1 80 On a Dynamical Theory of 



The last expression may be put under the form 



M sin fc\ sin (i z - i') 

 = sin 2 2 {cos 2 cos 0' 2 + sin 2 sin 0' 2 cos (i z - ') (40) 



+ sin 0' 2 sin ( 2 - ' 2 ) tan e) 

 - sinVajoos 2 cos 0' 2 + sin 2 sin 0' 2 cos (e a - / 2 ) 



- sin 2 sin (i z - ^2) tan c'} . 



Let the axes of x m y , s make with the direction of OP the 

 angles a y/ , /3 //? 7,,, and with the direction of OP' the angles 

 a '//> /3',/j 7'//- The cosines of these angles may be found from 

 the expressions (35) and (36) by supposing e and e' to vanish. 

 Therefore 



cos er /y = sin 2 cos i z , cos /3 7/ = - cos 2 , cos y f/ = - sin 3 sin 2 , 



(41) 

 cos a' y/ = sin 0' 2 cos ^2 / , cos jS'^ = - cos r 2 , cos 7'^ = - sin 0' 2 sin i\. 



If w be the angle which OQ makes with OP', and w' the 

 angle which OQ' makes with OP, so that 



cos u> = cos a 2 cos a' /y + cos j3 2 cos j3' /y + cos 72 cos 7' /y , 



COS O)' = COS a' 2 COS a /7 + COS )3' 2 COS /3 /y + COS y' 9 COS 7 /x , 



we find, by the formulae (35), (36), (41), 



cos w = cos efcos 2 cos 0' 2 + sin 2 sin 0' 2 cos (i z - i' z ) 



+ sin 0' 2 sin (i 2 - i' z ] tan cj, 



cos w' = cos s { cos 2 cos 0' 2 + sin 2 sin 0' 2 cos (i z - i' z ) 



- sin 2 sin (i 2 - i' z ) tan c'J. 



Hence, observing the relations (37), we see that the right-hand 

 member of the equation (40) is equal to the quantity 



sin 2 ^ (rs cos w - r's' cos a/). 



