170 GRAPHIC AND ANALYTIC [CHAP. XII. 



113. Inflection Tertlcals, Draw a line from P through 

 the support D, and through its intersection with cb draw a 

 vertical 3 (Fig. 78). This vertical we call the inflection ver- 

 tical. 



The equation of the line c b is 



where m l = D C, n I = C D, i = Ci'. The origin being at D. 

 For the line C, 



where % = D ^. 



If in this last equation we make x = <z, we have for the or- 

 dinate at P, 



mi fa a) 



in ' 

 and hence for the line P D, 



m l fa a) 

 y == H - -as. 

 ^a 



For the intersection of P D with b c then 

 m* mi fa ) 



7 .g + Wli = -^ - OJ. 



nl ^ !# 



Hence 



*!<*(**-*) . 



X = 77 - 7-7 j - ^ -- ; .... (1) 



fa a) (n I *) *! a 



"We see at once that the value of x is independent of m^ or 

 D C, hence the intersection of P D and c 5 lies always in the 

 same vertical, whatever be the position ofPC. In other words, 

 if the three sides of a triangle pass always through three fixed 

 points (i 1 , D, ^), and two of the angles (P and c) be always in 

 the same verticals, the third angle must also always lie in the 

 same vertical. 



For the distance of the inflection vertical on the other side 

 of the loaded span (beyond E), we have similarly 



a, = ^^( l ~ a ) , 2 x 



(I a) fa + *j) ^ 2 



where I is the loaded span and ^ the distance of the inflection 

 point to the right of E. 



Equations (1) and (2) give the distances of the inflection ver- 

 ticals from the supports D and E. 



