Taking the second member of ( 7) with 

 the first member of (4) and vice versa, and 

 forming corresponding determinants and 

 integrating, we have 



II 1 li 1, 

 9)c dj^^djL Al 



I! d t d t d t 



M 



- 'x, y, z) 



f (m, mi, nx,) 



where m 2 -fmi+m 2 =l and f is positive. 



Multiplying (9) by (1, l l3 1,) and adding, 

 we have 1 m+li nii+L m 2 =0, or (m, m 1; 

 m 2 ) is in the plane of motion. 



Take (11,11,, 110=1^^^11 forming 



the direction cosines of a third axis per- 

 pendicular to the two already found. 



Form with (1, L, 1,) and (9) correspond- 

 ing determinants, and we have : 



10) c r dx dv dz ' 



d t d t d t 



M || ] 1, 1, 



r iky /, 



f i n. n,, n, ) 



Multiplying the sec- 

 ond member of (7) 

 into the first member 

 of 1 4) and via versa 

 and integrating, we 

 have: 



. d/° M s , 

 CX dt = r ''~ f/t 

 where T f />-=£ 



Taking the scalar- 

 product by /., we find 

 S /. [i = 0, or ,". is in the 

 plane of motion. 



Take v=X :>■ forming 

 the rectangular unit 

 vectors /, ,«, v.. 



Multiply (9) by X and 

 we have : 



dt 



M 



f > 



This is the hodograph. It is a 



radius — and center 

 c 



(n, n,, n,) 



circle [remembering (8)] of 



f 

 = — >. The radius of this 



c 



hodograph is one right angle in advance of the radius vector of 

 the planet to which it corresponds. 



Transposing the f terms of (9) to the first member, squaring, 

 and using (6), we have: 

 c 2 M 



11 ) 



f* = M s or a = c J M | (M 2 — f 1 ). 



Multiplying (9) into (x, y, z) we have, Multiplying (9) into 

 by adding : /° and taking scalars : 



( 12) c'— M r=f ( m x-fm, y+m, z ). c J -Mj^-fS /<- I s 



