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first instance adopted. The selection has been made to the exclusion

of those values obtained from short sides.

Deduction.-Doom Dangi

Senchal
Thakoorganj
Senchal

Doom Dangi

Thakoorganj

}

= .07617.

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Tonglo

Tonglo

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13.2

69. With this value of f, the heights from Senchal and Tonglo were computed, and the mean of these values, as also the differences between each value and its mean, were next found. The heights were now corrected, in such wise, that when the heights deduced from Senchal are compared with the mean heights already mentioned, the differences should be numerically equal. The greatest and same process being gone through at Tonglo, H. S., there resulted the mean values of f, which have been employed for that station and for Senchal. These values will be found recorded in the heights herein given, and it will also be found, that they have been employed for all heights of the Sub-Himalayas observed at Senchal and Tonglo hill stations.

70. It may be useful to remember, that if there be two points A and B observed from O, whose heights respectively are ha and hɩ determined by a certain value of ƒ at 0 =fo. Also if da equal corrected geodotic distance O to A, and do O B.

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Then if fo vary,

so that ha (the height of A computed from O) changes by ± da, and

бъ

d2

ho byd, so will ±

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δια

da2

Hence should the foregoing method

for finding the value of ƒ at plain stations in terms of the observed value at hill stations, be hereafter ever adopted, it will be found advantageous to construct a table of the squares of the distances in miles, for this purpose.

71. The general principle of procedure is now apparent. But as

will be remarked, the process described is only applicable so long as a continuous connection is preserved, between the stations of observation and the points observed. In the observations under consideration, there occurs a blank space between points LII. and LIII whence the method described was no longer applicable beyond the former point. But it fortunately happens that LIII. and succeeding points are observed from hill stations, whereat, as already mentioned, the values of ƒ are liable to but trifling variation. The mean value of ƒ in these cases was deduced in the ordinary way as mentioned at para. 64. The following is an example of this method.

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Mean f adopted at Jagesar, H. S. .04630.

72. Values of ƒ tabulated.—The values of ƒ employed in these calculations may be tabulated thus.

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Sin

incidence

refraction

=1+ m in the mean state of atmosphere and at

the level of the sea, and also, since the quantity m varies with the density of the atmosphere, so that when the density of the air is only the nth part of what it is at the level of the sea, the refractive power is

there only 1 +

m

n

it might have been expected from these tabulated

results that in the first instance, fa

1

height of station of observation. No such law, however, is to be found unless the numerous exceptional cases be excluded to make a rule.

74. Wherefore it appears, that the law of variation in ƒ due to variation in the density of the atmosphere, consequent on variation in height, is completely absorbed and lost sight of in the irregular variations, arising from local causes and also from the unavoidable imperfections of observation to points so ill-defined as the apices of snowy mountains.

75. Finally it is to be noticed that the foregoing method is acknowledged to be imperfect and unsatisfactory, but compared with the ordinary mode of finding ƒ from reciprocal vertical observations,

it is believed that the values herein determined are a nearer approximation to the truth.

76. Notices certain refinements not appreciable in these operations. 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 neglected 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 = (va Cos λ K) Cos a K), wherein K

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N

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v¿ sin λ¿ — vɑ sin λ +− [ (M +va Cos λa) (M —va Cos λa) ]*

= { v2

N

M

va

M

À

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[(M + va Cos λ) (M-v2 Cos λ) ] Cosec & A.

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

is 7886" = about the th part of a circle described around our planet. And if the principal and mountain operations of the North East longitudinal series be taken together, they will be found to cover somewhat more than the portion of the entire earth's surface; or, taking the land at half the expanse of water, about 1061 such series would cover every portion of the former.

80. Accuracy discussed.-And with regard to the accuracy of the mountain results, it is evident that the same estimate cannot equally apply to a peak with a sharp conical apex, and to a mountain whose summit represents a saddle back or an even bluff. Prominent amongst the accurately determined points are XIII. Mont Everest or XV. and XLII. or Dhoulagiri, both in respect to geographical position and height above sea level, but though such points are far more numerous than those which exhibit comparatively large differences between the several values composing their mean results, yet it is suggested that the synopsis of latitudes and longitudes and the paper of heights should be consulted before adopting a point, if necessary for rigorous purposes.

81. The same estimated. It is estimated, that on an average, the points on the Himalaya mountains are correct in latitude to of a second and in longitude to about that quantity. The heights are probably true to 10 feet, but this last estimate must be qualified by the consideration that they are all too low from the deflection due to mountain attraction.

82. Why mountain attraction was not determined.—In the original design of these operations, it was intended that the deflections. in azimuth and in the meridian due to the attraction of the Himalaya mountains should be estimated along the principal series by suitable celestial observations, but this intention was relinquished owing to the considerable delay it entailed.

84. Area and cost. The area covered by these principal and secondary operations amounts to about 61,815 square miles. But the piecemeal nature of work, the long intervals which frequently occur, and the unavoidable employment of the North East longitudinal series partly on other duties, make it a difficult and unsatisfactory process to attempt finding the cost of these operations. As an approximation, however, it may be stated that this cost does not exceed Rupees 2 per square mile,

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