44 



II. PARTICULAR MOTIONS OF A POINT. 



Multiplying the second of equations 42) by x and subtracting from 

 it the first multiplied by y we obtain 



Fig. 12. 



M 



which expresses the fact 

 that the horizontal compon- 

 ent of the acceleration has 

 no moment about the origin, 

 as in 12. We may therefore 

 integrate, obtaining, 



47) / c ^_ ?/ ^ ==c 



where c is another constant 

 of integration representing 

 the moment of the horizontal 

 component of the velocity 

 about the origin and cor- 

 responding to the h of 20. 

 It will be convenient to 

 introduce polar coordinates 

 such that (Fig. 12) 



x = Zsin-frcosqp, 

 y = I sin # sin 9, 



z = 



48) 



Differentiating we have 



dx = I (cos # cos <pd& sin # sin cp d<p), 

 dy = 1 (cos & sin y d& + sin & 



dx 2 + dy 2 + dz 2 = P (d&* + 

 xdy ydx = I 2 sin 2 &d(p. 



Thus our first integrals 44) and 47) become 

 49) i 



50) 

 Substituting the value of 



derived from 50) in 49) we have 



