118 Statics. 



through the point C. The necessity of the first condition is evi- 

 88. dent. As to the second, it will be seen, with a moment's atten- 

 tion, that this is not the less necessary ; since, if the direction 

 AT) of the body p', for example, although perpendicular to the 

 plane, does not pass through the point of contact C', the resist- 

 ance of the plane, which cannot be exerted except according to 

 the perpendicular at C, would not be directly opposed to the 

 force AD, and consequently would not destroy it, even when it 

 137. is supposed equal to this force. 



197. If the body, instead of touching the plane only in one 

 Fig. 109, point, touches it in several points, it is not indispensable that the 



10 ' single force AD, which acts upon it, should pass through any one 

 of these points ; but it is necessary that it should be perpendic- 

 ular to the plane, and that it should be capable of being decom- 

 posed into as many forces perpendicular to the plane, as there 

 are points which rest upon it, and that they should be such as to 

 pass through these points. Thus if the bodyp, for example, were 

 Fig.109. in contact with the plane at the points C, C', and the force AD 

 were not in the plane which passes through the two perpendicu- 

 lars raised at the points C, C', an equilibrium would not take 

 place, because the force AD could not be decomposed into forces 

 passing through C and C', without a third arising which would 

 not be counterbalanced. 



198. Hence, if a body which touches a plane in one or in 

 several points, be urged by several forces directed at pleasure, it 

 is necessary, (1.) That these forces should admit of being reduc- 

 ed to a single one perpendicular to the plane ; (2.) That this, in 

 the case where it does not pass through one of the points of con- 

 tact, should be capable of being decomposed into as many forces 

 parallel to it, as there are points of contact, and that these should 

 pass each through one of the points of contact. 



1 99. If the single force which urges a body be gravity, it is 

 necessary that the plane should be horizontal ; and if the vertical 

 plane, drawn through the centre of gravity of the body, do not 

 pass through one of the points of contact, it is necessary, at least, 

 that it should not leave all the touching points on the same 

 side. 



