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 Trf the longitude of the perihelion of the projection of the varia- 

 ble section, reckoned in the direction of the motion from the axis 

 of X. 



Adding the squares of (26) and substituting the value of s, we get 



h'' (Mr - 60' =r-' {/' +f' -ff"' ) - {xfi-yf+zfY, (37), also 



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



shall have ds=Mdr, .'. substituting this value of ds in (25), also 



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



rdr 

 the products we get b^ -ir- =xf-\-yf'-{-zf", which reduces (37) to 



^=(Mr-60'=^</"^+/^+/"')-^^'^' (38). Put2c?R= 



,^ 1 2a2 



Ma-Jorf?a=— -yrdR, (39), then multiply (5) by '^dx, ^dy, 



^dz, take the integral of the sum of the products, and we get 



il -M ^---j =0, (40), this with (18) gives -X 



(rdr\ ^ 

 -^j 5 (41), this and (38) give ^^(Mr- S')'' = 



r'{f'+f'+f'')-b'[-(2ar-r^) ~b^], (42), which must be 



an identical equation, hence j-^ + — = -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 -17 =0, .*. (41) becomes at those points r'^ —2ar-\- 



-a6» ' .4. / ab^ a / «^^ 



^- =0, hence we have a-hv a'' — -^)a — \^ a^—^ for the 



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

 half sum = a = half the greater axis, and whose half difference 



= V a^ _ -|r|-= the excentricity ; and v 1 — ^== t^e ratio of 

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

 Ijj =sa(l —€^)=p'= the semi-parameter of the ellipse ; neglecting 



