Motion of Oscillation- 2 1 5 



341. As to the time employed in describing any arc AM or 



AB of the curve ; since d t = - , substituting for v its 



v 



equal V 2g-fc 2##, we have, 



__--<*'_ 

 * ' 



so that it would be necesary by means of the equation of the 

 curve to find the value of d s in x and dx, and having substituted 

 it in the expression for d t, we should have that of t by integrat- 



Of the Motion of Oscillation. 



342. We have seen that a heavy body having descended 

 through any arc of a curve AB must, sotting aside the resistance 

 of the air and friction, ascend again to the same height in a curve 

 BA 1 which has at the point B the same horizontal tangent with 

 BA. Accordingly, this body in returning would describe in a 

 contrary direction the whole extent of the curve A'EA ; and thus 340, 

 would continue to move backward and forward without end. 

 This kind of motion is called oscillation. We have seen what is 

 in general necessary to determine the duration of each oscillation 

 which must evidently be double the time employed in describing 

 the arc AB, if BA 1 is the same as AB. 



When the curve through which the body descends is circular, 

 and the oscillations take place through small arcs only, they 

 have this remarkable and important property, that their duration 

 is not sensibly affected by the extent of the arc AB-, so that the arc Fiff 164> 

 JIB being small, as four or five degrees only at the most, the 

 body will always arrive at B in the same time very nearly, wheth- 

 er it set out from the point A, or from some other point O, taken 

 between Jl and B. 



Thus, retaining the denominations used above, and designat- 

 ing by a the radius BC of the circle BAD, we shall have, by 

 the nature of the circle, Trig. 



101." 



y 2 = 2ax x 2 or y = 



