JM. Girard on Navigable Canals. 113 
But these opposite dynamical effects, although successive, 
operate in equal times, and aecording to the same laws ; for 
they are effected while the lock is filled or emptied ; and 
this filling and emptying takes place exactly in the same 
length of time, as may be immediately deduced from the 
formule which express those of spouting fluids in contigu- 
ous vases, separated by vertical diaphragms, whence it fol- 
lows that the difference of these dynamical effects is the rig- 
orous expression of the active force lost in producing them. 
This loss is therefore 
x (a—(t" —t'))+t"e—t c= oe. 
2dly, When the expenditure of water y, is null, it is evi- 
dent that the dynamical effect downward is reduced to t’x, 
the product of the mass of the boat multiplied by the fall of 
the lock. 
The dynamical effect upwards, as in the preceding case, 
is represented by by ?@'x. 
Consequently the difference of these dynamical effects, or 
the loss of active force is 
x (t'--t')=n 0; 
because when y = 0, we have always x = t—-1’. 
3dly, When the expenditure y of water is negative, or, 
which is the same thing, when by the passage of the boats, 
ascending and descending, a certain volume of water is 
raised from the lower to the upper level of a lock, we have 
—y = —x + (t"—7). 
The Dynamical action downward remains equal to the 
product of the boat multiplied by the height of the fall, and 
the dynamical action upward becomes equal to the product 
of that same height multiplied by the sum of the masses of 
the boat and the water raised, or 
’n+o(—2+@¢'—2)): 
The loss of active force is therefore 
tha —te—a2(—a2+ (t"'—t'))==20. 
Therefore, whatever be the expenditure of water, the 
lift of the lock, and ‘the draft of water of the boats which 
pass through it, the loss of active force, indispensable in 
effecting the double passage of the boats, is always in pro- 
portion to the square of the height of the lift. 
Therefore, if we designate by h the whole rise of a canal. 
from its lowest to its summit level, and if this rise he di- 
Vou. IV.....No. 1. 15 os 
