Pressure of Fluids. 303 



ing the middle points of the two opposite sides of the correspond- 

 ing face of the prism. The problem, therefore, is reduced to 

 finding what must take place in any polygon, when each of its 

 sides is drawn or pushed by a force applied perpendicularly to 

 its middle point, and represented as to its value , by this side. 

 We shall see that they will mutually destroy each other. 



Let us, in the first place, consider only the two forces/? and q, F 'g-205. 

 applied perpendicularly to the middle points of the sides AB, 

 AC, of the triangle ABC, these forces being represented as to 

 their values by these sides respectively. It is clear that their 

 resultant would pass through their common point of meeting /', 

 which, in the present case, is the centre of a circle in whose cir- G eom. 

 cumference the points A., B, C, are situated. We say, moreov- 

 er, that this resultant will pass through the middle point of BC, 

 to which it will consequently be perpendicular, and that it will 

 be represented in magnitude by BC. For, if we decompose the 

 force p into two others, one De parallel, and the other Dh per- 

 pendicular to BC, by forming the parallelogram D e g h we shall 

 have, by calling these two forces e and h respectively, 



p : e : h :: Dg : De : Dh :: Dg : De : ge. 



Now, by letting fall the perpendicular AO, the triangle g e D is 

 similar to the triangle AOB, since their sides are respectively 

 perpendicular. Accordingly, 



Dg : D e : ge : : AB : AO : BO, 

 whence 



p : e : h : : AB : AO : BO. 



But, by supposition, the value of the force p is represented by 

 AB ; therefore that of e is represented by AO, and that of h 

 byJ?0. 



If we decompose in like manner the force q into two others, 

 the one Im parallel, and the other Ik perpendicular, to BC, it 

 may be shown as above, that m is represented by A.O, and k by 

 CO. The two forces m and e are therefore equal, since they 

 are represented by the same line AO. Moreover they act in 

 opposite directions, and according to the same line DI parallel 



