Motion along curved Surfaces. 21 1 



it will not be an infinitely small quantity of the first order, it will 

 only be an infinitely small quantity of the second order. In es- Cal - 4 - 

 tablishing this, the question reduces itself to showing that the 

 versed sine of an infinitely small arc is an infinitely small quan- 

 tity of the second order; and this may be done thus. CD being pig.161. 

 any arc of a circle, and BD a perpendicular upon the diameter 

 AC, we have Geom. 



215. 



AB : BD :: BD : EC -, 



hence, if CD, (and for a stronger reason BD) be infinitely small, 

 BC the versed sine of CD, will be infinitely smaller than BD, 

 since it is contained in BD as many times as BD is contained 

 in the infinitely greater quantity JIB. Therefore BC is infinite- 

 ly small of the second order. 



337. Accordingly, if a body without gravity move along //ieFig.162. 

 curved surface ABC, it will have throughout the same velocity. For 



by considering this curve as a polygon of an infinite number of 

 sides, since the sides make angles infinitely small with each other, 

 the loss of velocity at the meeting of each two adjacent sides is 

 an infinitely small quantity of the second order with respect to 

 the original velocity. Consequently the sum of the velocities 

 lost in passing over an infinite number of sides, that is, in passing 

 over any arc ABC, can only form an infinitely small quantity of 

 the first order. Therefore the velocity is not affected by this 

 circumstance. Cal. 4. 



338. We come now to the motion of heavy bodies along 

 curved surfaces. We shall consider for the present only that 

 which takes place in a vertical plane. 



339. Accordingly, let AMB be a section of a curved surface, Fig.163. 

 made by a vertical plane, and the path described by a body 

 along this surface. Let us consider this curve as a polygon of 



an infinite number of sides, and let us suppose that the body has 

 just described the small side LM. As its meeting with the side 

 MN cannot occasion any loss of, velocity ; it will describe MN 335. 

 with the velocity which it had in M, gravity being supposed no 

 longer to act upon it. But the force of gravity being exerted 

 according to the vertical MO, urges the body anew as it would 

 urge one upon a plane of the same inclination. Consequently, 



