Section III, 1889. [ 7 ] Trans. Roy. Soc. Canada. 



II. — Notes on MatJiemafical Physics. 



By Prof. J. Loudon, M.A., University of ïoronto. 

 (Read May 7, 1889.) 



The relations between the snm of the moments {G) and the virial (F) of a set of 

 co-planar forces may be established, as follows : — 



The line for which V = 2£(Xx+ Yi/) vanishes may be obtained from the fact that V is 

 equal to the svim of the moments of the forces round the origin when each is turned 

 through a right angle. 



Hence if a set of co-planar forces be reduced to a resultant force R at O and a 

 couple G, the line for which the sum of the moments vanishes is, as we know, at a 



G 



distance 00' = — from the plane GOR, whilst the line for which the virial vanishes is 

 R 



V 



parallel to 00' at a distance 00" = — . The coordinates of C, the intersection of these 



R 



two lines, are, therefore, immediately seen to be 



V.Y-j-GY VY—GX 



where X, Y are the components of R. 



Let now C be taken as origin and CO', CO" as axes. Then C is an astatic centre ; 

 for, on turning each force through any angle ^, the sum of the moments round C becomes 



2,Pr sin (^9+a) = Cos 6 2.Pr sin rt+sin ft S.Pr cos a 

 = 0, 



where r is the distance of the point of application of P, and « the inclination of P to r. 



C may also be shown to be the astatic centre by compounding the original forces 

 with their equimultiples each turned through a right angle, for which equimultiples the 



