Assurance that Forsyth's aa is the complement of the geodetic latitude, 0o, of the great 

 elliptic arc is found from his expression, [18] page 106, which is 



tan tto = sin 2 0o/ i (tan Ij + tan Ij)^ - 4 tan li tan Ij cos^^o i''''^- 

 With equivalent substitutions from (103) and some trigonometric identities it will transform into 



tan 05 = (tan ^0, + tan ^02-2 tan (fi^ tan (j)^ cos A A) ''^/sin A A 

 which defines the vertex of the great elliptic arc. See equations (21) of this investigation. 



A cursory check of the equations just preceding (102) in Forsyth's treatise revealed that 

 the numerical coefficient of the second order term *1 in (102) should be 15/64 instead of 23/64. 

 Then by use of relations (103) and (95) it was found that (102) could be written as 



S = 



d - (f/4) (Xd - 3Y sin d) 

 + (fVl28) (AX - BY - CX' + DY^' + EXY + FX'Y + GX') 



(104) 



where A = 64d + 16d^ cot d, B = 96 sin d + 16 d' esc d - 48 sin^'AA esc d, C = 30d + 15 sin 2d 

 + 8d^ cot d + 12 sin'AAcot d, D = 30 sin 2d, E = 48 sin d + 8d' esc d - 36 sin^AA esc d, 

 F = 6 sin^A A esc d, G = 6 sin ^A A cot d. 



Note that the first two terms of (104) are exactly the Andoyer-Lambert form given by the 

 second of equations (80). But we apparently also have the second order term in the flattening. 

 Thus, Forsyth had both so-called Andoyer-Lambert approximation forms as early as 1895 but 

 they had not been recognized as such. 



Equation (104) was used to compute several lines of known lengths. On those in which the 

 term *2 of (102) was small, an improvement would be obtained by including the second order 

 terms. On others, the error introduced would outweigh the first order correction, which could 

 mean, since equation (104) is a power series in f, that the coefficient of the second order term in 

 f is erroneous. Now examination of the second order terms of equations (82) and (96) shows no 

 cubic terms in X and Y as are found in the second order term of (104). Hence Forsyth's paper 

 [18], was reworked from the beginning and it was found that indeed the term *2 in (102) actually 

 vanishes and reaffirmation was also made that the numerical coefficient of the term *1 of (102) 

 should be 15/64 rather than 23/64. These errors are the result of carrying throughout the 

 derivation the numerical factor 9/32 in the last term of the expression for §, [18], section 17, 

 page 98, when it should be 3/32. This affects the approximation equation for tan $, section 22, 

 page 104. In the last term, the factor -7 sin ^a should be +5 sin ^a. This continues to be reflected 

 through section 27, pages 111 to 115, until the term is actually seen to vanish in collecting the 

 terms together on page 115. Also on page 115, omission of a factor Vi in use of a trigonometric 

 identity in the third line from the bottom gave the printed value 54 for the numerical coefficient of 



76 



