HIMALAYAS ON THE PLUMB-LINE IN INDIA. 
61 
the earth at this part is a plane passing through the middle point of the elementaiy 
base qQ.q', and sloping upwards towards the side on which A lies. This is an extreme 
case ; for the actual height will be extremely small near the middle of qq', that is, in 
parts where the variations of cos are least ; and the actual heights will be greatest 
towards q and q', that is, in parts where the variations of cos are greatest. Hence 
the error incurred in replacing the actual heights by their average will be much 
greater in this than in a more general case. Now in this extreme instance 
h'={c—c cos -py) tan 
c being the distance of qQq' from A, and the inclination of the plane to the horizon. 
Hence the true value of the integral 
/(' cos \pd-4'=c tan f} ] ^ (cos -pz—cos^ 'p)d-4/ 
= ctan fi^2 sin^/3— 1(3 — 5 sin /3^ 
= ctan??^^ ^*—1920 
And the approximate value of the integral 
l 1 
= 2 sin 2 (3./i:=2 sin 5|8 . j (c— ccos -pz) tanfjd-p-hfB 
J— 
=^sin ^(3 tan ^(3—2 sin 
Hence the ratio of the true value to the approximate value =1 — If (3=30°, 
the width I intend to give to the lunes, this ratio differs from unity by the insignifi- 
cant fraction 0’00228, which we may well neglect; and if this is the smallness of the 
error in such an extreme case, we may consider that the approximation will in the 
general case differ from the true value by an inappreciable quantity. 
17 . To find the attraction of the whole mass comprised within the compartment, 
the expression (1) in art. 15. should be integrated with respect to ^ from 
to ^=a-}-<p. But to do this we ought to know what function k is of There is, 
however, no known law of connexion between them. We must therefore resort 
to some other means. According to the law of dissection, which I shall in the end 
COS^ i fl 
adopt, the value of j— does not vary much within the limits of a single compart- 
sin-« 
ment, that is, between the limits and ^=a-j-f). Indeed its middle value, when 
^=a +2 is only one-fifteenth part smaller than its greatest value, when ^=a, and 
MDCCCLV. 
K 
