342 Sir W. Rowan Hamilton on Quaternions. 



scalars ; whereas to multiply by a negative scalar, under the 

 same sign U, is equivalent to multiplying the versor itself by 

 — 1 : hence, 



U(x-x2*-0 = -U(jc2i-i-x)=-U(x-i-i-i). . (130.) 



If then we introduce two new fixed vectors, rj and 6, defined 

 by the equations, 



>, = TiU(.-Jt); 9 = TxU(x-»-i-»); . . (131.) 

 and if we remember that any quaternion is equal to the pro- 

 duct of its own tensor and versor (Phil, Mag. for July 1846); 

 we shall obtain the transformations, 



.-jc=>jT-^^; x-xV»=-flT-^— ?^; . (132.) 



which will change the equation of the ellipsoid (128.) to the 

 following : 



TV^?^^=T(»-x) (133.) 



70. To complete the elimination of the two old fixed vec- 

 tors, », X, and the introduction, in their stead, of the two new 

 fixed vectors, )3, 9, we may observe that the two equations 

 (132.) give, by addition, 



,-x2,-»=(>;-9)T'-^; .... (134.,) 



taking then the tensors of both members, dividing by T > 



and attending to the expression (81.) in article 56, (Phil. Mag. 

 for May 1848,) for the mean semiaxis h of the ellipsoid, we 

 find this new expression for that semiaxis : 



^^"^-^^^w^r^ (^^^•) 



Esq., of October 17th, 1843, printed in the Supplementary Number of the 

 Philosophical Magazine for December 1844), namely by the principle that 

 the square of every vector (or directed straight line in tridimensional 

 space) is to to be regarded as a negative number, this theory is not merely 

 distinguished from, but sharply contrasted with, every other system of alge- 

 braic geometry of which the present writer has hitherto acquired any know- 

 ledge, or received any intimation. In saying this, he hopes that he will not 

 be supposed to desire to depreciate the labours of any other past or present 

 inquirer into the properties of that important and precious Symbol in Geo- 

 metry, V — 1. And he gladly takes occasion to repeat the expression of 

 his sense of the assistance which he received, in the progress of his own 

 speculations, from the study of Mr. Warren's work, before he was able to 

 examine any of those earlier essays referred to in Dr. Peacock's Report: 

 however distinct, and even contrasted, on several fundamental points, may 

 be (as was above observed) the methods of the Calculus of Quaternions 

 from those of what Professor De Morgan has happily named Double Al- 



eSBRA. 



