Ill 



July 21, 1845. (Adjourned Extra Meeting.) 



SIR Wm. R. HAMILTON, LL.D., President, in the 

 Chair. 



Read, — The following Resolution of Council, passed on 

 16th July, 1845 : 



" Resolved, — That at the Adjourned Meeting of the 

 Academy, summoned for Monday, 21st July, it be recom- 

 mended to the Academy to authorize the Treasurer to sell 

 Stock to the amount of £400 sterling, for the purpose of dis- 

 charging existing liabilities." 



Resolved^ — That the Academy do approve of and adopt 

 the recommendation of Council, as now read ; and that the 

 Treasurer be authorized to sell Stock to the amount of £400, 

 for the purposes described. 



Mr. Petrie, V. p., having taken the Chair, the President 

 continued-a paper on the Applications of Quaternions to some 

 Dynamical Questions. — See Appendix, No. III. 



The President having resumed the Chair, the Rev. Charles 

 Graves read the following paper on two methods of solving 

 Biquadratic Equations. 



I. — An equation being supposed to have a pair of imagi- 

 nary roots, a -f \/ — 1.6 and a— \/ — \.b, if we diminish all 

 its roots by the quantity a, the transformed equation would 

 plainly have two roots differing only in their signs. This 

 consideration suggested the following mode of solving the bi- 

 quadratic equation, 



x^ -f- AjO;^ -\- K^iX -|- A4 = 0. (1) 



Let its roots be diminished by a, a quantity to be determined 

 by the condition that the transformed equation, found by sub- 

 stituting y + a for x, shall have two roots which differ only in 



