28 Sir W. Rowan Hamilton 011 Quaternions. 



obliging ourselves to retain here a term which has been used 

 in several other senses by writers on other subjects ; and the 

 word tensor has (it is conceived) some reasons in its favour, 

 which will afterwards more fully appear. Meantime we may 

 observe, as some justification of the use of this word, or at 

 least as some assistance to the memory, that it enables us to 

 say that the tensor of a pure imaginary, or vector, is the 

 number expressing the length or linear extension of the straight 

 line by which that algebraical imaginary is geometrically 

 constructed. If such an imaginary be divided by its own 

 tensor, the quotient is an imaginary or vector unit, which 

 marks the direction of the constructing line, or the region of 

 space towards which that line is turned', hence, and for other 

 reasons, we propose to call this quotient the versor of the pure 

 imaginary: and generally to say that a quaternion is the pro- 

 duct of its oison tensor and versor factors, or to write 



Q = TQ.UQ, 



using U for the characteristic of versor, as T for that of 

 tensor. This is the other general decomposition of a quater- 

 nion, referred to at the beginning of the present article; and 

 in the same notation we have 



T.TQ=TQ; T.UQ=1; U.TQ = 1; U.UQ=UQ; 



so that the tensor of a versor, or the versor of a tensor, is 

 unity, as it was seen that the scalar of a vector, or the vector 

 of a scalar, is zero. 



The tensor of a positive scalar is equal to that scalar itself; 

 but the tensor of a negative scalar is equal to the positive 

 opposite thereof. The versor of a positive or negative scalar 

 is equal to positive or negative unity ; and in general, by what 

 was shown in the 12th article, the versor of a quaternion is 

 the product of two imaginary units. The tensor and versor 

 of a vector have been considered in the present article. A 

 tensor cannot become equal to a versor, except by each be- 

 coming equal to positive unity ; as a scalar and a vector can- 

 not be equal to each other, unless each reduces itself to zero. 



20. If we call two quaternions conjugate when they have 

 the same scalar part, but have opposite vector parts, then be- 

 cause, by the last article, 



(TQ) 2 = (SQ + VQ)(SQ-VQ), 



we may say that the product of any two conjugate quaternions, 

 SQ + VQ and SQ — VQ, is equal to the square of their com- 

 mon tensor, T Q; from which it follows that conjugate versor s 

 are the reciprocals of each other, one quaternion being called 

 the reciprocal of another when their product is positive unity. 



