34- On the Distance of two Points of the Earth's Surface. 



d $ 



formula (4); and M, equal to -^- = 2A, is known by the 



formula (3) : it therefore only remains to compute the partial 

 differentials of M, which is easily accomplished in the manner 

 already explained. Omitting the detail of the calculation for 

 the sake of brevity, the result will be as follows : 



8B = <r(|-/3 ? (l-/3 9 )-|/3* + /3 9 (1-/3 2 ) ^- cos a cos a') 



+ (-|-/3 9 (l-^)-4 / 3 4 )sincrcos(a + a0 



+ ■— /3 4 sin 2c cos 2(a + a'). 



The values of A and B being now found, we have only to 

 substitute them in the assumed expression of s; but in doing 

 this I shall write sin i for /3, the symbol i standing for the in- 

 clination to the equator of the great circle of the celestial 

 sphere, which passes through the two given points. Thus the 

 following formula is obtained: 



b = «r . i I — sin- 1 -f —=• sin- i cos* i — tr sin 4 i 



L < 4 16 64 



+ 4- sin 3 ; cos 2 i ~— cos a cos a! [ 



'8 sin a > 



— { — sm- z rg- sin 3 z cos s z + -rr sin* * 5 sin <r cos (a + a!) 



15e* 



+ — g sin* i sin 2<r cos 2 (a -fa'). 



As this formula is perfectly general, it must comprehend 

 the case when the two points are on a meridian, that is, when 

 they are situated in a great circle perpendicular to the equa- 

 tor. On this supposition we have sin i = 1, cos i = 0, a = A, 

 a' = A', tr = A — x'; and the formula becomes, 



- (~- + Ir) sin ( A_A ') cos ( fl + a ')» 



+ -^ sin 2 (A- A') cos 2 («+«'), 



which coincides with the usual formula for the length of an 

 arc on the meridian of an oblate elliptical spheroid. 



The same method of investigation which I have here used 

 may be applied with advantage to other cases of spheroidical 

 trigonometry, as I may show on another occasion. 



June 14, 1830. James Ivory. 



VII. Notes 



