254 Variations of the Arbitrary Constants in Elliptic Motion. 



it is orthographically projected on the plane x> y, p* being the semi 

 parameter, e the ratio of the excentricity to half the greater axis 

 and zt the longitude of the perihelion of the projection of the varia- 



reckoned 



from the axi9 



of x. 





ib 



2 , 2 



b-(Mr-b*) =r*(p+f"+f" )-(xf+yf'+ z f") ,(37), also 



since m moves momentarily in the ellipse to which b belongs, we 



shall have ds=Ndr, .\ substituting this value of ds in (25), also 



restoring the value of s, then multiplying (25) by oc, y, z and adding 



rdr 

 the products we get b 2 -y- =xf+yf'-\'Zf" i which reduces (37) to 



b*(Mr-b*y =r>(f*+f <*+/"') -b* r ^> (38). Put 2dR 



dt 



1 2a 



2 





Md~> or da=- ^r^R, (39), then multiply (5) by 2efor, 2<fy, 



2dfz, take the integral of 

 dx 3 + dy 2 + dz* v/8 1 



M 



•ji -M[---j =0, (40), this with (18) gives -X 





lrdr\ 



(2<ir-r 3 )-6* = \j^j i (41), this and (38) give6'(Mr- §•)* 



-(2ar-r*) -6«), (42), which must be 



f- -+- f' 2 4- f" 2 M M a 



an identical equation, hence ^- 4 — — — +— =; -r^> (43). 



To find the greater axis and excentricity of the ellipse in which m 

 is moving at any instant, we shall have (supposing it to continue its 

 motion in the section,) when it arrives at the extremities of the 



dr 



greater axis tj =0, .'. (41) becomes at those points r a — 2ar+ 



_ _ 



-j^j- =0, hence we have a 4- V a* — -^r> a - v a a — «■ for the 



distances of the extremities of the greater axis from the focus, whose 

 half su m = a == half the greater axis, and whose half difference 



\/ °"^" / b 2 

 v a 3 — ^jt= the excentricity ; and VI % -.= the ratio of 



rtM 



the excentricity to half the greater axis, put this = e, and we get 

 b* 



jyf =a(l -e 2 )—p'= the semi-parameter of the ellipse ; neglecting 





