I20 JOURNAL OF THE 



It is more expeditious, however, to compute the success- 

 ive values of / by differences. Thus, if we change N to 

 N + I in (23), we get the angle / made by the chord from 

 sta. (N + i) to sta. N^ with the Y axis. Subtracting (23) 

 from this we have the angle made by the two chords, from 

 N to Ni and (N + i) to Nj equal to 



ist difference = 2 (Nj 4- 2 N + i) miimtes .... (24). 



x^s N increases one at a time, the first difference changes 

 four at a time, 



. •. 2d difference = 4 minutes. 



We observe from (23) that for N = o, z' = [\ = 2 N, ^, 

 which agrees with (19); also for N = N^, the right mem- 

 ber reduces to 6 N^, which by (7) is exactly equal to the a 

 corresponding to the station. 



Hence, starting with N = o in (23), which gives A, and 

 increasing N one at a time, we compute the corresponding 

 i^ s until N = Ni when the a at sta. Ni is found. As N 

 again increases one at a time, the following i' s are found. 



The formula (23) is found to be exact to the nearest min- 

 ute, when compared with exact results, and is more nearly 

 correct the less N and Ni differ and for N = Ni it is abso- 

 lutely exact. 



As an application, let Nj = 5, whence first difference by 

 (24) = 12 + 4 N and second difference ^= 4. 



For N = o, angle between chords 05 and 15 = 12'. 

 N ^ I, " '' " 15 " 25 = i6^ 



N ^ 2, " '' '' 25 '' 35 = 20^ 



Similarly for the others. 



Starting with /\^ = 2 Ni ^ — 50' and adding the success- 

 ive differences above, we find, i^_^ = 50 -f- 12 = 62^ /^-s 

 = 78', V5 = 98', Vs = 122', /3_3 = a^ = 150', /._e = 182, 

 and so on. 



As (23) reduces to <? for N := N,, the method of differ- 

 ences above evidently applies in finding a^ from i\_^ and /^.e 

 from a^. 



