on Change of Temperature. 295 
whence 
<£ = 44°59'13"-93. 
Also 
(a'c' + ac \ , a'c' — ac, ,. K0 ,. 
tan ( — ^ act J = tan — ^ — tan (45 + 9) ; 
whence 
tan (80° 27' 46"-aa)-=-tan4' 51"-2 tan 89° 59' 13"'93, 
_ tan4'51"-2 
~~ tan 46 // -07 
= tan 81° 0' 35"-7, 
and 
fl«=161°28 / 21 // -7, 
the arc being measured in the positive direction, namely ac. 
Also 
a =180°-a«=18° 31/ 38"'3. 
(b) Neglecting squares of small quantities, as is done by 
Neumann, the positions of the lines of greatest and least ex- 
pansion can now be immediately found. From Prop. XI. d, if 
T be the axis nearest to a in the arc act, 
aT=^-45° - 
= 35° 44' 10"-85. 
Neumann gives as result 35° 46'. 
(c) If 8, 8 1 be the principal expansions, we have seen (Prop. 
XIII.), that 
8 1 -8=k= 
sin Pa sin Po 
for in this case sin (Ta + Tct) = — 1 . 
If P coincide with c, we find 
s 5, 582*4 x circ. meas. of \" 
sin 80° 32' 37"-2 sin 80° 55' 44"-5 
= •0028987, 
where 8 refers to the axis nearest to the pole a in the arc 
act. Neumann gives as result "002892. 
Finally, by the method of Prop. XV., illustrated later by 
example (page 300), S 2 — S may be calculated. 
To ascertain to what extent we shall be justified in neglect- 
ing the small terms of the second order, we shall make some 
calculations founded on the more exact formulas given 
above : — 
