179-181] DISTURBED ELLIPTIC MOTION. 161 



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

 tangential acceleration / we shall have 



e = -*- - - D - - - 

 yu, v e\R a 



21 f efl 

 e sin 0r= = --- - - 



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 Sa 0. 



If p is the perpendicular from the focus 8 on the tangent at P, 

 meeting it in F, then the value of h is increased by PYSv, or we 

 have 



Hence pSl = 2hSh = 2pvBv *J(R* -p*) ; 



also SI = 2ae$e, so that 



Again, l/R = 1 4 e cos 6, so that 



=--+e sn 

 e 



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

 normal acceleration f we have 



pfv 



a=0 &amp;gt; * 



181. Examples. 



1. For a small tangential impulse prove that 



&?=2Sy(e + cos0)/v, SET = 28v sin 0/ev. 



2. For a small normal impulse prove that 



8e= r8v sin 6/av, 6nr = Sv (2ae + r cos &)jaev. 



3. For a small radial impulse prove that 



Sa = 2a?edv sin 0/A, Se = hbv sin d/^, STZT = - A5v cos 0/ep. 



4. For a small transversal impulse prove that 



da = 28m 2 (1 + e cos 0)/A, Se = dv (r (e + cos 6} + l cos ^}/A, SOT = 8v sin ^ (? + 



L. 11 



