360 



The Rev. Brice Bronwin on the 



and we have that of A 



Ids 



-^y. Whence, if we have respect to 

 (3.) and (4.), we find 



dt ^ dt dt ^ dt dt 



M 



di\ 



+ (A AS A + ACSC + AESE)^^ + (A ASB + AC8D + AESF)^^' 



dt 



dt 



+ (ABSA + AD8C + ^FSE)>)^ + (AB8B + ADSD + AF8F)>j^]. 



If in this we make A and 8 change places, we shall have the 

 expression of 



^ Ad/.r,s, Hdy . A^ 



But by (2.), 



dt ' '■^ dt 

 AA8B + ACSD+ AE8F = 

 AB8A + AD8C + AF8E = 



COS'' 2 



1 

 COS" I 



dt ' 



AE8F 



AFSE 



(5.) 



Whence by substitution, making — ^^^^ — - = h, and having 

 regard to (5.), (B.) becomes 



dt 



^^-dt-^^-dT-^ ^''-dt-'^'^-dr 



+ 



h 



y (c.) 



(AE8F-- AFSE) + ARrf^ = 0. 



cos"? 



This last divides into two. Make p = — E, gr = F, and 

 we have 



^.{d.qlp-d.plq) + ^Rdt^O.. . (D.) 



Or by a further and obvious transformation, 



^sini(A2 8fl - A98?) + ARrf^ = 0. . . (E.) 

 And we shall also have for the determination of the co-or- 

 dinates or elements on the plane of the orbit. 



This, from the nature of the characteristic A, is equivalent to 

 Id^ f/R „ 8r7>j , c?R ^ .^ ^ . ^ , . 



^1+71=^' 77+^ = ^'^^=^'^"=^- • (;•) 



These will give the four elements, a, e, ir and e. But with 

 any elements whatever, constant or variable, the values of 



