Mr. Ivory's Solution of a Geodetical Problem. 35 



I might describe other applications of the principle to the 

 preservation of iron, steel, tin, brass, and various useful me- 

 tals ; but I shall reserve this part of the subject for another 

 communication to the Royal Society. 



V Solution of a Geodetical Problem. By J. Ivory, Esq. 

 M.A. F.R.S. 



IT is proposed to solve the following problem: 

 The length of a geodetical line on the earth's surface, to- 

 gether with the latitude, the longitude, and the azimuth, of one 

 of its extremities, being given; it is required to determine the 

 latitude, the longitude, and the azimuth of the other extremity. 



The earth, being supposed an oblate spheroid of revolution, 

 we may put unit for the pola r semi- axis, and represent the 

 equatorial semi-diameter by • 1 +«»: then, if x, y, z be three 

 rectangular co-ordinates of a point in the surface, x and y 

 being parallel, and z perpendicular, to the equator; we shall 

 have 



*»+y* 



+ z 2 = 1. 



l+e* 



And if ds represent the element of a line traced in any manner 

 upon the surface, we shall further have 



ds= \/dx 2 + df + dz\ 

 Again, if we assume, 



x= i/l+e 2 . cos <p cos ty, 



y= a/ 1 +e 2 . sin <p cos \l/, 



2= sin 4/, 

 these values will satisfy the equation of the surface, without 

 supposing any relation between the angles £ and «|r. Diffe- 

 rentiate these values, and substitute the differentials m the 

 expression of ds; and, for the sake of brevity, put 



V= Vl-K sin'vl/ + (1 +<?) cos 2 vJ/. |£, 

 then, ds = d*V. 



It is easy to discover that v|/ is the true latitude; that is, it 

 is the complement of the angle which the perpendicular to the 

 surface of the spheroid makes with the polar axis. If A de- 

 note the given latitude of the beginning of the geodetical line, 

 and 8X the difference of latitude at any other point, then 

 J, = A + 8 X The arc f measures the angle between two meri- 

 dians • it is therefore the difference oi longitude reckoning 



Yj 2 from 



