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XIV. 
A LECTURE NOTE ON THE RELATION OF THE THEOREM 
OF WORK TO THE THEOREM OF MOMENTS. 
By THOMAS PRESTON, M.A., F.R.U.I. 
[Read Frsruary 22, Received for publication Fepruary 24; Published 
JuNE 13, 1893.] 
‘THE moment of a force with regard to any point (being the pro- 
duct of the force and the distance of the point from its line of 
action) is clearly of the same dimensions as work, and consequently 
the theorem of moments, which states that the moment of the 
resultant of two (or more) concurrent forces is equal to the sum of 
the moments of the forces, must be in some way related to, if not 
identical with, the theorem of work which states that the virtual 
work of the resultant in the same case is equal to the sum of the 
virtual works of the component forces. It is undoubtedly impor- 
tant that students should have the relation between these theorems 
pointed out to them at the beginning of their course, so that they 
may have a clear grasp of the principles which they afterwards 
employ in so many departments, and a full knowledge of the 
ground on which each rests, and of how far they overlap each 
other, or are altogether independent. 
I have consequently been induced by these considerations to 
bring under your notice the following lecture note concerning the 
relation of the theorem of work to the theorem of moments. This 
relation will appear evident at once if we remark y 
that the work done by any force OX, during any 
displacement OP, is equal to the moment with re- 
gard to P of an equal force OY drawn through O 
at right angles to OY. For the work of OX, 
when O is displaced to P, is OX . OM, and the 
moment of OY with regard to P is OY. PN, Fig. 1. 
and these are obviously equal since we have taken OY = OX, and 
PM and PW are drawn at right angles to OX and OY respec- 
SCIEN. PROC. R.D.S., VOL. VIII., PART II. N 
