The Rev. Brice Bronwin on the Theory of Planetary Disturbance. 51 



^^""^^ hdi^r^+hdAi^-^^''' ^^''^^'- ^^^° 



2 sin^ ^ = 2 sin= '-° + J cos iPdt = 2 sin^ ^ + j Pdt - 2 j sin^ ^ Pdt. 



But substituting this value of 2 sin" - under the integral sign of the second 

 member, 



2 sin^ ^ = 2 sin= ^" + cos i, \ Pdt - ^ {\Pdtf + 2 |Prf< J sin= | Pdt. 



By continuing these operations, we find 



2 sin^ 1 = 2 sin^l + cos i, | Pdt - \ cos i„ {\Pdif + ^^ cos j„ {\Pdty - &c. (15) 



This may serve to eliminate sin- -. But since 2 sin- - = 1 — cos i — \ — 



a/(1 — sin^ z) = \ sin^ * + i sin* i + -jJg- sin" i + &c., it may also serve to elimi- 

 nate sin" i and sin i, if we should find it convenient to do so. 



It is much easier to find r and v on the plane of the orbit, than to find the 

 values of the corresponding quantities on the fixed plane; but it is more diffi- 

 cult to find the latitude in the former case. It is very troublesome to find it 

 directly from the variable values of the elements i and 5 ; and yet we must 

 have the values of these quantities separately to a considerable degree of ex- 

 actness in order to find a. M. Hansen has found the latitude by means of sin i 

 sin 3-, sin i cos b ; which is a much better method ; and he has found a by means 

 of these latter quantities, or rather functions derived from them. But this is 

 attended with a great deal of trouble. I propose to pursue a different course, 

 and to find sin i, or s, separately, by the equation given above for the purpose, 

 and then to find i/. To do which I shall find 



y — sin (a- — 3 — fn£), 



X being a constant quantity. This is finding a function of 3 -(- ^pit. 



dy _ dij / d^ 

 dt~d^y'^^Tt 



r l-sWdR dRdA\-] dy ^ 



%f_ \-s^ fdR 



h2 



