OF THE SOLID CRUST OF THE EARTH. 
347 
Suppose that the zone is the rth, then 
2c(r+l) 2 cr 
W— ■ \ U = — . 
n n 
Substituting these, neglecting the fourth power of h -f- u, and introducing a subsidiary 
angle p, snch that 
n(h + mk) 
c[2r + 1) 
tan p, 
the effect of the attenuation 
Q/ji f . ____________________________ yh k rfili 1 
= 4^{ _2 +^ 4 ( r + 1 ) 2 +( 2r + 1 ) 2 tan 2 p 4r 2 +(2r + 1 ) 2 tan 2 p - — + 4r(r + I)c s 
The pair of radicals in this expression 
=s/\ d-(2r + l) 2 sec 2 p+2(2r-|-l) — \Zl + (2?’ + l)® sec 2 p— 2(2r+lj 
2(2r + l) 
(l + (2r+l) 2 sec 2 p)“ 
(2r + l) 3 
(l + (2r+ l) 2 sec 2 ?7 
+ ... by expansion 
cos 2 p ^ t cos 5 p , 
2(2r+l)V (2r + l) 2 ” 4 "’ ' ' 
= 2 COS ip- 
COS 3 ?- 
cos 5 ? 
(2r + 1) 2 
nearly =2 cos p 
2 cos p — cos 3? — cos 5? 
16(2r+l) 2 - 
substituting this in the expression for the effect of the attenuation, and adding it to the 
vertical attraction of the mass above the sea-level given in formula (3), 
Resultant Vertical Attraction for the zone =~- ^ 7 . R, 
2mw 360 ’ 
where (3 is the angular extent, at the station, of the part of the zone on which attracting 
matter stands, at the height k ; and R is given by the following formula : — 
R= — 1-j- cosp 
2 cos p — cos 3? — cos 5? 
32(2r+l) 2 “ 
n 2 m{k 2 — 2 hk) — A 2 
8c 2 r(r + 1) 
and tan <p= 
r 2r+l c 
The expression for R may be somewhat simplified for zones beyond a certain distance. 
For when p is sufficiently small to allow of its fourth power being neglected, 
V ~ c 2 \2r+l ) ’ 
and the part of R depending on p becomes by expansion 
pV-. 1 \ ? 4 /, 22 \ 
“ 2V + (2r+l) 2 / +24^ i " t_ (2r + l) 2 / > 
Neglecting p 4 , and substituting for p 2 , this becomes 
n 2 /h + mk\ 2 / 1 \ 
2c 2 \2r+l j y 1 "* _ (2r+l)7’ 
ri 2 j(h + mk) 2 / 1 \ m(k‘ 2 —2kh)—h?) 
2c 2 j (2r-f l) 2 y (2?’+ I)" 2 ) 4r(r+l) j 
and 
