76 



IV. " A Third Memoir on Quantics." By ARTHUR CAYLEY, 

 Esq., F.R.S-. Received March 13, 1856. 



The object of the author in the present memoir is chiefly to col- 

 lect together and put upon record various results useful in the theo- 

 ries of the particular quantics to which they relate. The tables at the 

 commencement relate to binary quantics, and are a direct sequel to 

 the tables in the author's second memoir upon Quantics, Phil. Trans, 

 vol. cxlvi. (1856). The definitions and explanations in the next 

 part of the present memoir are given here for the sake of conve- 

 nience, the further development of the subjects to which they relate 

 being reserved for another occasion. The remainder of the memoir 

 consists of tables and explanations relating to ternary quadrics and 

 cubics. 



V. "Elementary Considerations on the subject of Rotatory 

 Motion/' By W. GRAY ATT, Esq., F.R.S. Communi- 

 cated by the Rev. J. B. READE,F.R.S. Received March 12, 

 1856. 



(Abstract.) 



The author explains the subject of rotatory motion in a series of 

 propositions by the use of prime and ultimate ratios. He com- 

 mences with a simple problem, determining the law of the forces by 

 which a particle of matter is deflected into any given course, and 

 pursues the inquiry by a consideration of the effect of these forces as 

 referred to a sphere, going on to the investigation of the character of 

 the motion of any body enclosed within an imaginary sphere, such 

 sphere itself being supposed to revolve upon two axes inclined at any 

 angle to each other. Hence the author determines the position of 

 some point of the circumscribing sphere momentarily at rest, or in 

 other words, of the resultant axis, from which he insists that all 

 centrifugal forces must really be calculated. 



His first application of the law thus enunciated is to the motion of 

 the peg-top ; and upon the principles he has already laid down, he. 

 shows that there is in the first instance rotation round a momentary 

 horizontal axis, calling up rotation round a momentary vertical axis ; 



