Dynamics. 



or 



x * 



g h 



Gcom. Now, n being the circumference of a circle whose diameter is 1, 



291. 



1 : n :: h : <ZZM'B-, 



whence, 



ZZM'B 



and consequently, 



= 71 



or, 



We have thus an expression for the duration of an entire os- 

 cillation ; and as this quantity does not contain ft, or the height 

 from which the body falls, and which determines the extent of 

 the path described AB, it follows that the time tf does not sensi- 

 bly depend upon the extent of the arc, so long as this arc is very 

 small. Therefore, the oscillations -which take place in small arcs of 

 a circle are sensibly isocronous or of the same duration. 



343. This property belongs to the small arcs of all curves in 

 which the radius of the evolute at the lowest point is not zero ; 

 since the arcs are confounded with those of the circle by which 

 Cal. 79. their curvature is measured. 



If we would know the error liable to be committed by taking 

 this value of t' for the duration of a semioscillation in a circle, 

 we proceed thus ; 



Taking the value found above for d s, namely, 



adx 

 \/%ax x 2 



Cal we reduce it to a series, retaining the three first terms only, 

 We 2. which are abundantly sufficient for our purpose, and we shall 

 have 



