on Change of Temperature. 427 
£ = 1-0004452 ---0000376, 
v y 
^=0-9999155- + -0001522- + -0000419. 
v y y 
Substituting for - - the values given in § XX. (a) for the 
lines LMN respectively, we find: — 
log cos L'X, 9-9823170-0 I log cos L'Y, 9-0275653-8 
logcosM'tT 8-8979660-7 logcosMY, 9-8940381'5 
logcosN'X, 94287729-9 | log cos N% 97869209-6 
whence 
L'M'=90° + 6"-35, M'N'=90 o + 16"-08, N1/ = 90° + 31"-65 
(c) To find the positions L" M" W at 140° C. 
For 20°— 140° C, 
^-^=+•0009507, /=--0000706, 
r _2 = _ -0000012, e=+ -0004701, d= + -0000888; 
whence 
i= 1-0009507 ---0000706, 
log cos L'Z, 9-4125578-3 
logcosM'Z, 9-7898228-7 
log cos N'Z, 9-8714128-4 
v y 
£= 0-9999988 - 
v y y 
£= 0-9999988 -+ -0004701- + '0000888, 
and 
log cos L"X, 9-9822981-8 log cos L"Y, 9-0273255-7 log cos L"Z, 9-4128578-9 
logcosM"X, 8-8983082-1 logcosM"Y, 9-8940190-7 logcosM"Z, 9-7898481-1 
log cos N"XT 9-4289631-7 logcosN'% 9-78692434 logcosN"Z, 9-8713857-5 
from which we find 
L"M"=90°-16"-89, M"N" = 90° + 7"-53, 
N"L"=90° + 35"-01. 
(d) To find the positions L'" M'" W" at 200° C, given the 
positions LMNat 20° C. 
This calculation will serve merely to test the influence of 
small quantities of the second order on the reversal of the 
formulae. 
For 20°— 200° C, 
p-q=+ -0017255, /=--0004525, 
r-g=+ -0000565, e= + -0010007, d= + -0001257; 
whence 
? = 1-0017255-- -0004525, 
v y 
^ = 1-0000565 - + -0010007 - + -0001257, 
v y y 
