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Sir W. Hamilton examines, in great detail, the composition of 

 the two conjugate quantities H4, He, which are each of the 

 thirtieth dimension relatively to the five original quantities 

 a;', . . . x^; and arrives at the conclusion that neither H4 nor 

 He is a symmetric function of those five quantities x', . . . x^, 

 though each is symmetric relatively to four of them. He 

 finds also that these two quantities H4 and He are not gene- 

 rally equal to each other, but difier by the sign of an imagi- 

 nary radical, namely, 



when they are fully developed, in consistency with Professor 

 Badano's definitions. Analogous results are obtained for 

 the two quantities H3, H5 ; and these general results are veri- 

 fied by applying them to a particular system of numerical 

 values of the five quantities x', . . .x^. It is shown also that 

 the three quantities H7, Hjs, Hig, are neither independent of 

 the arrangement of those five quantities x, nor (generally) 

 equal to each other. And thus, although Hi is symmetric, 

 and H2 vanishes. Sir W. H. conceives it to be proved that 

 Professor Badano's expressions, for the twenty-four values 

 of Lagrange's function t^, give no assistance towards the 

 solution of the general equation of the fifth degree, and 

 therefore that the same method could not be expected to 

 resolve equations still more elevated, even if we were not in 

 possession of an a priori proof that no root of any general 

 equation above the fourth degree can be expressed as a func- 

 tion of its coefficients, by any finite combination of radicals 

 and rational functions. 



DONATIONS. 



Three Silver Coins, found at Rockingham, the seat of Vis- 

 count Lorton. Presented by C. T. Webber, Esq. 



Copper Medal, " The glorious attempt of LXIV. to pre- 



