442 KRETZ. 



Hence, after slight reductions, from (13) and (14) 



X=sin (a — a^) tan/ [ Ftan/g + l] cos/q > 

 X 



SO that 



or 



and 



tan/ = 



cot 6 



sin ( a — Ofl ) [ F sin /g + cos /q] 



X 



sin Aa [ F cos Jq -|~ sin (Jq] 



/j> , A t^ sin Aa f F4- tan rffl] , ^ 



Expanding and reducing we obtain finally : 



sin Aa f F+ tan dfJ — X sec ^n tan Jn . , 



tanA(5 = — , , . — , Vt^ . r-? r' (16) 



A^ sec c\ -f- sin Aa [ V-\- tan Oq] tan Oq ^ ' 



This may be written, substituting for the sine, 



A(5=tan- 



/ Aa" Aa^ \ 



/ Aa -_ -I- — J [ F+ tan (5^] — X sec Jq tan Jq 



7 Aa" AaS \ ' 



X sec (^0 + ( ^« y~ -\ ) [ ^+ tan rf^] tan 6q 



and if we replace here z/a by its value from (12), divide the 

 numerator by the denominator, and then apply the formula for 

 expanding the arc-tan., we get, to terms of the fourth order, the 

 following series for J3 : 



A6= Y^D^{X sec SqY Z^^ = — ^ sin 2 (^q, 



+ I?,{Xsecd,yv D.,^ — \, 



+ i?^( A^sec S^)'^ F^ D^^—l sin^ 6^ tan 6^, 



-^ I)^{X see 6qY Z'5 = l(3sin JqCosSJo 



-(- sin' ^Q cos Sq). 



(I7)> 



When Aa and Ad are known approximately from meridian 

 observations a still more convenient form may be deduced from 

 ( 1 2) and (17) by inversion of series. It is preferable in several 

 ways : the labor involved in the calculations is slightly smaller 

 and the results are somewhat more accurate, as the Aa and Ad 

 used are free from errors of scale-value and orientation, which 



1 See footnote, p. 443. 



(102) 



