28 LECTURES ON THE LUNAR THEORY. [LECT. VI. 



These are numerically equal to the quantities denoted by the same 

 symbols in Lecture IV, but 6, bears the contrary sign. We further find 



~m : Ma4=- ^ {(\ + 3m-2m 2 ) a., + (8 - 15m + 6;/r) &}, 

 2 / 4 1 TO l 



or reduced to numbers 



a,= -'00002,210, 



b,= -00005,414 = 11"-17. 



Finally let us exhibit the relation between the constants employed 

 in this investigation and those of Lectures IV, V ; to distinguish them, 

 attach accents to the latter, so that 



tit + + b.f sin 2\fj + &/ sin 4\jj, 



a' 



- = 1 + ,, cos 2i/; + / cos 4</.-, 



and, omitting the constant /3 as before, 



( 1 - m) = \]t + (I m) b.f sin 2i// + ( 1 m) b t ' sin 4i/>. 



Then 



2\fj = (2 - -2m) - (2 - 2m) b.! sin (2 - 2m) 0, 



sin 2ift = sin (2 - 2m) - ( 1 - m) &.,' sin (4 - 4m) 0, 

 cos 2i// = ( 1 - m) b./ + cos (2 - 2m) - ( 1 - m) b.f cos (4 - 4m) 0. 

 Substitute in the equation for ; we find 



nt + e = - &./ sin (2 - 2m) - [&/ - ( 1 - m) 6,' 2 ] sin (4 - 4m) 0, 

 and similarly 



,,'u = 1 + ( i _ TO) 0,'ft/ + a! cos (2 - 2m) + [/ - ( 1 - m) </;/] cos (4 - 4m) 0. 

 We observe that a' differs from a by quantities of the fourth order. 



