43 



we may apply it merely to finding the sets of terms 

 a 4 , a 3 b t a 3 c, &c. 



2 b\ 



Nothing can be more convenient than the process when the entire 

 series of columns is required ; but it is very desirable to have a pro- 

 cess for the formation of any column apart from the others ; and the 

 object of the memoir is to investigate a rule for the purpose. It 

 is found that there is, in fact, a rule analogous in some measure to 

 Arbogast's, but which consists in operating simultaneously upon cer- 

 tain pairs of letters : the pairs which may be operated on are four in 

 number, but the conditions are such, that, as regards two of these 

 pairs, when the one is operated on, the other is not, and for the 

 same term the number of pairs operated on is therefore three at 

 most. 



II. " On the Method of Symmetric Products, and on Certain 

 Circular Functions connected with that Method." By the 

 Rev. ROBERT HARLEY, F.R.A.S., Corresponding Member 

 of the Literary and Philosophical Society of Manchester. 

 Communicated by ARTHUR CAYLEY, Esq. Received Oct. 

 18, 1860. 



(Abstract.) 



After briefly adverting to his own and to Mr. Cockle's earlier re- 

 searches on the subject, the author proceeds in the first section of 

 his paper to give a concise systematic exposition of the method of 

 symmetric products, and to indicate a new application of the method 

 to the solution of the lower equations. Hitherto, in applying the 

 theory to the solution of cubics and quartics, an auxiliary equation 

 has been introduced, the symmetric product of which equation has 

 been made to vanish. The peculiarity of the process here employed, 

 is that the auxiliary equation is dispensed with, and the symmetric 

 product remains finite. 



In the second section the author deals with circular functions. 

 The structure of such functions is considered, and a calculus devised 

 by means of which operations upon them may be materially abridged. 

 The new cyclical symbol (S f ) is defined ; some of its applications are 



