14 THEORBITOFURAXUS. 



Let us, for convenience, replace x and y by two other variables ^ and y; connected 

 with them by the equations 



a? = a^, 



y z=zay] cos i]/. 



^ and y; are then functions of the eccentricity and mean anomaly only, and may be 

 developed according to the multiples of the latter. Substituting the last three 

 expressions in the preceding value of ^'o^^p it becomes 



■"^^p^iT^j'^/'^^'^-^^^^'M 



+ 



If we put ^i^ for the value of v'o when tlie mean distance of the planet is put equal 

 to unity, so that r^, like ^ and y; contains only the eccentricity and mean anomaly, 

 "wc shall have 



8p = ^^ { vf^a Qdt - ^fy!a Qdt | (13) 



+ 



We must now express | and ri in tcrms^ of the time, or of the mean anomaly. 

 Putting for the present ii, for the eccentric and v for tlie true anomaly, we have, 

 by the theory of the elliptic motion, 



X = ;• cos V ^a (cos u — e), 

 y ^r sin v = a cos ^ sin m, 



from which follow 



^ = cos 11 — e, 

 Yi = sin u. 



As ^ and yi are to be expressed in the form 



»7 = I ^(li sin Uj, 



the finite integrals extending to all values of i from — oc to -j-oc, we shall deduce 

 general expressions from Pi and q^ arranged according to the power of the eccen- 

 tricity. Since 



u^ g -\- e sin u, 



we have by Lagrange's theorem 



■ n e^ d sin^ q e^ 6" sin* q . 



cos u r= cos q — e sm" q ^ — etc. : 



•^ •' 2! dg 3! %2 



or 



cos 



using the notation 



w! = 1.2.3 n= r(», + 1). 



We then have 



0! = 1! = 1. 



