168 Scientific Proceedings, Royal Dublin Society. 
tively. We have then the general elementary theorem that the 
work of a force is equal to the moment of an equal perpendicular 
force. Hence if we take the case of two 
forces OP and OQ, and their resultant R&, 
forming the sides and diagonal of a 
parallelogram, and if three others OP’, y 
OQ’, OR’, be drawn at right angles to . \ 
them, and equal to them respectively; NY ye 
then if O be displaced to O’ the works of - a 
the forces OP, OQ, OR, will be equal to 
the moments of OP’, OQ’, OR’, respectively, with regard to the 
point 0’; but since OP’Q’R’ is a parallelogram, it follows that 
the moment of &’ is equal to the sum of the moments of P’ and Q’; 
therefore the work ot R& is equal to the sum of the works of 
P and Q. 
By some such demonstration as this I think the beginner 
might be shown how these two theorems are related, and that 
when the theorem of moments is established the theorem of work 
follows as a corollary; or, in fact, when if has been shown that 
the triangle, having the diagonal OF of a parallelogram for base, 
and any point as vertex, is equal to the sum of the triangles 
having the same point for vertex, and the sides OP and OQ for 
bases, then we may write down as corollaries to this geometrical 
theorem—(1) the theorem of moments; (2) the theorem of work ; 
(3) the theorem (or parallelogram) of angular velocities, the latter 
following from the fact that, if a body rotates round any axis OP 
with an angular velocity measured by OP, then the velocity of 
any point O’ will be measured by the product of OP and the per- 
pendicular from 0’ on OP, that is, by the moment of OP with 
regard to 0’. Similarly the velocity of O’ due to a rotation round 
OQ will be equal to the moment of OQ, and the sum of these is 
equal to the moment of OF; therefore, &e. 
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Fig. 2. 
