﻿196 ON SOME APPLICATIONS OF THE METHOD 



As it is of importance that z/ 1 x , J x y , and J 1 z„ should be known beyond the last 

 decimal place of x , y , and z„, the following equations, or others of a similar nature, are 

 useful in determining them. 



The projections of the radius vector in an ellipse upon the principal axes are 

 r sin. v = \/ap sin. u, and r cos. c = a (cos. u — e). 



And since these axes have a fixed direction, the coordinates of a body moving in an 

 elliptical orbit, referred to any plane passing through the sun, are 



(17) x = A -\- Ai sin. u-\- Ai sin. 2 A u, y = B -f- B v sin. u -\- B. z sin. 2 Jk, z = C -f- C\ sin. u-f- C 2 sin. 2 £ u; 

 whence 



J 1 x = x l — x = 2 A t cos. £ (u, -j- «o) sin. £ (u, — w<>) -j- A 2 sin. £ («! -(- u„) sin. £ {ui — u ) ; 



(18) 4* y<, = yi— Vo =2 -B, cos. J («i + u ) sin. £ («! — u ) -}- #i sin. £ («! +«o) sin. £ («! — « ) ; 

 ^'20 = 2, — z = 2 C, cos. £ («, -j- u ) sin. £ (j^ — m ) -j- C 2 sin. £ (u, -f- u„) sin. £ («! — m„) ; 



.4,, ^„ &c, being constant quantities, determined by giving particular values to u. 

 Putting for u its values when ^ = 0°, and # = 90°, we find 



A>=*, ^=-===« A, — -; 



Z' 2Z 



C ° = Z > C i = vT^7 8 ' C 2 = — YZTe 



X, Y, and Z are the coordinates of the perihelion point of the orbit; X', Y', and Z', 

 those of the point of intersection of the parameter with the orbit, that is, a point 90° 

 from the perihelion. 



In the parabola the above expressions become 



(20) a=.X+.X' tan. £» — Xtan. 2 £c, y= Y+ Y tan. $ c — Ylan. 2 ^ c, z = Z+Z' tan. £c — Z tan. 2 £c. 



., , sin. £ (c, — c„) v sin. £ (c, — c ) sin, j- (c, + c ) . 



cos. £ c, cos. £ c cos. 2 £ o, COS/ £ c 



*° yi y ° COS. i », cos. I C COS. 2 £ C, COS. 2 £ c 



. _, sin. X (c, — Co) „ sin. J (o, — c ) sin, j- (d + Cp) 



COS. £ C t COS. £ C COS. 2 i Vj COS. 2 £ c 



By differentiating the above equations, and substituting for |? and £ the proper 

 values, we find the following expressions for the differential coefficients of x, y, and z : 



(22) Yx=(Acos.u+^ 2 sin.u)-^=, Vy = (£ lC os. u+ £.B 2 sin.M) — -j-, Vz = (C 1 cos.u+$ C 2 sin.«)^-^- 

 And in the parabola, 



(23) Vx={X'— 2X tan. i c) — ^=, Fy = (Y' — 2 Y tan. $ c) -^=, Y2 = (Z' — 2 Z tan. i v) j^=. 



