GEOLOGICAL CHANGES ON THE EARTH’S AXIS OF ROTATION. 
311 
so that 
d P dQ, . . n 
rr COS at -rr Sin at= U. 
We find then 
at dt , 1 
dt wo ^ dt 
P __i f (ta. 
r ~ W J \di 
Q= l§(f t - m ) cos “ tdt J 
For the case considered in Part I., where u and v are constant, 
P=— ^cos wt-\-C, Q=— ^sin at+C', 
and therefore by (7) 
( 9 ) 
x= — -g+ C cos at-\-C' sin wt 
(10) 
The solution expressed in equations (5), (7), (8), (9) is convenient for discontinuous 
as well as for continuously varying and constant values of u and v. 
Consider, then, the case of u — 0 and -y=0, except at certain instants when u and v 
r c*t 
have infinite values, so that I,, udt and I vdt express the components of a single abrupt 
JT' Jt' 
change in the position of the instantaneous axis ; where T and T' denote any instants 
before and after the instant of the change, but so that the interval does not include 
more than one abrupt change. 
Therefore, if t 0 be the instant of the change 
C T 
? v sini*>£ dt =sm at a 
Vvdt ] 
Jt' , 
Jt' ! 
C T 1 
J V cos at dt = COS at 0 
\r Vdt J 
( 11 ) 
Hence tbe part of x depending on v vanishes at the instant immediately after the 
abrupt change when t=t 0 . Also we have by integration by parts, 
sin at dt=u sin at — a^ucosatdt, J 
J^cos at dt—u cos at a J* m sin at dt J 
. (12) 
And, therefore, taking the integrals between the prescribed limits, since u — 0 both 
when #=T and when t= T', we have 
^ ^ sin at dt= — a cos at 0 J udt , ~| 
^ ^ cos at dt— a sin ut 0 J* udt J 
(13) 
2 x 
MDCCCLXXV1I. 
