304 M. P. S. Girard on Xavigahle Canals. 



nates of certain points of the negative branch of a logarith- 

 mic curve. 



From the nature of this law it follows that, continuing to 

 pass boats through this lock, the surface of (he upper level 

 will continue to rise, though without ever arriving at a 

 point where the difference between its level, and that of 

 the water below the lock, will be equal to the difference of 

 draft between the ascending and the descending boats. If 

 that limit could ever be attained, the relative height of the 

 two levels would change no more, and an indefinite num- 

 ber of boats might pass through the lock in both direccions 

 alternately, without producing either a loss or gain of wa- 

 ter io either level. 



This leads us to remark that, in all cases where we can 

 raise a portion of water from a lower to a higher level, it 

 is proper to profit of this faculty as far as we can dispose 

 of it to advantage — for we raise so much the less as there 

 is already a greater quantity above, as we may see from a 

 singie £flance at the law of its augmentations. 



We have thought it expedient to develope, with consid- 

 erable minuteness, the consequences of the double passage 

 through a single lock, because those consequences are sim- 

 ple and easy to understand ; but this supposition differs so 

 widely from ordinary circumstances that it was not permit- 

 ted to confine ourselves to the examination of this question 

 alone. We have therefore, taking a general view, embra- 

 ced the case of a navigable canal, consisting of an indefi- 

 nite number of unequal levels, connected by locks of une- 

 qual lifts ; supposing first that the double passages were ef- 

 fected successively through each lock, beginning with the 

 highest, we have sought the expression of the rise of water 

 on any level whatever of the canal, and we have found that 

 it depended not only on the extent of that level and the 

 fall of the lock which terminates it, but also on the ex- 

 tent of all the levels, and the fall of all the locks situated 

 above the one in question. 



In general, the rise of water in any given level, the 

 length of that level, and the fall of the lock which termi- 

 nates it, may be considered as the three co-ordinates of a 

 curved surface, so that from the equation of that surface, 

 the value of one of these three variable quantities may be 

 immediately determined, when the other two quantities 

 are known. 



