KUTTER'S FORMULA 179 



The unwieldy nature of the formula given above has led to 

 almost general use of graphical methods of solution. In the 

 first notice of the formula in the Swiss exhibit in Philadelphia, 

 1876, there was shown with the printed exposition a diagram 

 familiar to us in its English units, by means of which c could 

 be graphically determined. It is printed in the back of the 

 translation of Kutter's book by Hering and Trautwine, and can 

 be found in some of the pocket-books. Mr. Hering in his 

 translation gives tables for x and y, by means of which the 

 diagram can be replotted at any time. Similar tables are 

 given in Jackson's translation. 



But even with this diagram to aid in finding c, several 

 algebraic reductions need to be made before the real purpose 

 of the formula, that is, the value of v, is known. Trautwine 

 in his pocket-book devotes four pages to tabulating c for 

 different values of S, R, and n, when values of v might have 

 been given. After c is known the square root of the product 

 of R and S must be multiplied by c to get v. 



In Vol. VIII of the Transactions of the Am. Soc. C. E., p. i, 

 Mr. Hering gives a method by which the velocity can be at 

 once read from the diagram constructed for c. His reasoning 

 is very simple. The equation 



can be written 



1) C 



VR~V^/S' 



or the four terms are in a simple proportion, so that by plot- 

 ting the values of c and of Vi/S on one side, and of v and of 

 V R on another side, of an angle, the corresponding relations 

 will be represented by similar triangles. In Kutter's diagram 

 the coefficients c are already plotted on the vertical and the 

 values \^R on the horizontal axis; by plotting an additional 

 scale of grades on the former and of velocities on the latter 

 axis the graphical solution is complete by merely drawing 



