OF THE SOLID CRUST OF THE EARTH. 
341 
of matter round A, reckoned positive downwards, 
_2^j d / 
I C 2 fid \ 
= 2-rr sin 5 . dz . n?5 . § 
c — r cos i 
r — c cos i 
=£(«-*)■ 
(c 2 + r 2 — 2cr cos fl)f \ U c 2 + r 2 — 2cr cos I 
d / 2c sin 2 ifl — ^ \ . 
dQdz. 
dkJz 
dd\ v / 5t 2 + 4c(c— ^ sin 2 ^ 
Integrating from 5 = 0 to 5 = 5, and then putting 2c sin ^5 =m and u 2 =2cv, 
Total Attraction of the Cap 
(v — z)(c—zY 
V z 2 + 2(c — z)v 
dz 
2 ng [c 3 — (c — ty 
— ,2 
( v — z) (c — zfdz 
Vi 
■2vz + 2 cv 
Integrating by parts, the integral becomes 
= — (c— zf\/ z 2 -2vz-\-2cv—2§(c—z)\/ z 2 -2vz-\-2cv . dz 
= — (c— z) 2 \/ z 2 — 2vz-\-2cv + f(T 2 — 2vz-\-2cv)i-\-2{v— c)J\/ z 2 —2vz-\-2cv . dz 
= — (c — z) 2 \/ z 2 — 2^ + 2e"w -f %(z 2 — 2wjs + 
— (T— <?){(«— ;s)\/ 2 2 — 2-y3+2cy— (y 2 — 2cv) lo g K (v—z + \/ z 2 ~2vz-\-2cv)} 
= — {i z2 —( c ~~h v )z- c 2 — %cv-\-v 2 }\/ z 2 -2vz-\-2cv 
-\-(y—c)(y 2 —2cv) log e (y -z-\~s/z 2 — 2 vz + 2 cv). 
Putting this for the integral, replacing 2 cv by u 2 , and taking the limits, 
Vertical Attraction of the Cap below the station-level 
1 _ (1 _ a V-- 1 S+ ?! 
3 j \ c J c 2 c 6 4c 5 
This is the exact expression. 
9. Secondly. Suppose the Cap is immediately above the station-level. The above 
formula requires in this case some modification. In the first place, when the limits of 5 
are taken and 5 is put=Q, the radical in the denominator is now —z, and not 2 as before. 
This will change the sign of the first term ( c—z ) 2 in the integral with regard to 4, and 
will change the signs of the first and second terms within the brackets of the final inte- 
gral. Again, the limits of integration with regard to z must be taken from z — — t to z— 0, 
which is the same as putting —t for t and also changing the sign of every term of the 
