176 Major P. A. MacMahon. On the Symmetrical [Feb. 0, 



February 6, 1890. 



Mr. JOHN" EVANS, D.C.L., Treasurer and Vice-President, in the 



Chair. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Papers were read : — 



I. " A New Theory of Colour-blindness and Colour-perception.'' 

 By F. W. Edridge- Green, M.D. Communicated by Dr. 

 Lauder Brunton, F.R.S. Received January 28, 1890. 



II. "Memoir on the Symmetrical Functions of the Roots of 

 Systems of Equations." By Major P. A. MacMahon, 

 Royal Artillery. Communicated by Professor Greenhill, 

 F.R.S. Received January 30, 1890. 



(Abstract.) 



The object of the present memoir is the extension to systems of 

 algebraical quantities of the new theory of symmetric functions which 

 has been developed by the author in regard to a single system in 

 Volume 11 and succeeding volumes of the 'American Journal of 

 Mathematics.' In the theory of the single system the conceptions 

 and symbolism are to a large extent arithmetical, and are based upon 

 the properties of single integral numbers and their partitions into 

 single integral parts. In this sense the theory may be regarded as 

 being unipariite. In the present generalisation to the case of nv 

 systems of quantities the fundamental ideas proceed, not from a 

 single number, but from a collection of m single numbers. In regard 

 to number, weight, degree, part, and suffix, the collection of m 

 numbers invariably replaces the single number of the theory of the 

 single system. In this view the theory of the m systems is ra-partite. 



The quantities, to which the symmetric functions relate, may be 

 regarded as the solutions common to m non-homogeneous equations 

 each in m variables. Schlafli, in the ' Vienna Transactions ' (Denk- 

 schriften) for 1852, added another linear non-homogeneous equation 

 in m variables, anc then forming the eliminant of the m -f 1 equations, 



