OF DEFLECTIVE FORCES. 137 



^ ' J7 XX XX XX 



d^BT, andardy — y(\x:=.(x^ +i/^)d':iT=: (r^ — z^) d'zsr : and since 



it has been shown that xdy — ydx = c^df, we have d-sr i= 



c^dt 

 ; hence, substitutins: for z and dt their values in 



terms of 0, we shall have the relation of 'zsr and 6, which is 

 sufficient for determining the place of the moving body. 



If we call the time occupied by the body, in its passage 

 from the highest to the lowest point of its motion, a semi- 

 oscillation, or i T, we may determine it by finding the 

 fluent of the value of d^ taken from 0=0 to d=:^7r=9(y ; 



first resolviner — -. : — into a series by means of 



the dinomia! theorem, which gives us — — =1 +~ j;^ 



*y \M. -—.2? ) ^ 



1.3 1.3.5 



+ ^-jj:* + Q-r-^x^ + . . ., and then takmg the particular 



fluents of dd multiplied by the powers of y^ s\n-d, by 



r .u r 1 nr • oAf 1 1.3.5..(2m — \)w 



means oi the lormula 7 sm ^m ^dz = — -— — -—; 



^^ 2A. . 2m 2 



I . . r^ ^ r 2r(a + b) C /In 

 whence we obtain Ti^tt ^/ - V ^r-r < 1 + I pt )^r^ 



/1.3\ / 1.3.5 N 7 



Corollary 1. Supposing the point to be suspended 

 by a thread, without weight or inertia, and fixed at its 

 upper extremity, its length being r, the motion will be 

 exactly the same as if it rested on a spherical surface; and 

 the greatest deviation of the thread, from the vertical 



direction, will be the angle of which the cosine is — . If 



the velocity, in this situation, be supposed to vanish, the 



