S'28 On the Polygon of Forces^ &fc. 



another double set oF side b!nck<? similar and equal he ap- 

 plied to the central one; and if these latter be applied at 

 right angles to the former, there will be a combination of 

 blocks, having a central spire and four surrounding ones 

 cu the haunches of the side blocks, resembling in a consi- 

 derable degree (if I forget not) the slucture at the top of 

 the old church at Newcastle on Tvne ; and showing how 

 such a construction mav be effected without the aid of con- 

 cealed iron work, provided the horizontal thrust at the feet 

 be resisted. 



The following problem being naturally connected with 

 this subject, I take the liberty here to add. 



When three forces conspire in their action on a point, 

 ■which is thereby kept at rest, the proportional quantity of 

 each is ascertained by the well known '■^ triangle' of forces,'* 

 vhose sides are parallel to the directions of the forces; the 

 present problem is to ascertaiii, when any greater number 

 of forces act on a point (in the same plane) whose quan- 

 tities nnd directions are given, what their united effect upon 

 that point is — or, what other force aetini: in that point will 

 cnunteracl the given forces so as to keep the point at rest. 

 This is performed by a ligure which I think may be aptly 

 called t\i{; ptih/gori ofJorc(-s. 



Let SA, SBI^^ SC, and SD, fig. 4, be the directions, and 

 their lengths the quamities of four forces acting on the 

 pomt S ; it is required to know what their united effect is 

 on the point S: or, what other force acting on that point, 

 and in vi hat direction, will counteiact them, and keep it at 

 rest. 



From file outer end of anv oP the lines as SA draw pa- 

 rallel to the direction of the adjoining force SB the line Ab, 

 equal in len'ith ; from b, parallel in direction to the next 

 forte SC, and equal in Icnirth draw be; from c in like 

 irianner draw c d parallel and equal in length to SD ; com- 

 plete the polygon by drawing the line d S ; which is the di- 

 rection, and its length the ineasure of th^ force that wil^ 

 counteract the four given forces; and it is theretore the 

 measure of the effect of those forces, and the direction of 

 that effect is Sd. Therefore if the line dS be continued to 

 K making SE = dS, it will represent the required counter- 

 acting hirce in relation to the given point S, in the same 

 ananner as the given forces are represented. Thus the 

 forces SA, SB, SC, SD and SE acting on the point S with 

 energies proportional to and in the directions of those lines 

 will keep it at rest, or be in equilibrio. 



If 



