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XVI. On a particular Transformation of the Differential 

 Equations of Motion in the Theory of Planetary Perturba- 

 tion. By the Rev. Brice Bronwjn*. 



THE transformation employed in this paper leads to results 

 of the same form as Hansen's theory, with the exception 

 that logarithms are not introduced. Its object is to dispense 

 with his constant time. The known equations of motion, re- 

 ferred to the plane of the orbit, are 



d*r dv* p dR _ „dv fdR 



To transform these, suppose p and u determined by the equa- 

 tions 



I = -^{l +e cos( M - OT }, P * d £=h=x/pa(l-e*); 



where a, e, and zr are constant, and t is a function of t to be 

 determined. 



Now make ra^, v = mu, and k=mh. Then 

 dv du _ du dr __ mh dr 

 dt =m dt ~ m lhdt ~ Ydl ; 

 and the second of (a.) becomes 



ai dr . 1 fdR .. .. [ 



Also 



dp ~ P dP + dt dt ~*~ p df- ~ p dr*dP ^ p dT dt* 

 . 9 d ± dtdr_ ffifi _o^t^ l±±(o,dr\ 

 + dr dt dt +P dt 2 ~ P dr* dP T fidrdtV dt) 



d^_ R (fydf_ d?P 1 dpdR 



+ P ~dp- p dr* dP +? dP mhf3 dr dv * 



If m* = 1 + £, we have 



dv* 2;2 Mt 2 i^dr* Lf;2 /3c?r 2 



r dP ~'" " p 3 dP " " p* dP ' "" p 3 dP 



and 



M 



r 



p*p*-P p * dP "*" p 2 \/3 2 p dP J' 



Substituting these values in the first of (a.), remembering that 



<Pp _h* » 

 d^ pt+f- ' 

 * Communicated by the Author. 



