of' the Trigonometrical Survey of JLngland. 58 



Now Mr. Ivory contends that the quantity on the right-hand 

 ought to be multipUed by A + -|^ cos (-^^ ) ); thus add- 

 ing to the former value of tang ^ w a quantity multipHed by 

 the square of the excentricity, which increases the value of 

 CO deduced from the formula by a quantity nearly equal to 



— cos ( ^ \ sin w. In the case in question this quantity 



amounts to 7", and would thus leave only 11" of the difference 

 in longitude to be accounted for by the errors in the data of 

 the Survey, principally by the errors in the sum of the azi- 

 muths. I have endeavoured to prove that in the development 

 of the value of tang \ <o there is no such term, involving the 

 square of the excentrictiy, and Mr. Ivory's last paper has not 

 proved the contrary. At the same time I have fallen into the 

 mistake of contending that the formula for tang ^ w is rigorously 

 correct ; while it really contains terms proportional to the 

 fourth and higher powers of the excentricity. The mistake 

 has arisen from my confounding the azimuth of the station ob- 

 served with the azimuth of the geodetical line connecting the 

 stations at the place of observation. For the azimuth of the 

 geodetical line at B we may take that of the point of intersec- 

 tion of the vertical line at D with the horizon of B. Calling 

 B and D the azimuths of the geodetical line at B and D, it 

 will be found that nearly B — ?« = nj'— D = 



c* . (sin \— sin X') sin m . cos X . tang \ /3, 



, , , e* , . • (\Q sin (■m-\-'m') cos X . cos \' 



and f* + /*' = OT + OT -— . (sm A-smX')'- '^^^. ' 



This value of /x + jw,' ought to be substituted in the exact 

 formula 



^ 1 cos(^)^otang(^|^ 



Tang \ m = . / x+V \ 



The three quantities B + D, m + m', and ju. + /x', are conse- 

 quently only equal as far as terms involving powers of the ex- 

 centricity below the fourth are taken into account. As Mr. 

 Ivory intends to deduce the value of w expressed by the quan- 

 tities m, m', \, k', e from the equations A of his paper, he will, 

 no doubt, decide whether Dalby's expression for tang ^ w or 

 the same with his correction is more accurate. 



Dec. 12, 1828. J. L. TiAKKS. 



\'1II. No- 



