366 Dr. Tiarks on the Longitudes of the 



w the angle formed by the planes of the meridians of the 



two points on the spheroid B and D. (fig. 1.) 

 X, x' their latitudes ; and let X be greater than X'. 

 Z, /' the reduced latitudes, or the angles dependent on X, x' 



by this equation tang./ = */(d — ef) tang. X. 

 m 9 m' the angles formed at the intersection of the geodeti- 



cal line and the meridians between the former and 



each of the latter. 

 ju,, f/J the angles in a spherical triangle, two sides of which 



are 90°— X and 90° — X', and angle between them =a> ; 



p! being opposite to the side 90°— X, and p opposite 



to the side 90°— x', and 

 /3 the third side of the spherical triangle opposite to oo. 



Let PDF and PEB represent the two meridians whose in- 

 clination to each other is = w, and let the lines BC and DA 

 be perpendicular to the meridians at B and D. Draw AE 

 parallel to CB and CF to AD, and join A and B and C and 

 D by straight lines. Let CB be = r, and AD = r 1 ; the an- 

 gle EAB == ABC = y, and the angle FCD = CDA = *. 



It is then clear, that the inclination of the plane BCD to 

 that of the meridian PEB is = m, and that the inclination of 

 the plane BCF to the plane of the same meridian, is = jx, and 

 that ft>m. If we now assume that the three lines CB, CP, 

 CF determine on a sphere described about C as a centre, the 

 angular points of the spherical triangle PB'D' (fig. 2.), and 



Fig. 1. 



that on the arc 



DD' is made equal to x 9 we shall 



have 



