GEOMETRY. 
575 
790] 
36. It is important to express the nine coefficients in terms of three independent 
quantities. A solution which, although unsymmetrical, is very convenient in Astronomy 
and Dynamics is to use for the purpose the three angles 6, cf), t of fig. 19; say 
6 = longitude of the node; <f> = inclination; and t = longitude of x 1 from node. 
Pig. 19. 
The diagram of transformation then is 
X 
y 
z 
»1 
COS T COS 9 — sin T sin 9 COS cf) 
COS T sin 9 + sin T cos 6 cos <f> 
sin T sin (f) 
Vi 
— sin T cos 0 — COS T sin 0 COS 4> 
— sin r sin 9 + cos r cos 9 COS 4> 
COS T sin 4> 
Si 
sin 6 sin cf) 
— COS 9 sin 
COS cf) 
But a more elegant solution (due to Rodrigues) is that contained in the diagram 
X 
V 
z 
1 + X 2 - fl? - v 2 
2 (A/a — v) 
2 (Av + /a) 
Vi 
2 (A/a + v) 
1 — A 2 + /A 2 — v 2 
2 (/av - A) 
«1 
2 (vA — /a) 
2 (/av + A) 
1 _ ^ 
(1 + A. 2 + /a 2 + v 2 ). 
The nine coefficients of transformation are the nine functions of the diagram, each 
divided by 1 + A 2 + ¡j? + v* ; the expressions contain as they should do the three arbitrary