GEODESIC INYESTIGATIOA^S. 153 



The triangle S,,II),^ is evidently such that, — 



angle IS„D„ + angle ID„S,, -7 ir 

 but angle PB„S, + angle IB^^S,, = tt 



angle IS,,D„ 7 angle FD,,S, 

 or A" -7 D„. 



If we apply Serret's theorem (already mentioned) to the triangles S^ID^ , 

 SjjIDj, , we easily find — 



tan 2 (^/ — ^0 ^^^ ^^^ 2 ^ 



tan i (J."— D,,) = tan i A 



sini{z,-a2"-i20] 

 cosi[;;„-a2"+i20} 

 8ini[s,4-a2"-i2')| 

 cosi|z,-(i2" + ^20 j 



intimating that D, — A' and A" — !)„ are both positive ; and .•. that D,"? A' \ 

 and A!' T -D,/ . 



And, from these, we have — 

 tani(A-^0 ^°^> {.-a2- + i20} sin ^ {..-g 2^-1 20 } 

 tanH^"-A,) - eosi {.,-(4 2"+i 2') } ' sin i {.,+ a 2"-^ 2') } 



That A' is greater than A!' is evident from equations (12) given in the 

 sequel. 



^^ In the " Account of the Trigonometrical Survey of G-reat Britain and 

 Ireland" (see page 249 of that work) it is assumed that — 



i>, is greater than A' . 



and that D„ is also greater than A" 



It is easy to perceive that the formulae there given to express the differences 

 D, — A! , A" — I>^i is erroneous, viz. : — 



D -A' = i -^ . cos2 v. sin 2 A' . «/ 

 1— e- 



B^.—A" =.\-^^. cos2 I"- sin 2 A" . 2/ . 

 1 — e^ 



For, assuming the difference of longitude of the two stations to be 1°, and 

 I' = 50° , it is evident that we can have values for A' and A" each less than 90°, 

 and .'. the right-hand members of these equations positive : — thus intimating 

 that D„ is greater than A", which we know to be absurd. 



If we were to assume A' = 90°, then it is evident the above formulae would 

 intimate that 



I>, = A' = 90° ; 



which is also absurd : for from the point I, which is not the pole of the great 

 circle PS, , we cannot have two arcs IS, , ID, perpendicular to this circle. 



If we suppose the latitudes I', I" to be equal to each other, then we know 

 that in all such cases 



D, — A' = 0', A" — D„ = 0; 



but in such cases, the above formiilae would intimate that both J), — A' and 

 A" — Dj, have real finite values. (See note to problem 5) . 



