^14 Sir W. Rowan Hamilton on Qiiaternions, 



the same direction as the body decomposing or entering into 

 union, but under conditions in which its own affinity cannot 

 always be gratified. The catalytic body is therefore a sub- 

 stance which acts by adding its own affinity to that of an- 

 other body, or by exerting an attraction sufficient to effect 

 decomposition under certain circumstances, without being 

 powerful enough to overcome new conditions, such as elasti- 

 city and cohesion, which occasionally intervene and alter the 

 expected result. 



At the same time the theory is far from being fully proved ; 

 but if I have succeeded in rendering probable that the ca- 

 talytic force is only chemical affinity recognised under an 

 aspect which chemists have not been accustomed to view it, 

 and exerted under conditions which can only be developed 

 by close attention to details, it will not have been useless to 

 direct increased study to this interesting class of phenomena. 



XXXVI. 0« QiiaterJiions / or on a Nexv Si/stem of Imaginaries 

 in Algebra. By Professor Sir William Rowan Hamilton, 

 LL.D., V. P.R.I. A.^ F. R.A.S., Correspofidhig Member of the 

 Institute of France^ and of other Scientifc Societies in British 

 and Foreign Countries^ Andrews^ Professor of Astronomy in 

 the University of Dublin ^ and Royal Astronomer of Ireland. 

 [Continued from vol. xxx. p. 461.] 



33. T^OR the sake of those mathematical readers who are 

 •*- familiar with the method of co-ordinates, and not with 

 the method of quaternions, the writer will here offer an inves- 

 tigation, by the former method, of that general property of 

 the ellipsoid to which he was conducted by the latter method, 

 and of which an account was given in a recent Number of 

 this Magazine (for June 1847). 



Let X y z denote, as usual, the three rectangular co-ordi- 

 nates of a point, and let us introduce two real functions of 

 these three co-ordinates, and of six arbitrary but real con- 

 stants, I mnV m' n', which functions shall be denoted by ic and 

 V, and shall be determined by the two following relations: 



u{ll' -f mm' + nn') = Vx + fri'y + n'% ; 



i^{ll' 4- mm' + nn^Y = [ly — mxf + [mz — nyf + [nx — hf ; 



then the equation 



u^^v'^=i\ (1.) 



will denote (as received principles suffice to show) that the 

 curved surface which is the locus of the point x y zhtm ellip- 

 soid, having its centre at the origin of co-ordinates ; and con- 

 versely this equation u'^ + i^=l may represent any such ellip- 



