GEODESIC INVESTIGATIONS. 65 



„ sin i {A„-\-i") . „ „ 



cot i a" = -. — , ) ." ■ — ^ tan i (y—p) 



sin ^ (^/, — i") 



log tan i {y"-p") = 10-0018527 

 log sin i (^„ + r) = 9-9626464 



19-9644991 " .-. i «" = 44° 50' 51"-853 

 log sin i (A^—i") = 9-9621908 



10-0023083 .-. a" = 89° 41' 43"-706 



A = i" - i" — 0° 8' l7"-800 — 0° 8' 13"-309 

 = 0° 0' 4"-491 

 = The angle between the " Normal Chardal planes." 



From this it is evident that the normals (at S' and S") at their nearest 

 points (points \ (p' + p") distant from the respective stations) have their 

 distance asunder denoted by 



D = (p' + p" — versin d) x sin 4"-4-91 

 2 

 = 20933670 X • 00002177298 

 — 455-78 feet. 



Test of accuracy of calculations : — 



The angle C which the chord subtends at the centre of earth has been 

 obtained from the plane triangle, and tht angles a' , a" which the chord makes 

 with the central radii to its extremities have been obtained from the first and 

 third spherical triangles respectively, so that, if the work be correct, the sum 

 of the three must be equal 180°. 



We have C = 0° 52' 9"-036 

 a' = 89° 26' 7"-268 

 a" = 89° 41' 43"-706 



180° 0' 0"-010 



And this is very satisfactory, considering that the tables used are to seven 

 places of decimals only. 



The accuracy of the work may also be tested by " Dalby's Theorem," 

 which gives results very near to rigorous accuracy : — 



*^" ' " = sinH^"+0 ' ^ ^ ^ 



log cot i {A"+A') = 7-8028362 

 log cos i (I" -I") = 9-9999940 



17-8028302 .-. i w =r 0° 28' 51" ■ 857 



log sin i {l" + l') =9-8787330 



7-9240972 ' a; = 0° 57' 43" • 714 



which differs only by about ^^ oi a, second from the result already obtained 

 for the difference of longitude of the stations. 



E 



