M. Girard on Navigable Canals. i0g 
The first boat finds the lock filled, and on entering it, for- 
ces back into the upper basin a volume of water equal to 
that which it displaces. The first expense is therefore neg- 
ative, and =—St,. 
The water being let off from the lock the boat is taken in- 
to the lower basin. 
The second boat finds the surface of the water in the lock 
on a level with that of the lower basin. Water must there- 
fore be let in from above to raise its surface to a level with 
the upper basin, this requires a volume of water = Sa, 
which, after the introduction of the second boat is reduced 
to to (e—ty). 
The expense of Sena ay third boat is 
a—tyi) 5 
And the expense for ie whole convoy, if we denote by 
m’ the number of boats of which it is composed will be ex- 
pressed by 
S(@ (m'—1)—t,,—ty—ts—We. ) 
The expense of water for a second convoy composed of a 
number m” of ascending boats will be 
ie 8 (net 4", 44)" +, &e.) 
That of a second descending convoy composed of 1m’ 
boats will be 
S (2 (m"—1)—1,’—1,,-?%,— &e. ) 
Therefore if we designate by Sy’, Sy’, Sy’”, &c. the par 
tial quantities of water expended by each ascending convoy, - 
and by Sz’, Sz”, Sz2’”, &c. the partial quantities expended 
by each descending convoy, we shall have, for the whole 
quantity expended from the upper level of a lock for any 
number K of convoys of boats ascending and descending al- 
fernately : 
S (yy +y"+, &e.b2/ +242!" 4, &o= 
S(we+ie4+¢ +t, he ° - 
$8 (n’ete 40", 46)", Be, 
+§ (aot! +t," +6" -+, &c. 
+S (x (m’—1)—t',, —ty—t';—, &e.) 
+S (2 (m"—-1)—t",,—tg@—t" 5 —, Se.) 
LL S (x (m" — 1) ei — too eas (Ce ; &c.) 
