374 
PRESTON. 
and multiplying by a common factor. The complete inte¬ 
gral is * 
F=h 5 h (sin a 2 — sin af) Nap. log ( ^ Y. (8) 
' r x -f v rf + h 2 * 
When the difference between the radii is taken so small that 
we may neglect terms containing higher powers of r 2 — r 1 
the attraction may be written 
F = Tc 8 ( r 2 — 7*j ) (sin a 2 — sin af) 
h 
V r 2 + h 2 
( 9 ) 
where r is the mean value between the two radii above con¬ 
sidered. In cases where h 2 may be neglected, we have 
F— Jc d (r 2 — rf) (sin a 2 — sin aj-- (10) 
We shall show later that a sensible error would be committed 
by throwing out h 2 in some compartments of ITaleakala. 
Equations (7) and (10) offer two distinct ways of calculat¬ 
ing the attraction. The former is the more rigorous and is 
approximate only to the extent of omitting the square of 
the heights. The accuracy of the latter depends on the re¬ 
lations between the horizontal distances as well as the values 
of h. The fact that both formulae can be applied under 
certain conditions furnishes a method of checking the work. 
This check will be applied to the leeward side of the moun¬ 
tain, which gives by far the greater part of the attraction. 
Let us suppose that from Kaupo (Ka Lae o Ka Ilio) a 
meridian and a parallel of latitude are drawn (Plate 8). 
On each side of the meridian draw 9 radial lines, making 
angles with the meridian whose sines have a common dif¬ 
ference of r V. Draw also 10 concentric circles, the radius of 
the first being one mile, and the succeeding radii increasing 
each by one mile over the preceding one. From the 10 mile 
point the radii increase in a geometrical ratio of The 
* Clarke (col. A. R.) Geodesy. 8°. Oxford, Clarendon press y 1880, p. 295. 
