on Change of Temperature. 289 
change of temperature defined by the above quantity e, then, 
as before, 
tan 6== cot Pa- (1 + e) cot Pa 
cot Qa — (1 + e) cot Qa 
whence 
(n_±\_ cotQa— cot Pa — (l + e)(cotQa — cot Pa) 
n ^ ^ ~ V + [cotPa — (1 + ejcotPa] [cotQa — (1 + e)cotQa] ' 
If the planes P, Q are isotropic for a change of temperatures 
defined by the small quantity/ corresponding to e, it -will fol- 
low in exactly the same way that 
cot Qa— cot Pa — (1 +/)(cot Qa — cot Pa) = 0. 
Substituting in the above value of tan (9 — 6), we get 
tan (0-6)= g(/-g)(cotQa-cotPa) 
v rj e 2 +[cotPa— (1+g) cot Pa] [cotQa— (l+g)cotQa] : 
and to the second order of small quantities, 
g(/-g)(cot< 
[cot Pa — cot Pa] 
sin PQ sin Pa sin Qa 
, ( a_,\_ //•_ n e (f— g)(cot Qa— cot Pa) 
{ V>- eK J e) [cot Pa -cot Pa] [cotQa -cotQa] 
Unless sin ma is very small and comparable with e and /, the 
variation in the inclination of P to Q is a small quantity of 
the second order. 
Prop. XL We shall now consider the results which are 
obtained on neglecting the squares of small quantities. 
(a) We have already seen in Prop. VI. that the displace- 
ment of any pole P is in this case given by the relation 
9 =■ k sin Pa sin Pa, 
where k is some small quantity independent of the position 
of P. 
(b) If 7, 8 be the variations of the arcs ac, ad, then 
7 = k sin ca sin (aa—ac), 
S = k sin da sin (aa — ad), 
whence both aa and k can be determined. Practically, how- 
ever, it is quite as easy to use the rigorous formulas of Pro- 
positions II. and VII. 
(c) From the symmetry of the expression k sin Pa sin Pa 
Phil. Mag. S. 5. Vol. 16. No. 100. Oct. 1883. Y 
