On Practical Geodesy. 35 



If it were possible (and it is usually supposed so in applying 

 Legendre's and Delambre's processes in the solution of 

 questions pertaining to the spheroidal triangles of a trigo- 

 nometrical survey) to find a sphere such that a spherical 

 triangle described on its surface can have sides equals in 

 length to the sides of a spheroidal triangle, and chords equal 

 to the chords of the spheroidal triangle; then, it is obvious 

 that by putting D^, D.,, D3, for the angles of this spherical 

 triangle which correspond to the angles Ki, K^, K3, of the 

 chordal triangle, we should have — 



cos K 



cos D^ = 7—. ; r ^—, — : 



1 COS 1 (a^3 + a3 J • COS ^ (a^3+ a, J 



— tani (a^g + a^J -tan J («] 2 + ^21) 

 cos K„ 



cos D, 



cos D. 



cos i («2 1 + «i 2) cos 1 (a, 3 + ttg J 



— tani(a3^ + a^J ' ^^n J (a,3 + a^^) 

 cos K„ 



)(]33) 



COsi (a3 3 + a, 3) C0sl(a3, + a^3) 



— tanl(a3^ + a,3) ' tan J (a^^ + a^3) 



By comparing the values of the angles D^, D^, D3, of the 

 imaginary spherical triangle as given in the formulae (133), 

 with the correct values of the corresponding angles C^, C,,, 

 C3, of the spheroidal triangle as given in formulae (132), it 

 is evident that, with due respect to the utmost accuracy 

 required in practice, we have — 



cos C^ — cos D^ = tan J (a^3+ a^J tan J {a^^-\- a^^) 





— tan a^ 3 tan a^ ^ 



cos C, 



— cosD, = tanl {a,.^+ a^^Jtan 1 {a,^^+ a^^) . 



— tan a^ ^ tan a^ 3 





cos C3 



— COSD3 = tan J(a33-f- a^ 3) tan i (a3^ + a^3) 



— tan a^^ tan a^^ 

 their logs being the same to at least 8 or 9 places of decimals. 



From these it is evident that cases may occur in geodetic 

 surveying in which one of the angles of the spherical triangle 

 is greater than the corresponding angle of the spheroidal 

 triangle, and that another angle of the spherical triangle is 

 less than its corresponding angle of the spheroidal triangle. 



