262 THE BEACED AKCIT. [CHAP. TJV. 



and give rise to tedious and laborious computations in practice. 

 A method combining simple analytical results with graphical 

 construction similar to the preceding, will, however, obviate 

 these difficulties, and bring the subject fairly within the reach 

 of the practical engineer. 



In the present case, as before, the common intersection of the 

 weight and the reactions lies in a curve, the equation of which 

 may be found, and the curve itself thus plotted for any given 

 case. 



But this curve, or locus, ILK [PI. 24, Fig. 92] being con- 

 structed, in order to find the directions of the reactions which 

 now no longer pass through the ends of the arc A and B, it is 

 necessary to find and construct also the curve enveloped by these 

 reactions for every position of P ; that is, the curve to which 

 these reactions are tangent. If, then, these two curves are con- 

 structed, we have only to draw through L [Fig. 92] lines tan- 

 gent to this enveloped curve, and we have at once the reactions 

 in proper direction, and by resolving P along these lines, can 

 easily find their intensities, and therefore V and H, as before. 



\st. PARABOLIC AKC. 



For a parabolic arc we have for the locus ILK, y = h 

 that is, the locus is a straight line at tyh the rise of the arch 

 above the crown since we now take y as the ordinate to the 

 locus measured above the horizontal tangent at the crown. The 

 origin is, therefore, at the crown instead of at the centre of the 

 half span, as in the previous case. 



For the second curve, or curve enveloped by the reactions, we 

 have,* taking v as the abscissa and w as the ordinate of any 



. ArTO 2a 2 (23a 2 + 20ao:-f 5o*)A 



point [-big. 92], v = , w = ..., , . fn 'r } 



3 a + x ' 15 (a + x) (3 a + x) 



where, as before, a is the half span, h the rise, and x the dis- 

 tance of the weight from crown. For x = 0, v = f a, and 

 w = ff h. For x = a, v = -y a, and w = h. For x a, 

 v = a, and w =. oo. Eliminating x from both equations, we 



5 a 3 5 a v + 2 v* . 



have =-? 7 h. 



15 a (a v) 



Hence the curve enveloped by the reactions is on each side an 



* For the proof of all the expressions assumed, see the Supplement to this 

 chapter. 



