112 JM. Girard on Navigable Canals. 
sis sufficiently justifies its maintenance. In what follows 
we shall therefore suppose it established. 
Hitherto we have only considered the actual expense of 
water, positive, null, or negative, which is occasioned by the 
passage through the lock or canal; but at every double pas- 
sage at the lock there is a quantity of movement given not 
only to the water expended, but also to the ascending and 
the descending boats. There is produced, then, a certain 
quantity of dynamical action, which we shall now endeavour 
to appreciate. 
I shall denominate, according to the common custom, 
dynamical action or effect, the product of a certain weight 
by the vertical distance which it goes through, either in as- 
cending or in descending, whether its movement be uniform 
or uniformly accelerated, during the unit of time. 
This dynamical action or effect is always equivalent, as 
may easily be shown, to the active force (force vive) of a 
certain mass, which should move with a certain velocity ; 
thus, in other words, it remains for us to determine the ex- 
pense of active force necessary to raise one boat and lower 
another, through the lock of a canal. 
The general equation which expresses the relation be- 
tween the lift of a lock, its expense of water, and the draft 
of water of the boat, is, as we have seen above, 
=r—(t’—t’). 
From this we shall deduce the value of the dynamical ac- 
tion employed at each double passage for each of the three 
cases, where the expense y of water is positive, null, or neg- 
ative. 
1°. The quantity y being positive, itis evident that the 
volume of water «—(t’’—t’) which it represented, descends 
from the upper to the lower level, that is, from the height x: 
the dynamical action of this volume of water is therefore 
x (a—(t”—t’)). 
Moreover the boat ¢’” descends the same distance: its dy- 
namical action is consequently = ¢’ «. 
The sum of these two actions which operate downwards, 
estimated in a vertical direction, is therefore : 
x (w—(t’—t))4+t'"2x. 
The only mass which is raised by this operation is that of 
boat t', which passes from the lower to the upper Jeval: its 
dynamical action, upward, is therefore f’ x. 
