40 



OF PRESSURE AND EQUILIBRIUM. 



point of contact is less or greater than a 

 fourth proportional to tlie sum of the radii, 

 and the radii taken separately. 



Let the sphere or cylinder roll from the 

 vertical position into a position infinitely 

 near to it on either side: then the point A 

 of the upper cylinder, which was origi- 

 nally in contact with the lower, may still 

 be considered as in the vertical line BA : 

 and if CD be the vertical line passing 

 through the actual point of contact, BE : 

 AE ; : BC : AD, and if the centre of gravity be at D in the 

 Uhc AE, the point of support being immediately under it, 

 the equilibrium will remain : but if the centre of gravity be 

 below D, the sphere Will return towards AB ; if above, it 

 will retire further from it. 



For example, if CE be infinite, and the lower surface of 

 the moveable body be a plane, the equilibrium will remain 

 stable, while the height of the centre of gravity above the 

 point of contact is less than the radius of the sphere. If the 

 fixed body have its upper surface horizontal, the equilibrium 

 of any body will be determined by its radius of curvature, 

 as the equilibrium of an egg placed on one end is tottering, 

 but stable when placed on one side. 



289. Theorem. If any other equivalent 

 forces be substituted for weights, acting at the 

 same distance from the fulcrum, and with 

 the same inclination to the rods or levers, 

 the phenomena of equilibrium will be pre- 

 cisely the same. Also if either of the forces 

 be transferred to an equal distance on the 

 other side of the point of suspension or ful- 

 crum, and act there in a contrary direction, 

 the equilibrium will still remain. 



For the arguments derived from the composition of pres- 

 sures are equally applicable to all these cases. 



290. Theorem. If a force be applied 

 obliquely to a lever, its effect in turning the 

 lever will be diminished, in the ratio of the 

 sine of the inclination to the radius. 



For instance, if two levers be connected by a rope, two 

 forces applied perpendicularly to the levers, at the ends of 

 the rope, will be in equilibrium when the forces are as the 

 perpendiculars let fall on the respective levers from the op- 

 posite tads of the rope. For the action of each force in the 



direction of the rope, and its absolute strength, are as the 



sides of the triangle formed by the 



lines of direction, or as the length of 



the rope and the perpendicular falling 



from its end on the lever : therefore, 



each perpendicular representing the absolute force, the 



length of the rope will in both cases express the relative 



action. The forces are represented in the figure by arrow 



heads, and the fulcrums by little circles. 



291. Theorem. If two threads, or per- 

 fectly flexible and inextensible lines, be 

 wound in contrary directions round two cy- 

 linders, moveable on the same axis, there 

 will be an equilibrium when the weights at- 

 tached to them are inversely as the radii of 

 the cylinders. 



For every section of the cylinder perpendicular to the 

 axis, is a circle, and the threads 

 being tangents to the circles, will 

 be at thedistancesof the radii from 

 the vertical plane ; therefore, by 

 similar triangles, (he right line 

 joining the weights will be divided in the ratio of the radii, 

 and the centre of gravity will be in the vertical plane ; and 

 the point of the axis immediately over it is a centre of sus- 

 pension ; therefore there will be an equilibrium (285). 



292. Theorem. When the direction of 

 a thread is altered by passing over any per- 

 fecth' smooth curve surface, it communicates 

 the whole force acting on it. 



For the resistance of the curve is alvrays in a direction 

 perpendicular to that of the thread, and therefore does not 

 impair its action, as is obvious from the composition of 

 forces. 



293. Definition. A pulley is a cylinder 

 moving on an axis, in order to change the 

 direction of a thread without friction. 



Scholium. The comparison of a pulley to a lever is 

 both unnecessary and imperfect. 



294. Theorem. By me^ns of a single 

 moveable pulley, each portion of the thread 

 being vertical, a weight may be supported 

 by two forces, each equivalent to half the 

 weight; or by two threads, each passing 

 over a fixed pulley, and connected with ano- 



