Sir W. Rowan Hamilton on Qtiaternions. 283 



adding therefore x^ to both members of (15.), dividing by — .r, 

 and then eUminating x by (13.), which is done by merely 

 changing xp + xp'^ to a-p, we find the equation 



<xp + pK = 0; (16.) 



and finally 



(T=- —pKp~'^'. (17.) 



SO that the expression already assigned for the vector from c 

 to D, presents itself as the result of this analysis. And in fact 

 the tensor of this expression (17.) is equal to Tx, by the ge- 

 neral rule for the tensor of a product, or because (— pxp~')^ 

 =pxp~Yxp~i=px^p"' = x'^, since x^ is a (negative) scalar; while 

 the product ((r — x)p, being = — (xp + px), is equal, by article 

 20, to an expression of scalar form. 



42. Conversely if, in any investigation conducted on the 

 present principles, we meet with the expression —pxp~^, we 

 may perceive in the way just now mentioned, that it denotes 

 a vector of which the square is equal to that of x ; and that, if 

 X be subtracted from it, the remainder gives a scalar product 

 when it is multiplied into p : so that, if we denote this expres- 

 sion by cr, or establish the equation (17.), the equations (11.) 

 and (12.) will then be satisfied, and the vector o- will have the 

 same length as x, while the directions of <r — x and p will be 

 either exactly similar or exactly opposite to each other. We 

 may therefore be thus led to regard, subject to this condition 

 (17.) or (16.), the two vector-symbols o- and x as denoting, in 

 length and in direction, two radii of one common sphere, such 

 that the chord-line cr— x connecting their extremities has the 

 direction of the line p, or of that line reversed. Hence also, 

 by the elementary property of a plane isosceles triangle, we 

 may see that, under the same condition, the inclination of o- 

 to p is equal to the inclination of x to — p, or of — x to p ; in 

 such a manner that the bisector of the external vertical angle 

 of the isosceles triangle, or the bisector of the angle at the 

 centre of the sphere between the two radii o- and — x, is a new 

 radius parallel to p, because it is parallel to the base of the 

 triangle (acd), or to the chord (ad) just now mentioned. 

 And by conceiving a diameter of the sphere parallel to this 

 chord, or to p, and supposing — x to denote that reversed 

 radius which coincides in situation with the radius x, but is 

 drawn from the surface to the centre (that is, in the recent 

 construction, from a to c), while <r is still drawn from centre 

 to surface (from c to d), we may be led to regard <r, or — pxp~\ 

 as the reflexion of — x with respect to the diameter parallel to 

 p, or simply with respect to p itseUi as was remarked in the 

 38lh article ; since the vector-symbols p, cr, &c. are supposed, 



