1862.] The Trigonometrical Survey of India. 41 



it is believed that the values herein determined are a nearer ap- 

 proximation to the truth. 



76. Notices certain refinements not appreciable in these opera- 

 tions. — In concluding the remarks on these computations, it may be 

 interesting to notice certain refinements in calculation which have 



. not been deemed applicable to these operations. For instance, the 

 spheroidal excess and the contained arc might have been computed 

 by more rigorous processes, but that the refinement would have been 

 purely of an arithmetical nature. Again the formula for latitude 

 and longitude has not been employed beyond its fourth term, because 

 the remaining terms are difficult of arithmetical expression and 

 would besides have given no results commensurate with the labour 

 necessary to compute them. Similarly the chord correction is neg- 

 lected in these heights, amounting as it does in the extreme case of 

 Menai to Mont Everest, or XV, to no more than a foot. 



77. There remains to notice one other correction also herein not 

 taken into account, of which it may be remarked, that, under existing 

 circumstances it would partially cancel the chord correction, if both 

 these refinements were introduced. This correction may be stated 

 thus. 



78. Ordinarily, in the formula for computing difference of height, 

 it is sufficiently accurate to assume the given arc (or distance) to 

 belong to a circle, whereas in reality, it is a portion of an ellipse. If 

 the correction due to this assumption = x b, then it can be shown 

 that x b = (ya — Cos X b K) — (vj — Cos X a K), wherein K 



( N 



= 1 v h sin A. 6 — v a sin X a + — [ (M + v a Cos X a ) (M — v a Cos X a ) ] 2 



( M 



N i ) 



[(M + v a Cos X b ) (M — v h Cos X b ) ¥ [ Cosec 8 X. 



m ; 



It is sufficient to remark in this place, that in the extreme case of 

 Menai, T. S. to Mont Everest or XV. the correction x b = only 0.3 

 of a foot. 



79. Magnitude of these operations illustrated. — Lastly it may be 

 interesting to notice, that the area of the largest triangle to points 

 on the Himalaya mountains (No. 297) is about 1706 square miles, 

 its spheroidal excess being 106". The longest side, Anarkali, T. S. to 

 XXXIX. is equal to 151 miles, and its corresponding contained arc 



G 



