124 Statics. 



fore the body is in contact with each of the planes only in a sin- 

 gle point, and perpendiculars IO, KO, be drawn through these 

 points, they must meet in some common point O, of the vertical 

 passing through the centre of gravity G, in order that the weight 

 of the body may admit of being decomposed into two other forces 

 having directions perpendicular to these planes. The compo- 

 nents IO, KO, will represent the pressures exerted upon the 

 planes. It hence results, that the plane which passes through the 

 points of support and the centre of gravity must be vertical, or perpen- 

 dicular to the inclined planes, or to their common intersection, which 

 loill consequently be horizontal. 



What is here said is not peculiar to the case of a body urged 

 by gravity simply. Whatever be the forces acting at /, K, their 

 resultant must conform to what we have said of the vertical 

 passing through the centre of gravity. 



Let XZ be a horizontal plane passing through the intersec- 

 tion B of the inclined planes ; and through the point K, draw KH 

 also horizontal ; and let the weight of the body KIG be repre- 

 sented by p, and the pressures exerted upon the two planes AB, 

 BC, by p, q, respectively. In order to obtain these pressures, 

 we must suppose the weight $ of the body to be a vertical force 

 applied at O ; thus regarded, it may be decomposed into two 

 others, directed according to OJ, OK-, we have accordingly the 

 following proportions, 



48 ' 9 : p : q : : sin IOK : sin GOK : sin JOG, 



or, since the angle CBZ = GOK, and AEX = /OG, and the 

 Geom. angles IBK, IOK, are supplements of each other, 



80. 



Trig. 13. p . p . q . . s [ n ABC : sin CBZ : sin ABX, 



: : sin HBK : sin BKH : sin KHB. 

 HK : HB ;. BK. 



Trig. 32. 



212. These principles are sufficient for determining, under 

 all circumstances, the conditions of equilibrium, where plane- 

 are concerned. By means of them we are enabled to explain 

 the strength of arches, and in general why hollow bodies, whose 

 exterior surface is convex, are better fitted, on this account, to 

 resist a compressing force. If. for example, a body is composed 



