Mr. J. Bridge on the Gyroscope. 343 
Whence 
dp ec 
ie Gt aa p), if r= when u=p; 
dv ae gr > 
a ee 
sae 2a (sin y— sin alee 
cw* 
—u?*, if w=p when v=0. 
v is a Maximum or minimum when 
c ac 
y=p— Tre? =e oa+or pe very nearly ; 
then 
__ _% COS ( ak 
aaa —_———, and v= + { p— Ato)" 
From this it follows that @ continues to be of a magnitude so 
small (for p and = are supposed very small) that sin @ may 
still be regarded as equal to @. 
Again, 
to = 8 sinuicu (+0; 
dv \? : i 
3) =(°(1+¢)?+a¢ sinn.)(p _ sta —0); 
and if it is supposed that ¢=0 when v=0, 
_(,_ #cosp ; 
v=(p Hea) cosa +e) 
where w! is very nearly equal to a, = being very small. Also 
_ wcosp = (  acoop\. | { 
= +e) ( alto) sin w'(1+¢)¢; 
whence 
dp_« ca (ae \si ; 
dt alte? COs arte 7) jena 
The motion of the extremity of the principal axis is given by 
dj's a op a ) Ue 
dt o(1 +c) Sie ates +c)@ Baett et 
dy! _ op ot ; 
= cos woe recs, cos w!(1+c)t. 
