28 FORCES IN THE SAME PLANE. [CHAP. II. 



is the resultant of the forces preceding, and we call such a 

 polygon the mean polygon of equilibrium. 



If we wish to find the mean polygon for P^ we have only to 

 take the new pole C' at 2 in the force polygon (a). According 

 to the preceding Art., each side of the new polygon must pass 

 through the intersection of the corresponding side of the first 

 with the line S 2 which passes through a and is parallel to C C'. 

 Thus S' 4 must pass through V and o. S' 5 through c' and n, and 

 BO on. S' 7 is the resultant of P 37 , and since 83 is the resultant 

 of P!^ ; S 7 , the resultant of P 1J7 , must pass through the intersec- 

 tion m of S' 7 and 82- 



We observe here again the influence of the couple P 5 and P 6 . 

 S 4 and S' 4 are simply shifted through certain distances, without 

 change of direction, to S 6 and S' 6 ; and as we have seen above, 

 knowing the direction of rotation, and the moment of the couple, 

 we might have omitted it in the force polygon and still obtained 

 87 and S' 7 as before. 



2. Line of pressures In an arch. The practical applica- 

 tion of the above will be at once seen in the consideration of 

 an arch. Thus with the given horizontal thrust applied at a 

 given point of the arch, and the forces P w , we construct the 

 force polygon C o 5, and then the line of pressures abed. 

 [Fig. 16, PL 4.] 



Required with another thrust H' = o C' acting at another 

 point, and the same forces P w , to construct the corresponding 

 line of pressures. To do this we have only to lay off o C' equal 

 to the new horizontal thrust, then choose a point of the force 

 line, as 3, as a pole and draw the corresponding polygon, 

 k op k ; the point of intersection, &, is a point upon the line 

 m n parallel to o C, and upon this line will be found the inter- 

 section of corresponding sides of the two polygons. Thus from 

 the intersection of the side a p of the first polygon with m n, 

 draw a line to o and we have a'. From the intersection b of 

 the second line of the first polygon draw a line to a', and we 

 have b' a', and so on. 



29. The preceding articles comprise all the most important 

 principles of the Graphical Method which can be deduced in- 

 dependently of its practical applications. Future principles 

 will be best demonstrated, and at the same time illustrated, by 

 considering the various special applications of the method, and 

 to these applications we shall therefore now proceed. 



