83 



8. The above formula will also serve for an easy investi*-- 

 gatiou of an expression, for the correction of the distance.. 

 We have cos a — cos t/ = (1— e sin 1") (cos A — cos D) 

 putting a+i^+y+i? = e sin 1" 

 Let « = A + i & d =D + a? 

 Then cos a .= cos A— 6 sin 1" sin A — ~ sin* 1" cos A. 



cos d = cos D — .r sin i'' sin D — '^ sin* 1" cos D . 

 Hence 



X sin D — b sin A + {i* cos D — h* cos A) i sin 1" = — e. 

 (cos A — cos D) 



From Avhicli equation we obtain 



J sin A e (cos A- — cosB) j^ • ,„ , -t-v /isinA— f (cos A — cos D)\2 



iir= — f: '■ — T\ — 2 sin JL cot u i ^-w i 



siiiD sin D ^ -^ \ sin D ' 



I 1 • 1 // ; ^ cos A 

 + * Sin 1" b ^^, 



and because 



i sin A + I sin 1" b" cos A = & sin (A + *) 

 also cos A — cos D = 2 sin (~2~) ^^^ \2~) 

 Ave have, sufficiently near the truth 



b sin (A+h -2e sin ('+-P) sin (?=^) . . • -,„. o . T\ 



X = '' \ "- } V ■i ^ (c) — i sm 1" c- cot D 



sin D 



In this formula c is easily computed by the assistance of 

 proportional logarithms, logarithmic tables, to four or five 

 places of figures, and tables for the value of e similar to the 

 ttlble for N. The term J sin 1" c* cot D may be had by a 

 table sufficiently convenient. This method of finding the 

 correction of the distance is shorter than those in the requi- 

 site tables, and than that of M. Legendre in the Memoirs of 

 the National Institute. It is, notwithstanding the difference 



in 



