382 Sir W. Rowan Hamilton 07i Quaternions. 



therefore that these two vector-factors are either exactly 

 similar or exactly opposite in direction ; since otherwise their 

 product would be a quaternion, having always a vector part, 

 although the scalar part of this quaternion-product (o-— x)p 

 might vanish, namely by the factors becoming perpendicular 

 to each other. Such being the immediate and general signi- 

 fication of the equation (12.), the justification of our establish- 

 ing it in the present question is derived from the consideration 

 that the radius vector p, drawn from the centre a to the sur- 

 face E of the ellipsoid, has, by the construction, a direction 

 either exactly similar or exactly opposite to the direction of 

 that guide-chord of the auxiliary sphere which is drawn from 

 A to D, that is, from the end of the radius denoted by x to the 

 end of the radius denoted by cr. For, that the chord so drawn 

 is properly denoted, in length and in direction, by the symbol 

 (T — X, follows from principles respecting addition and subtrac' 

 tion of directed lines, which are indeed essential, but are not 

 peculiar, to the geometrical applications of quaternions; had 

 occurred, in various ways, to several independent inquirers, 

 before quaternions {as 2)roducts or qnotieiits of' directed lilies 

 in space) were thought of; and are now extensively received. 

 41. The two equations (11.) and (12.) are evidently both 

 satisfied when we suppose (r = x; but because the point d is 

 in general different from a, we must endeavour to find another 

 value of the vector o-, distinct from x, which shall satisfy the 

 same two equations. Such a value, or expression, for this 

 sought vector <t may be found at once, so far as the equation 

 (12.) is concerned, by observing that, in virtue of this latter 

 equation, o- — x must bear some scalar ratio to p, or must be 

 equal to this vector p multiplied by some scalar coefficient x, 

 so that we may write 



(rz=zx. + xp; (13.) 



and then, on substituting this expression for cr in the former 

 equation (11.), we find that x must satisfy the condition 



T(x+^p) = Tx, (14.) 



in which this sought coefficient x is supposed to be some 

 scalar different from zero, that is, in other words, some posi- 

 tive or negative number. Squaring both members of this last 

 condition, and observing that by article 19 the square of the 

 tensor of a vector is equal to the negative of the square of that 

 vector, we find the new equation 



-(x + jrp)2=-x2 (15.) 



But also, generally, if x and p be vectors and x a scalar, 



(x + a;p)^ = x^ + ir(xp -f px) -f- a;'^^^ ; 



