THE IMPROVEMENT OF RIVERS. 



Then 



IfP,-P, 



,, 



X.Z- \Z> , p i v 



x-z ''"' ' 



aX . Z . - Z* 

 X-Z 



For the gate to move, P, must be greater than P,. 



Let the relation be represented by P, = P,, being a proper fraction. Then 



Combining this with an equation for the length of the gate, which is taken as unity, 

 we find 



n 

 i 



(i- n) 



MAR TRAP OM THE KARNF 



- Y+ i. 



FIG. 24. OLD BEAR-TRAP CURVES. 



By substituting values of n between zero and i we find the corresponding values of 

 X and Y. From this was platted the accompanying curve (Fig. 24) showing the apexes 



of the ordinary bear-trap when at 

 full height. From it the lengths 

 can be scaled, and it will be noted 

 that the range is very wide. To 



secure safe results, however, it is 



"*"*<> \ 



X \ best that n should not exceed -fy 



X x\ 



X \ 



or 



, 



Y// 



FIG. 25. 



Parker Bear-trap. The general 

 solution for this type is given by 

 the equation 

 P 4 P, .cos P, cos j-.sin = o, 



the nomenclature being as shown in Fig. 25. From this as a basis the accompany- 

 ing tables were deduced, and the curve of proportions platted (Fig. 26), the analysis 



