PATHS OF MOVING PARTICLES SPRUNG 69 



by ds the differential of the path, then will the principle of living 

 force, vis viva or mechanical energy be represented by the following 

 equation : 



d (£ V 2 ) = no 2 sin <p ds cos 



dr 

 But since, as is evident at once geometrically, — 3 — — . = sin <p, 



b ds cos a 



then the same equation can be written: 



dQV 2 ) = - rco 2 dr 



from the integration of which results 



V 2 =D - r 2 co 2 (13) 



where D is a constant. Again the principle of the conservation of 

 areas gives 



Vsinfl = - (14) 



r 



In these two equations the general problem is contained, and to a 

 certain extent already solved. From (14) there is first derived 



y 2 cos 2 0= V*- — 

 r 2 



and by introducing the value of V 2 from (13) 



V cos = J D - r 2 ur - — (15) 



The expressions (14) and (15) contain the west-easterly and south- 

 northerly components of the absolute velocity as functions of the 

 distance r from the axis; we have only to subtract from these the 

 velocity of the surface of the earth at the place in question to obtain 

 the corresponding components v sin d and v cos d of the relative 

 velocity; in this way we obtain 



• /i C 



v sin o = — — rco 



r 



-<Jd- 



C 2 



v cos d = il D — r 2 co 2 — — 



(16) 



r 



By squaring and adding these equations the following is obtained: 

 v 2 = D - 2 Ceo = v 2 (17) 



