X '= X [1 + f"(2 - X) il + (f/2) (3 - 2X)!] 

 Y'= Y[l + f(2 - X)] + (f/2) (X^ - Y^) cos d 



(153) 

 (154) 



+ (f78) 



4Y (2 - X) (3 - 2X) 



+ (X^ - Y^) !(11 - 5X) cos d + Y (1 - 3 cos 'd)\ 



From Figure 22 we have 



X = 0.2752704532, Y = 0.1603011198, 



sin d = 0.97057512, cos d = 0.24079852, (155) 



T = 1.367856856, f = 0.0033900753, 

 f/2 = 0.00169503765, fV8 = 1.436576317 x 10"' 

 Using the values from (155) to compute d ', X', Y ' from (152), (153), (154) find: 

 d'= (0.97057512) (1.367856856 - 2.717164 x 10"" - 1.2634 x 10*') 



= (0.97057512) (1.367583876) = 1.327342885; (156) 



X'= (0.2752704532) (1.005871239) = 0.27688663; 



Y'= 0.160301120 + 9.37275 x 10'" + 2.0440 x 10'= + 4.068 x 10"'= 0.16126290. 

 From Figure 21, d'= 1.327342885, X'= 0.27688668, Y'= 0.16126298 and comparing 

 with the values from (156), we have a "flat" check for d', 5 in the eighth place for X ' and 

 8 in the eighth place for Y '. Now the first significant figure of f^ is 1 in the 5th decimal 

 place and of f is 4 in the 8th decimal place. If seven place tables are used, terms in f^ 

 are sufficient. If eight figure tables are used, as Peters trigonometric functions, there is 

 some uncertainty in the 8th place of decimals. 



Now the corresponding formulas for d, X, Y in the terms of d ', X', Y'are found similarly 

 to be, to second order terms in f inclusive; 



d = sind' lT'+(f/2) Y'+(f78) [2 Y' (X'- 1) + (2Y'' - X'^) cos d']! 



X = X'[l + f (X'-2)!l + (f/2) (2X'-l)n (157) 



Y = Y'[l - f (2 - X')l - (f/2) (X'^ - Y'^) cos d' 



(fV8) 



4Y'(2-X') (1-2X') 



+ (X'^- Y'')1(5-3X') 2cosd'+ Y'(l-3 cos 'd')! 



From Figure 21 we have 



X '= 0.276886675, Y'= 0.161262981, 



sin d'= 0.97051129, cos d'= 0.24105566 



T'= 1.367673822. 

 With the values of X', Y ', sin d', cos d ', T'from (158) and of f, f/2, fVS from (155)' 



(158) 



92 



