of a Degree perpendicular to the Meridian, 243 

 Thus we finally obtain, 



Cos tV = — cos x ' , cos M sin w = sin (3 sin m, 



Cos \I/ = — , cos \I/ sin « = sin B sin w'. 



Suppose now that a spherical triangle is constructed of 

 which the base is equal to the arc /3, and the two sides to the 

 arcs 90 — if/ and 90— 4/' : I say, that the angle of this triangle 

 opposite to the base /3 is not sensibly different from w, the 

 difference of longitude, when the two latitudes are nearly 

 equal; and is exactly equal to it when the two latitudes 

 are equal. In order to prove this, it is to be observed that 

 a cos 4', a cos \[/, are the respective distances of the two stations 

 from the polar axis of the spheroid ; and a sin fy V 1 — e 2 , 

 a sin \(/ \/ 1 — e 2 , are their distances from the plane of the 

 equator. Wherefore, because co is the angle between the two 

 meridians, we have this expression for the square of the chord, 

 viz. 



— = (cos \J/— cos v[/ cos co) 3 + cos 9 ^' sin 2 co + (1 — e z ) (sin \f/ 



—sin \[/) 2 ; 

 or, which is the same thing, 



-^- as 2 — 2 (cos\(/ cos ty cos w + sin ty sin ty) — <? 9 (sin\J/ — sin \f/) 9 : 

 but £* = 4 sin 2 — = 2 (1 —cos /3) ; and, hence, 



. cos /3 = cos \J/ cos v|/ cos co + sin \[/ sin \J/ + — (sin \j/ — sin \[/) 9 . 



Now, from this equation and the relation that is known to 

 subsist between an angle of a spherical triangle and the three 

 sides, it follows that w may be reckoned equal to the angle of 

 the triangle opposite to the base /3, whenever the latitudes are 

 so nearly equal that the term multiplied by e\ has no sensi- 

 ble value; and when the latitudes are exactly equal, the equa- 

 lity affirmed is rigorously true. But for the greater precision 

 let us inquire, what variation the term multiplied by e 2 will 

 produce in the arc /3. For this purpose suppose that /3 be- 

 comes /3 + 8 6, then, 



Cos (/3 + 8 13) = cos v[/ cos 4/ cos co + sin \[/ sin \J/, 



~ „ { (sin X— sin X f ) a (sin X— sin A.') 



8 — _i i-.x _i 



^ sin 1 y 



the true latitudes being written for v[/ and \J/, and — for sin j3, 



in the expression of the small variation. In the instance of 

 Beachy Head and Dunnose, we shall find 8 /3 = 0"*1 1, a quan- 



2 I 2 tity 



