212 On a Theorem analogous to the Virial Theorem, 



that occurs is that which acts along the diagonal where 

 sin 2<£ = — 1. In fig. 1 opposed forces P act at the middle 

 points of the sides, but since in each case + a = O, the terms 

 containing P disappear. Hence R = 0. 



Fig. 1. 



Fig. 2. 



In fig. 2, where external forces P act diagonally at the un- 

 connected corners, sin (6 + a) = — 1, and since p = 2r, R= — P, 

 signifying that the diagonal piece acts as a tie under tension P. 

 In neither case would the weight of the members disturb the 

 conclusion. 



The forces exercised by the containing vessel upon a liquid 

 confined under hydrostatic pressure p contribute nothing to 

 the left-hand member of (4). The normal force acting inwards 

 upon the element of boundary ds is pds, so that 



X= -pdy, Y=pdx, 



and accordingly 



t[*Y + yX] = ±p$d(.v*-f), 



vanishing when the integration extends over the whole 

 boundary. 



Abandoning now the supposition that the particle at x, y 

 is at rest, we have 



d 2 y d' 2 x 



d?(xy) „ dx dy 



~dtT ~ dTdt " dt 2 



so that if m be the mass of the particle, X, Y the components 

 of force acting upon it, 



9m dx dl J _ m d \*y) jl t \ + „y . (r\ 



2m didt- m ~d?~ + * Y+ y x '' ' ' (6) 



or with summation over all the particles of the system, 



»s.^^-^x(«,)+S(-y+^.. . (7) 



We now take the mean values with respect to time of the 



