on Change of Temperature. 287 
Peop. VL To find an approximate formula for the displace- 
ment of a given pole P. 
Xeglecting in the last formula higher powers of 6 than the 
square, we find that 
(1 + tan 6 cot Pa)(l -tan 6 cot Pa + tan 2 6 cot 2 Pa) = 1 + e, 
whence Q e .. , _ 
tan = — sin ra sin Pa + tan - a cot Pa. 
sin a a 
If 6 X be the first approximation to 6, obtained by neglecting 
the term of the second order tan 2 6 cot Pa, we have 
0^/jsinPasinPa, 
where k= — , and is thus a small quantity independent of 
sin aa/ i J e 
the position of P. 
If 6 2 be the second approximation, 
3 = X + tan 2 X cot Pa. 
Peop. YII. To determine the value of the coefficient h. 
Let be the variation of any given angle Qa ; then 
(Prop. Y.) 
- _ 1 + tan (j> cot Qa 
1 + tan <p cot Qa 
whence 
_ (cot Qa— cot Qa) tan </> 
1+ tan <f> cot Qa 
and ,,_ e _ sin<£ 
1 sin aa sin Qa sin ( Qa + <f>) ' 
a logarithmic formula by means of which k can be readily 
found. 
Peop. YIII. To find the maximum and minimum values of 
the term of the second order tan 2 6 X cot Pa. 
This proposition is convenient for determining whether or 
not the term can be neglected in comparison with experi- 
mental errors. 
Let a = 180° — a a, 
then 
tan 2 #i cot Pa=/j 2 sin 2 Pa sin 2 ( Pa + a) . ~ 
v 'sin Pa 
= ^sin2Pasin 2 (Pa + a). 
, 1? 
The maximum value of the term can never exceed y 
Differentiating with respect to Pa, we have for a maximum 
