46 Mr. Mac Cuuiacu on the dynamical Theory of 
If we substitute, in the first of the equations (31), the values just given for 
COS a,, cos a;, along with the above values of cos a,, cos aj, and attend to the rela- 
tions 
r cose = 5, TECOS Ep = Si, sin?, =s sini, sin?,=s'sin?,, (37) 
we find, after multiplying all the terms of the equation by sin7,, 
(7, sin 0, — 7; sin 6;) sin 7, cos 7, = 
7, (sin @, sin 2, cos 2, + sin’, tan e) + 7; (sin 6; sin 2 cos 2, + sin? 2; tan €’). ) 
Joining this equation to the equations (34) we have all the conditions that are 
necessary for the solution of the question. 
Multiplying the first of the equations (34) by the third, also the second of 
these equations by the equation (38), adding the products together, and then 
dividing by sin 7,, we obtain 
By (7° — Ty) = pp Ty $F pa 72 FMT, To, (39) 
where we have put 
4, = COS2,, H, = $(cos 2, -++ sin 6, sin @, tan €), 
1, = s’ (cos 7, ++ sin 6; sin 2; tan e’ ), 
Msin?, = sin (2, + 2) {cos 0, cos 6; + sin @, sin 6; cos (2, — 2) } 
++ sin 0; sin? 7, tan e + sin 6, sin® 7; tan e’. 
The last expression may be put under the form 
M sin @, sin (7, — #%) = (40) 
sin’ 2, {cos 6, cos 6; + sin 6, sin 6; cos (7, — 7) + sin 6, sin (7, — @;) tan e} 
— sin* #, {cos 0, cos 6; + sin 6, sin 6; cos (7, — 2) — sin 0, sin (7, — @;) tan €’}. 
Let the axes of «,, y,, z, make with the direction of OP the angles a,, B,,. y,, 
and with the direction of OP’ the angles a’, 6’, y,. The cosines of these angles 
may be found from the expressions (35) and (36) by supposing ¢ and ¢’ to vanish. 
Therefore 
cos #,, = sin @, cost, cosf,,—= —cos6,, cosy, = — Sin @, sin 2,, 
cos a, = sin 6, cos %;, cos B,, = — cos 6;, cosy,, = — sin 6; sin %,. 
(41) 
