1 80 On a Dynamical Theory of 



The last expression may be put under the form 



M sin ! sin (i z - i'z) 



= 8in%{cos 2 cos 0' 2 + sin 2 sin 0' 2 cos (it - e' 2 ) (40) 



+ sin ff 2 sin (it - i\) tan E } 



- sinV 2 {cos 2 cos 0' 2 + sin 2 sin 0^ cos ( 2 - i' z ) 



- sin 2 sin (i z - " 2 ) tan e' } . 



Let the axes of # , y , s make with the direction of OP the 

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

 a '//> $'//> T'//- The cosines of these angles may be found from 

 the expressions (35) and (36) by supposing c and t' to vanish. 

 Therefore 



cos er y/ = sin 2 cos / 2 , cos /3 /7 = - cos 2 , cos -y /y = - sin 2 sin 2 , 



(41) 

 cos a',, = sin 0' 2 cos iz', cos jS^ = - cos 0' 2 , cos y' = - sin 0' 2 sin i\. 



If to be the angle which OQ makes with OP', and a/ the 

 angle which OQ' makes with OP, so that 



cos o> = cos a z cos a' y/ + cos /3 2 cos /3' y/ + cos y 2 cos 7'^, 

 cos a/ = cos a' 2 cos a /y + cos j3' 2 cos /3 y/ + cos 7' 2 cos 7 y/ , 

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



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



+ sin / 2 sin (i z - i'*} tan e } , 



cos <*)' = cos e' {cos 02 cos 0' 2 + sin 2 sin 0' 2 cos (^ 2 - i\] 



- sin 2 sin (i z - i' z } tan /} . 



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

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



sin 2 e'i (rs cos w - rs' cos u>'). 



