.) 



Hence if we write the equation 



p«=p, (242.) 



to express the conception of a closed polygon of n sides, in- 

 scribed in the sphere, we shall have the general formula, 



Pqn={-^T9J>'^ (243.) 



which is immediately seen to decompose itself into the two 

 following principal cases, according as the number n of the 

 sides is even or odd : 



pq2m=+qimp\ (244.) 



P5'2m+1=— 5'2m+lf (245.) 



The equation (244.) admits also of being written thus, by the 

 general rules of quaternions, 



= y.pYq,^; (246.) 



and the equation (245.) resolves itself, by the same general 

 rules, into the two equations following : 



= S^2m+i; = S. qum+ip. . . • (247.) 

 We shall now proceed to consider some of the consequences 

 which follow from the formulae thus obtained. 



83. An immediate consequence of the equations (247.), or 

 rather a translation of those equations into words, is the fol- 

 lowing quaternion theorem: — If any rectilinear polygon, mth 

 any odd number of sides, be inscribed in a sphere, the contitmed 

 product of those sides is a vector', tangential to the sphere at the 

 Jirst corner of the polygon. It is understood that, in forming 

 this continued product of sides, their directions and order are 

 attended to ; the first side being multiplied as a vector by the 

 second, so as to form a certain quaternion product; and this 

 product being afterwards multiplied, in succession, by the 

 third side, then by the fourth, the fifth, &c., so as to form a 



