48 The Rev. Brice BRomviN on the Theory of Planetary Disturbance. 



„ ■ o 2 



V Sill" 



2 cos- 2 



A 7 2 



3ut L — sin- i cos" (w - S-) = sin^ i - sin^ i sin^ (r - 3-) = sin^? - <r. 

 Therefore, putting this value in the preceding, we have 



^^= 1 1 (^^^\.A(l + .iA^)-i!l^l + iA=). (9) 



2 COS^ q 



But the formulas we have obtained are not convenient for actual applica- 

 tion ; it may be well, therefore, to give them in series. For this purpose we 

 make sin z = s, sin (u - 5) = »? ; then a = si]. But as, in the first diiferentials, 



.da dn . . dn ,._ 



the elements do not vary, we must make "3: = « "jTi ^^^"^ i° 'Tt '^^ ^ 



ferentiate 3-. 



Now, expanding \/(l - <r) in (7), we find 



2Acos^^ '^^ 



nedectin^f smaller terms. Or, 



o o 



1h cos^ - 



It 



But 



1 2 sin^- 2 sin^ s i • i //i 2\ 



J 2 2 1-cosi l-v/(l-s^_, , 1 . , 1 „« 



: = : . : = -2 ■ = ;:2 -2 + H'5 +T6"* • 



:os*' - 4 sm- - cos'' - 

 Substituting this value, we have at length. 



