TRANSACTIONS OF SECTION J. 493' 



e2 + (e - d) (2rt + 46 + 4c + Sd) = (4^ - 3-) ^^'^ ^^\ 



On subtracting these two equations and simplifying, we get 

 e" + {e- d) (2rt + 46 + 4c + M) = 

 Similarly, to determine d, we have — - 



d-" + {d - c) (2a + 46 + 3c) = (3^ - 2«) ^^^JLl , 

 and similarly for the other widths. 



Thus, writing x for the unknown quantity in each case, we 

 have the following equations to solve in order to determine the 

 proper widths of the wall at each point in succession, starting 

 from the top : — 



a;2 + (x-a)a = 62-5 — ... ... ... (1.) 



The solution of this equation determines h. Then, we have, 

 to determine c, — 



x^ + {x - h) (2a + 36) = 62-5 (2^' - I-'') ^t ... (2.) 



To determine d, — 



x" + {x - c) (2a + 46 + 3c) = 62-5 (3^^ - 0:') _ ... (3.) 

 To determine c, — 



^2 + (a; - d) (2a + 46 + 4c + 3f7) = 62-5 (4-' - 3") ^t (4.) 



&c. 



The equations when put into this form lend themselves 

 very easily to a graphical solution. The typical form of the 

 equation is, — 



x" + {x — p) q = r. 



The positive root of this equation is the same as the positive 



value of X, which satisfies the two simultaneous equations, — 



lOy = x" — r. I 



10?/+ {x-p)q = o.) 



The first of these equations represents a parabolic curve. 



The curve lOy = x^ is very easily set off, as in Plate XY. This 



curve, once plotted, answers for all the equations, as the 



equation 10?/ =x^ — r represents the same curve with the axis 



of X moved upwards to a distance - , as represented by one of 



the red lines in the plate. The equation lOy -\- {x — p) q = o 

 represents a straight line which passes through the points- 



x = o,y= -^ Siud x=p,y = 0. The abscissa of the point of 

 intersection of the curve and straight line gives the positive 



