extend the results to second order in c and subsequently to second order in f since c can be 

 expressed as a series in f. 



The quantities needed to achieve the approximation are found in or derived from the results 

 of Forsyth's work, pages 86, 97—105. We list them here for reference in the development. 



$ = (yS + — u'sec a tan a [1 +-^{l - 6 tan^a)] Ilia 



z o 



u'=i.'+cU + c'V 111b 



<?!,= </,'+ cfl + c'«P 111c 



a = Oo + cA cot Oo + c^B Hid 



en u = cos u ' i 1 - /i c sin^u ' sin^u ' (7 + 4 cos^u 0! Hie 



64 



c = (e^ cos'a)/(l - e^ sin'a), e^ = 2f - f\ e^ = 4f^ 



c = 2f cos'a + f cos'a (3 - 4 cos'a) 11 If 



cos = en u cos a IHs 



tan $ = tan u'csc a [1 + %c + (l/64)c' (9-2 sin V- 4 tan^Oo)] Hlh 



-=(l-e^sin^a)'/^E(u) 

 a 



= u' + — [sin 2u'- (1 + 2 tan^a) u'] llli 



4 



2 



+ — [sin 4u'+ 4 sin 2u'(l - 2 tan ^a) + i8 tan^a (1 + 3 tan^a) - 3! u'] 

 64 



sin a = sin ao[l + c A eot^Oo + c^ cot Oq (B - H A^ cot Oo)] IHj 



cos a = cos Co [1 - c A - c^ tan a^ (B + ViK^ cot^Oo )] lUk 



tan a = tan Oo [1 + c A esc ^Og + c^ csc^a,, (A^ + B tan Oo)] 111m 



see a = sec ao [1 + c A + c^ tanao (B + A^ cot Oq { 1 + /^cot^aoH] lUn 



esc a = cse Co [1 - c A cot^ Oq - c^ cot Oo 1 B - ViP^ cot Oq (1 + 2 cot^Oo) IHo 



sin u'= sin i/'[l + c U cot v' + c^ (V cot v' - U72)] IHp 



cos u'= cos v'{l - c U tan v' - c^ (V tan v'+ UV2)] Hlq 



tan u'= tan v' + c U sec^ v' + c^ sec^ i/'(V + U^ tani^O Hlr 



sin 2u'= sin 2i^'(l + 2c U cot 2v') (to first order in c) 



tan (f>' = tan v' esc a^, \ + tan^i/' esc ^a^ = sec ^(f)' Ills 



U = -(A cot i/'+ (1/8) sin 2v'), A = - {v'/D tan a^ tan v' 



lilt 

 fl + (p''/2) sin a a sec ^Qq = — A esc a^ cot <^ ' 



82 



