Sir W. Rowan Hamilton o« Quaternions. 513 



53. The equation (52.) gives 



(|a + ,)T = T. + JT = V-10, .... {55.) 



this last symbol V~'0 denoting generally any quaternion of 

 which the vector part vanishes; that is any pure scalar, or in 

 other words any real number, whether positive or negative or 

 null. Hence /«, + » and t denote, in the present question, two 

 coincident or parallel vectors, of which the directions are 

 either exactly similar or else exactly opposite to each other ; 

 since if they were inclined at any actual angle, whether acute 

 or right or obtuse, their product would be a quaternion, of 

 which the vector part would not be equal to zero. Accord- 

 ingly the expression (53.) gives this equation between tensors, 

 TjL. = T,; [56.) 



so that the symbols jtx, and i denote here two equally long 

 straight lines ; and therefore one diagonal of the equilateral 

 parallelogram (or rhombus) which is constructed with those 

 lines for two adjacent sides bisects the angle between them. 

 But by the last article, this bisector has the direction of t (or 

 of — t) ; and by one of those fundamental principles of the 

 geometrical interpretation of symbols, which are common to 

 the calculus of quaternions and to several earlier and some 

 later systems, the symbol jm< + i denotes generally the interme- 

 diate diagonal of a parallelogram constructed with the lines 

 denoted by ju, and » for two adjacent sides : we might there- 

 fore in this way also have seen that the vector ]«, + « has, in 

 the present question, the direction of +t. This vector /* + j is 

 therefore perpendicular to v, and we have the equation 



= S.v(/x, + i), or S.viu,= — S.v*. . . . (57.) 



But by {56.), and by the general rule for the tensor of a pro- 

 duct (see art. 20), we have also 



T.v/* = T.v«; (58.) 



and in general (by art. 19), the square of the tensor of a qua- 

 ternion is equal to the square of the scalar part, minus the 

 square of the vector part of that quaternion ; or in symbols 

 (Phil. Mag., July 1846), 



(TQ)2=(SQ)2-(VQ)2. 



Hence the two quaternions vjw, and vt, since they have equal 

 tensors and opposite scalar parts, must have the squares of 

 their vector parts equal, and those vector parts themselves 

 must have their tensors equal to each other; that is, we may 

 write 



(V.Vi*)2 = (V.v.)S TV. via = TV. VI : . . (59.) 

 Phil. Mag. S. 3. No. 2 1 1 . Suppl. Vol. 3 1 . 2 L 



