36 Mr. Ivory's Solution of a Geodetical Problem. 



from the meridian that passes through the beginning of the 

 geodetical line. The azimuths at the extremities of the in- 

 tercepted part of the line will be denoted by p and [/, of 

 which the first is supposed to be known. 



The foregoing expression of ds is general, and will apply 

 to any line that can be traced on the surface of the spheroid. 

 But a geodetical line is the shortest that can be drawn be- 

 tween any two of its points ; and we may employ this pro- 

 perty to investigate its equation. Now, by making s and <p 

 vary, we get 



(IP ,» 

 v ' ay 



d^s^s. y ; 



and, by integrating, 



u = g o x v — d * -fl? x d - % . 



Hence the condition of a minimum length between two fixed 

 points, is expressed by this equation, viz. 



dp 

 d. ^— 



and, consequently, 



dtp 

 V 



— a, 



a being a quantity which is constant in the whole length of 

 the geodetical line. 



Let ds' be what ds becomes when x and y only vary, and z 

 remains constant; then, 



ds' = \/ 1 +e 2 . cos \[/. d$: 

 and, because ds = \ T dty, we get, by substitution, 



X^l +c'. cos -4<ds' 

 -ds - "■ 



But ds is perpendicular to the meridian on the surface of 

 the spheroid, and ds cuts the same meridian in the azimuth 

 angle fi'; whei'efore ds'= ds sin ft.'. Consequently, 



Cos \J/ sin ix,' = — : • 



This equation expresses a distinguishing property of the 

 lines of shortest distance upon the surface of a spheroid. The 

 product of the cosine of the latitude and the sine of the azi- 

 muth remains constantly the same through the whole length 

 of every such line. In the sphere, which is the extreme case 



when 



