273 



Nineteentli Report of the Council of the Leeds Philosophical and 

 Literary Society at the close of the Session 1838-39.—% the 



Society. 

 Annuaire Magnetiqup et Meteorologique du Corps des Ingenieurs 



des Mines de Russie. Par A. T. Kupfer— Par k Ministre 



des Finances. 

 The American Almanac and Repository of Useful Knowledge for 



the year 1840.— 2?y the Amer. Phil. Society. 

 Voyage dans la Russie Meridionale, sous la direction de M. A. De- 



midofF. Livraisons 1-23. Atlas au meme, folio.— .By Count 



Demidvff. 

 Pilote Fran9ais Quatrieme partie comprenant les Cotes Septentrio- 



nales de France depuis I'lle Brebat jusqua Barfleurs.— Par 



la Ministre de la Marine. 

 Maps of the Ordnance Survey of England and Wales. Nos. 71, 



and 74. — By the Board of Ordnance. 

 Maps of the Ordnance Survey of Ireland (County Kildare), 45 



sheets By the Board of Ordnance. 



The following communications were read : — 

 1. Solution of a Functional Equation, with its application to 

 the Parallelogram of Forces and to Curves of Equili- 

 bration. By Dr Wallace, Emeritus Professor of Mathe- 

 matics. 



The functional equation to be resolved is this — 



f{x:)f{x,)=c{f{x^+x:)+f{x-x,)} 



In this equation x^ and x, denote any two values of the indefinite 

 variable x, and c is a constant quantity. The object of inquiry is 

 the forms of the function. 



It is proved that there are two forms of the function, which alike 

 satisfy the proposed equation : viz. 



/(a?)=ocos- (1) 



c 



./(^)=|{«"^r^} .... (2) 



In the first of these functions x increases, while /(a:) decreases ; 

 and in the second, x and/(j:) increase together. The first of these 

 functions is applied in the paper to the theorem of the Parallelogram 

 of Forces, whilst the theory of Curves of Equilibration is deduced 

 from the second function : these last, in their most general form. 



