M P. S Girard on M'avigable Canals. 29S 



Whence we see, further,that the passage of a boat will 

 always produce a real elevationof water in the lock, whence 

 it descends, unless the number of boats already gone through 

 is infinite; and in this case the nunnber of terms 



(b+S/ +VB+SJ +VB+S>' +VB + SJ + 



^B+S/ ~ S 



Therefore 



S B + S 



«t'+M"+t«'"+M'^-f...M(„-,)=gT:g(D — a;)-- g— =D-a:; 



as we have already found. 



The fall of the lock then becomes x+D — x—T). Con- 

 sequently, the descent of an infinite number of boats 

 through the lock which terminates the level B, would ren- 

 der the fall of that lock, equal to the common draft of wa- 

 ter of the boats which pass through it ; but it is evident that 

 the fall can never attain that limit since the number of 

 boats can never become infinite. 



Substituting t^^ — t, for D in the general expression 



«(")=b+s(^-^KbTs) 



of the rise on the level of B by the passage of the nth boat, 

 it becomes 



«cn) = B+s(c'" - ^')-^)(Bq:^) ' 



and all tliat we have hitherto said of a simple succession of 

 descending boats, will apply to the double passage or to 

 the case of boats passing alternately up and down the locks. 

 If all the boats are equally charged, or in other words, if 

 t,,=t , the preceding expression becomes 

 L So; ^ B x"-' 



which is always negative, and indicates that the height of the 

 Vipper level diminishes instead of increasing. 



After a certain number of double passages, the primitive 

 height of the water in the level h will evidently, in this hy- 

 pothesis, be represented by 



Sx . B ' B^ B^ 



^~B-fSV^ + B+S'^(B+S)'+(B-}-S)=''^ ••■" "' • 



