HIMALAYAS ON THE PLUMB-LINE IN INDIA. 
97 
I shall use arbitrary factors with these to give the opportunity of introducing any 
changes in the deflections that may be thought necessary on a further examination of 
the contour of the surface of the attracting mass ; so that 
15"-885(1-m), 5"-059(1 and 20"*944(1 -m;) 
will be the differences of deflection in the meridian at the extremities of the three 
arcs I., II., III. As the third of these must be the sum of the other two, we have the 
I’elation 20-944 m;= 15-885M-f5-059y, 
or z4;=0'7584M+0'2415y. 
63. Also, by comparing the above values of the deflections with those in art. 47, 
we have the relations 
M=0-1883z-f0-4257«/+0-3861^, 
i;=0-0949«+0-27732/+0-6278x, 
M;=0'16572 + 0-38983/+07044a?. 
64. Before proceeding, I will remark that the amplitude of the arc II. determined 
astronomically, as given above, is somewhat greater than that deduced by calculation 
from the length of the arc. Unless this can be accounted for by the form of the 
assumed ellipse, it intimates that there is some disturbing cause north of Damargida 
which increases the inclination of the plumb-line to that at Kalianpur. Should this 
be the case, the correction for it may be effected by adding a small quantity to the 
deflection between Kalianpur and Damargida; that is, by increasing 5"'059(1— y), 
or by diminishing v by some quantity v'. 
65. We must now add the meridian deflections to the astronomical amplitudes. 
The results are the true amplitudes of the three arcs I., II., III., viz. — 
5° 23' 52"-943— 15"-885m, 6° 4' r''032 — 5"-059?;, and 11° 2/ 53"-976 — 20"*9442^. 
A comparison of these with the three values of X, viz. X', X", X'" in art. 62. deduced by 
computation, shows that s and a have been chosen so as to make A in each case too 
small. Let dl!', dx'" be the three errors of A', A", A'" ; and suppose and a-\-da 
are the values of z and a which will by computation bring out the true amplitudes ; 
fl?x'_ ll"-147-15"-885M 
X' “■ 5° 23' 4l"-796 ■ 
7"-746-5"059v 
: 0-0005739 — 0-0008 1 79m, 
X" ■ 
dd" 
6°3'5d"-286 — 0‘0003548 - 0-00023 1 7v, 
17"-676-20"-944w; 
X'" “ lU27'36"-300 
By the formula of art. 62. we have 
ar( 
:0 0004284 — 0-0005076m;. 
rc=aA'|l—^g^ 14-3^^^ cos 
If we differentiate the logarithm of this, we have 
_ da , dx' 1 / . „ sin x' ^ , 
0=-+^--(^l+3 — cos2^’jrfs. 
