HIMALAYAS ON THE PLUMB-LINE IN INDIA. 
87 
It appears, in short, to be quite hopeless by any admissible hypothesis to reduce the 
calculated deflection so as to make it tally with the error brought to light by the 
Survey. In the conclusion of this paper, however, it will appear that such a reduc- 
tion is not necessary for reconciling the discrepancy. 
47. I will here write down the formulae in their most general shape. Let 1— « be 
a factor (similar to 1— a? and 1—?/) for the part of the Known Region not including 
the Plateau. Suppose also the whole of the heights of the Known Region are 
reduced by a hundreds of feet, and those of the Doubtful Region by h hundreds (in 
this way I bring in the correction E at the foot of the six Tables). Then 
A= ( 1 — ;2) 4"-200 + ( 1 —y) 8"‘772 + 4"-88 1 — 0"-3 1 2a — 0''-260b. 
B = (l-;2)l"-209-+-(l-7/)2"-010-l-(l-a?) 8''749 — 0"-059a-0"-158^. 
C = (l-«)0"729-l-(l-y)0"'607+(l— x) 5"-573-0"-022a-0"*100i. 
The method of using these arbitrary symbols is this. If it appear on examining the 
contour of the earth’s surface more carefully, that the heights in the Known Region 
and south of the space I call the Plateau, ought to be reduced in a certain ratio, we 
have but to give the value of that ratio to : the same is the case, as I have already 
shown, with y and the Plateau itself. If, on the other hand, we do not wish to reduce 
the heights in the Known Region in a certain ratio, but by the same given quantity, 
z and y must be put =0, and a= the number of hundreds of feet by which we wish to 
reduce. Thus if we Mush to reduce the whole Known Region by 1000 feet in altitude, 
we must put a=10. We may, moreover, combine these methods of reduction, and 
both reduce the general I’atio of the heights, and afterwards cut these reduced heights 
down by a constant quantity by giving y, and a the proper values. The same 
things may be done for the Doubtful Region by assigning proper values to a? and h-. so 
that the formulae here given admits of adaptation to various hypotheses of reduction. 
48. We may use these last formulae for comparing the masses which stand on the 
Known and Doubtful Regions with each other, and with the mass of the earth. In 
doing this I shall suppose that the heights have been rightly assigned in the present 
paper. Therefore 2 = 0, y—0, x=Q. Then if a=4T58, the part of A wliich arises 
from the Known Region is reduced to zero. Hence 4158 feet is the average height 
of the mass standing on that part. In the same manner, by making Z»=57'30, the part 
of A which arises from the Doubtful Region vanishes ; and therefore the average height 
of the whole mass standing on that portion of the Enclosed Space is 5730 feet. It will 
be observed that this is greater than the former. The obvious reason of this is, that 
the mass of the Known Region is highest at its furthest parts from A, whereas the re- 
verse is the case with the mass of the Doubtful Region. The superficial extent of each 
and long. 7G° 45', Y in lat. 30° 30' and long. 80°, and Z in lat. 31° 30' and long. 83°. Join W and Z by a 
circle about Kaliana as centre, and join these and the other points by arcs of great circles of the sphere. 1 take 
the average height of the enclosed mass to be 10,000 feet above the sea, a mean altitude which I conceive is 
rather under than over the mark. 
N 2 
