154 



GEODESIC INVESTIGATIONS. 



Putting a' and a" to represent tlie angles PS,S^^, I^S^^S^, of tlie triangle 

 S,PS,^ ; it is evident, from what has been already shown, that in all 

 cases of mutually visible stations S', S", on the earth's surface we have, to 

 a very minute accuracy — 



A' + A" = a' + a" 



And since we can, by one of Napier's analogies, express the angle S^PS^, or w 

 in terms of the sum of a', a", and the sides l^, l^, ; therefore, by substituting 

 A! + A!' instead of a' + a" in such analogy, we have — 



tan \ cc = ^0^ i (y-h\ eot i (A' + A") ; 



cos i ilj,+ ly) 



tan i CO 



_ cos i {I'—l") 



cot i {A' + A") .. 



sin i- {I'+l") 

 This important formula is known as Dalby's theorem. 



(1) 



By applying Delambre's analogies to the triangle 8,PS,y, and assuming 

 A' + A" := a' + a", we have — 



sin i {A'+A") = cos i (I'— I") 

 cos i (A' + A") = sin ^ {I' + l") 



cos i o) "" 

 cos i J- 

 sin i 0} 



tan i (^' + ^'0 - °°' f Sf~S • eot i a, 



(2) 



siui {I' + l") 



cot ^ (^' + ^'0 = ?i^4i|±S . tan i a> 

 cos ^ (Z — I ) 



Again, by applying fonnulse (1), page 157 of Serret's Trigonometry, to the 

 triangle S^PS^, we have — 



cos ^ (A' + A + OO) . ,.-0 1 7,^ , /,-o 1 7//N 



— ; ; ^/ — 777-^ — ( = tan (45°— I V) ■ tan (45° — i T) 



cos t (^ + ^ — '•' ' 



■0,) 



cos i (A' + A" + Q}) 



cos i (A' + A"—cc) ~~ 



tan Y ?/• tan i l„ 



(3) 

 (4) 





From the triangles 'S'^P/, S^^PI, we have — 



sin A' cos i 2' = cos sin (p, ) 



sin ^" cos I 2" = cos sin ^^, j 



sin A' cos Z' z:^ cos 6 sin )3, "i 



sin A" cos /;" =: cos 6 sin i8„ j 



sin e = sin i 2' sin I' — cos i 2' cos I' cos A' ") 

 sin e = — sin i 2" sin Z" + cos i 2" cos Z" cos A" ) 



tan i 2' cos ?' := cot <p, sin ^' — sin I' cos ^' ") 

 tan i 2" cos ?" =: cot ^„ sin ^" ~ sin I" cos ^" / 



tan I' cos ^ 2' = cot fi, sin ^' — sin ^ 2' cos ^' ") 

 tan I" cos i 2" = cot ;3,, sin ^" — sin i 2" cos A" j 



... (5) 



... (6) 



- (7) 



... (8) 



.. (9) 



