(153) 
Since £ is very nearly unity, we have to a very close 
degree of approximation 
A 
A 
tan IZ = — sin æ COS&. 
‘ 
Pl 
In the case of the Earth « is known to be about 0/’.15 
and = is about _. (or _—. when A’ replaces A). Hence 
IZ is about 0//.0005 (or 0//.00054). Thus for observa- 
tional purposes the instantaneous axis is practically 
identical with the axis of resultant moment of momen- 
tum. Hence the free nutation does not sensibly affect the 
obliquity of the ecliptic as derived from observation, and 
there is no observable nutation of this kind in longitude. 
However in order to examine the subject fully [ conti- 
nue to maintain the distinction between the instanta- 
neous axis and that of resultant moment of momentum. 
[ now desire to find the relationship between the 
astronomical and geographical poles and equators. 
Hitherto the investigation has been rigorous, but [ shall 
now proceed approximately. 
If À be the geographical latitude of the place of obser- 
vation P Iying on the cirele CA, IP being the astrono- 
mical colatitude may be put equal to ST — À —ÔÀ, 50 
bat tà 1s the variation of latitude. 
Let us replace L by C’ as designating the instantaneous 
axis, then 1f the great circle IP or CP be produced 
until PA’ is equal to À + àX, C’ and A’ are the extre- 
mities of two axes at right angles to one another. Adding 
Lo these a third axes B’ at right angles both to A’ and C’, 
we have a new set of rectangular axes such that C/ is the 
astronomical pole and A'B/ the astronomical equator. 
