179-181] DISTURBED ELLIPTIC MOTION. 161 



If the particle is subject to a disturbing force producing a small 

 tangential acceleration / we shall have 



LL ' v e \R a) ' 



a- 1 



e sm 0tzr = -= J --- -= - 1 

 R v e\R 



180. Normal disturbing force. Suppose the particle to 

 receive an impulse imparting to it a velocity Sv in the direction 

 of the normal outwards. Then the resultant velocity is, to the first 

 order, unaltered, and consequently a is unaltered, or 8a = 0. 



If p is the perpendicular from the focus S on the tangent at P, 

 meeting it in F, then the value of h is increased by P Y$v, or we 

 have 



Sh =</(&-&) to. 



Hence pSl = 2hSh = 2pv$v V(# 2 -p 2 ) ; 



also Bl = ZaeSe, so that 



fa 



Again, Z/.R = 1 + e cos 0, so that 



- 2aeSe/R = ( 4 - l) 

 \i J 



If the particle is subject to a disturbing force producing a small 

 normal acceleration f we have 



*=<> *- 



- + e sin 

 e 



181. Examples. 



1. For a small tangential impulse prove that 



&?=2&y(e + cos0)/v, fozr = 25v sin 6 lev. 



2. For a small normal impulse prove that 



8e=- r8v sin 0/av, 8or = dv (2ae+r cos 5)/oev. 



3. For a small radial impulse prove that 



da = 2a?edv sin 6/h, 8e = hv sin ^//z, Sor = - A8v cos 0/ep. 



4. For a small transversal impulse prove that 



da = 28m 2 (1+ e cos 0)/A, Se = d; (r (e + cos 6} + 1 cos 0}/h, 8ar = 8v sin 6 (I -f r)/eA. 



L. H 



