160 



aEODESIC INYESTIGATIOFS. 



All these expressions are rigorously accurate for any tvro stations on the 

 earth, or any ellipsoid of revolution whatever ; and from the expressions for 

 either the cosines, sines, or tangents, we can obtain those for all the elementary 

 functions of i 2', ^ 2", in terms of the implicated entities. 



^^° By equating the values of tan J 2' given in (44) and (45), we can 

 easily express the sine of the diifereuce of longitude of the two stations as a 

 direct and explicit function of the two latitudes and two azimuths, — the 

 resultiug formula being rigorously exact for any two stations on any spheroid. 

 This has not been hitherto effected; for Dalby's Theorem is applicable only 

 to mutually visible stations on the earth, and is even then but a close approxi- 

 mate. 



^^° By means of the equations (46) we can easily express the squares of 

 the sines, cosines, and tangents of the angles of depression, as rational 

 functions of the latitudes and azimuths only ; but the forms so arrived at 

 take the indefinite form — when H^ =i H^^ (which is the case on any spheroid 

 when the latitudes of the stations are equal, , and always the case on the 

 sphere, no matter how the stations may be situated). They are not adapted 

 for calculations in obtaining the angles of depression, unless the latitudes of 

 the stations differ considerably. However, they are often of use in trans- 

 formation of equations. They are — ■ 



cos i 2')^ = (-^/ cos I" sin A'Y - {R„ cos I" sin A"y ^ 

 {Bj cos I" sin A")'^ — {E, cos V sin A')'^ 

 ^ {R , cos V sin A')~ — {B„ cos V sin A')"^ 

 {R„ cos I" sin A"y — {R^^ cos V sin A')"^ 



{R„ cos I" sin A")- — {R^ cos V sin A'Y 



{R, cos I" sin A")~ ■ — {R^ cos If sin A'Y 

 ^ {R „ cos I" sin A"Y — {R, cos V sin A'Y 

 {R^, cos I" sin A")'^ — {R,^ cos V sin A'Y 



/p {R„ cos I" sin A"y- — {R, cos V sin A')~ 



(R, cos I" sin A")^ — {R^^ cos l" sin A"y 



tan J- 2")- = (-^// ^°^ ^■" si" ^"Y — (^/ cos I' sin A'Y 



{R, cos I' sill A')"^ — {R^^ cos I' sin A')'^ J 

 Expressions for the angle between the normals at the stations. 

 cos V == sin I' sin I" -f cos I' cos I" cos co ^ 



gin- i y = sin- i {I' — I") -\- cos I' cos l" sin= Jo, j •'• 

 Expressions for the angle between the normal chordal i3lanes. 

 sin ^' sin 2Z' R^, sin ^-1" sin 2 Z"\ 



) 



OS i s.y 



sin i 2')2 

 tan i- 



(47) 



(48) 



sm A 



-*<l 



sin A = e' 



sin A = e-'k 



cos i2" 

 o R, sin l'—R„ sin I" 



1 



(i?/ - R,;Y 

 siu A' sin A" 



R, cos i2' 



(cos^ I" sin- A" — cos- I' sin- A'Y j. (49) 



( silt V sin l"\ 

 X 'r7) 



f sin i (r + 12' -f i2'0 . sin 



(v 



cos i2' . cos i2" 



1^] 



(50) 



In the " Account of the Principal Triangulation of Grreat Britain and 

 Ireland," an erroneous expression is given for the angle A, viz. : — 

 A = e2 sin 2A' cos^ {V -j- l") . i2 

 A—e- Bin 2A" cos^ (I' -f /") . |2 



