Mr. Ivory on the Longitudes of the Trigonometrical Survey. 433 



Using tbe same symbols as in this Journal for October, the 

 two following equations, which are rigorously exact, contain 

 the full solution of this problem ; viz. 



sin X sin X' 



A= •!-<?' sin* A, A'=-/l-e*sin 2 A', Q=-^- 



sin u . ., cos X 



-f- cos on sin X— cos A tan A' = 7 . c* A' Q / (A) 



tan m cos A' 



si n u • . i , i A » cos x.' 9 ~ 



7 + cos a) sin A' — cos A' tan A = .rAQ 



tan wi' ' cos x *-* ^ 



I 



These equations express the condition that two vertical planes, 

 one at each station, intersect in the chord joining the stations. 

 As the investigation is merely elementary, it may be omitted. 

 The two equations, although very simple, are alone sufficient 

 for the solution of the problem, since the excentricity and the 

 difference of longitude are the only unknown quantities they 

 contain. In order to simplify, I shall put x = sin A— sin A'; 



and I shall write r . e* x and . e 2 x, for the quantities 



cos X' cos X 7 ' 



on the right-hand sides. The equations coincide with the 

 surface of a sphere when e 1 =s 0, and when x = ; and as 

 x is always very small in the practical application of the pro- 

 blem, it is with difficulty that the quantities on the right-hand 

 sides enable us to distinguish between the sphere and a sphe- 

 roid of small oblateness. It is here indeed that the difficulty 

 of the problem lies ; and it will easily be conceived that, with- 

 out nice discrimination, one case is apt to be confounded with 

 the other, as it actually is in the method of the Trigonometri- 

 cal Survey. In order to solve the equations so as to give full 

 effect to the excentricity of the spheroid, it would be requi- 

 site to free them from the almost evanescent factor x ; but this 

 is what I shall not at present attempt to accomplish. As 

 I write in haste, I shall not inquire how the value of the ex- 

 centricity is to be deduced, but shall confine my attention to 

 the difference of longitude, supposing that the observations 

 have been made upon a spheroid of a known figure. 



The two equations may be brought to a form fit for calcu- 

 lation by the following transformation : viz. 



tan u = : i , tan u' = 



sin \ tan m ' " sin x' tan m' ' 



cos(u-\-u) ' tan X' 

 cos u ' tan X 



\ n sin X' /' 



cos (u' — to) tan X / e^x \ 



cosw' " tanX' \ sin X /' 



(B) 



New Series. Vol. 4. No. 24. Dec. 1828. 3K With 



