On Practical Geodesy. 17 



and .*., obviously, we have z^ = — , — — 



•^' R^ ' sin 1" 



And, I,- = Ps/ + R,r — 2 R/ • cos z,, 



= R,2 _|_ Q^2 _ 2 K^^ . Q^ . cos z, 



= P 



s 

 :., obviously, we have ^a = ^ — - — rr, 

 *^' R^^ sm I" 



Nevertheless it is evident that the perpendicular let fall 

 from the station S^ on the line S^^Z^, lies inside the triangle 

 S^Z^S^^, and that the perj)endicular let fall from S^^ on the 

 line S Z lies inside the triano-le S Z S ; and .". that 1 ~7 h, 



O OO _ O O 00 00 ' / ' 



and also l^^i h; and that, with respect to absolute accuracy, 

 we have — 



"' R, sin 1" ^ " R,^ sin 1" 



However, the values of z^ and z^^ as given by (75) are such 

 that for a distance of a degTee along the meridian they 

 cannot differ from the absolutely true values by as much as 

 Tu of an inch of error in the length of s would cause. (See 

 "Account of," &c., page 247.) 



It is no easy matter to guard against inferring that 



z,, can never be greater than -. — ^, ot (a + a). But 



that z^^ can be greater than a^^ + a^ may be easily seen in 

 the following manner : — 



It has been already shewn that in all cases in which l^ is 

 greater than l^^, we must have D^ greater than A^. Now if 

 we suppose the point S^ fixed on the spheroidal earth (and 

 .'. S^ also fixed on the unit sphere), and that the point S^^ 

 (which has S^^ as corresponding point on the unit sphere) 

 assumes at first a position such that I, = l^^, and then moves 

 continuously along the meridian in which it is situated, 

 makino: I less and less until the ano-le A, becomes = 90°, 

 then of course D, from being equal to A, at the commence- 

 ment must have increased continuously until at length it 

 exceeded 90°. And it is evident that at one state of the 

 implicated entities, the angle D^ was 90°, and A/ less than 

 90°, and .*. that in such state sin A, was less than sin D,. 

 But if we were to assume that z^^ should be always less than 

 ^// + <^/> 01' never greater than a^^ + a , then ID^ should be 

 always greater than IS^, and .*. sin A^ always greater than 

 sin I)„ which we know to be absurd. 



D 



