Sir W. Rowan Hamilton on Qualernions. 341 



which we are allowed to do, because the tensor of a product 

 is equal to the product of the tensors, we may observe that 

 while the denominator of the fraction in the first member is a 

 pure scalar, the numerator is a pure vector ; for the identity, 



i/> + l?x=S.(» + x)|5 + V.(«-x)p, . . . (126.) 

 gives 



S.(ip+p)(»-x)=0: .... (127.) 



the fraction itself is therefore a pure vector, and the sign T, 

 of the operation of taking the tensor of a quaternion, may be 

 changed to the sign TV, of the generally distinct but in this 

 case equivalent operation, of taking the tensor of the vector 

 part. But, under the sign V, we may reverse the order of 

 any odd number of vector factors (see article 20 in the Philo- 

 sophical Magazine for July IS'tS); and thus may change, in 

 the numerator of the fraction in (125.), the partial product 

 <p(i — x) to (i — x)pj. Again, it is always allowed to divide 

 (though not, generally, in this calculus, to multiply) both the 

 numerator and denominator of a quaternion fraction, by any 

 common quaternion, or by any common vector; that is, to 

 multiply both numerator and denominator into the reciprocal 

 of such common quaternion or vector: namely by writing the 

 symbol of this new factor to the right (but not generally to the 

 left) of both the symbols of numerator and denominator, above 

 and below the fractional bar. Dividing therefore thus above 

 and below by i, or multiplying into »~^, after that permitted 

 transposition of factors which was just now specified, and 

 after the change of T to TV, we find that the equation (125.) 

 of the ellipsoid assumes the following form: 



^y (-.V + f(x-xV') ^^ . . (128.) 



(< — x) + (x — x^» *) ^ ' 



the new denominator first presenting itself under the form 

 >?r^ — i, but being changed for greater symmetry to that 

 written in (128.), which it is allowed to do, because, under the 

 sign T, or under the sign TV, we may multiply by negative 

 unity. 



69. In the last equation of the ellipsoid, since 



X — x^<~*=x(» — x)«~^, 

 we have 



T(x-x2i-0 = TxT(i-x)Ti-»; . . (129.) 



and under the characteristic U, of the operation of taking the 

 versor of a quaternion, we may multiply by any positive 

 scalar, such as —x,~^ is, because x^ and x"^ are negative* 



• By this, which is one of the earliest and most fundamental principles 

 of the whole quaternion theory (see the author's letter to John T. Graves, 



