280 



dicular to the plane of the triangle, and in magnitude repre- 

 sents the double of its area; while the numerator is, as we 

 have just seen, in direction tangential to the circle at a, and its 

 length represents the product of the lengths of the three sides, 

 or the volume of the solid constructed with those sides as 

 rectangular edges. We may add, that this tangential line 

 ABCA is distinguished from the equally long but opposite tan- 

 gent ACBA to the same circle abc at the same point a, by the 

 condition that the former is intermediate in direction between 

 AB (prolonged through a) and ca, while the latter in like 

 manner lies between ac (prolonged) and ba : or we may say 

 that the line abca touches, at a, the segment alternate to that 

 segment of the circle abc which has ac for base, and contains 

 the point B ; while the opposite line acba touches, at the 

 same point, the last mentioned segment itself. The condition 

 for the diameter aa^ becoming infinite, or for the three points 

 ABC being situated on one common straight line, is 



V. ABC - 0. (17) 



This formula (17) is therefore, in this notation, the general 

 equation of a straight line in space; (15) is the general 

 equation of a circle ; (14) of a plane ; and (6) of a sphere.* 



which he inadvertently used as interchangeable in his first communication to the 

 Academy : and to make them satisfy the two separate equations, 



Q X Q-' q' = q'; 



q' 

 — X Q = Q • 

 Q 



He proposes to confine the symbol q' -r Q to the signification thus assigned 

 for the latter of the two symbols which have been thus defined, and which, on 

 account of the non-commutative property of multiplication of quaternions, 

 ought not to be confounded with each other. 



* The simpler equation of scalai- form, s . abc = 0, also represents a spheric 

 surface, if b be regarded as the variable point; but a plane, if b be fixed, and 

 either a or c alone variable. 



