﻿The Calculation of Centroids. 



247 



It is an important point in the treatment of the two 

 problems, "complex solvents " and "solvation," presented 

 in this and a succeeding paper, that in the " practical " or 

 "experimental" expression for yfr finally obtained the 

 "linear terms" by themselves satisfy the Gibbs fundamental 

 relation, for in perfect solution these terms alone remain. 

 And this is preserved in the successive approximations. 

 The relation also serves as a useful check upon the correct- 

 ness of the detail. 



Balliol College, 

 March 1922. 



XVII. The Calculation of Centroids. By J. G. Gray, D.Sc, 

 Car gill Professor of Applied Physics in the University of 

 Glasgow *. 



THE position of the centroid of a plane arc or area is 

 usually determined by the application of one or other 

 of the two theorems of Pappus. The methods described and 

 illustrated below seem, however, to be novel ; they are useful 

 in a great number of cases, including many to which the 

 theorems of Pappus do not apply. 



Fi<r. 1. 



Fig. 2. 



Consider a system made up of two masses M and m 

 (fig. 1) . Let the centroids of m and of the system M and m 

 be at a and G respectively. Now suppose the mass m moved 

 so that its centroid is brought to a'. G moves to G', where 

 GG' is parallel to aa' ; and we have (M + m) GG' '= m aa' . 



As a first example, consider the case of a circular arc AB 

 mass per unit of length m (say). Let be the centre of the 

 circle of which the arc forms part. Now suppose the arc 



* Communicated by the Author. 



