GEOLOGICAL CHANGES ON THE EARTH’S AXIS OF ROTATION. 
307 
In the elevation and subsidence given by lit sin 20 sin 2<p from 0=0 to and from 
tp=—y > to Hi and H 2 are clearly zero, under a like supposition as to the nature of the 
internal motions accompanying upheavals. 
Appendix B. (See p. 292.) 
To reduce the integrals i cos ^ arc cos C ° S 2 “ dx and f cos x \/ cos 4 x — cos 4 a dx to 
Jo cos Jo 
elliptic functions. 
Call the former A and the latter B. 
Then integrating A by parts, 
a = -j> z d ( arc cos Ss) • 
Put #=sin x, and cos 2y=cos 2 a, then we get 
A=2 cos 2yi*^ 8inY ( 0 
/ J 0 \1 —A ) Vx* — 2a? 2 + sm 2 2y 
and if x =\/ 2 sin y sin <p, this becomes 
a/‘ 2 C0 ^m i (-2sin 2 y)-F 1 L where the modulus is tan y. 
^ COS y ' v ' ' 5 
Again, integrating B by parts, 
C a . 4 cos 3 y sin y dy 
B=\ sin^. — * * x 
„ f* V 2 sin y 
2 \/cos 4 ^ — cos 4 a 
(1 — x z )x z dx 
0 Ax 4 — 2A + sin 2 2y 
But B is also = (”*’ 
, , . - . „ — dx from the expression before partial inte- 
sjx 4 — 2x 2 + sin 2 2y r 1 
[ration. Multiplying the latter expression by 2 and adding to the former, 
35=2 
sin- 
V sin 2 2y — 2-r 2 + x 4 
dx ; 
and substituting the above value for x , 
§B= 
sin* 2 y 
-C 2 cos y 
F'+\/ 2 cos y(E‘ — I *), 
O a/9 
5=-- J— cos y[E' — cos 2y F 1 ], the modulus being tan y. 
B may be calculated from this form by means of the tables in Legendke’s ‘ Fonctions 
Elliptiques,’ tom. ii. But A is not yet in a form adapted for numerical calculation. 
