on Change of Temperature. 281 
OP', OP be the corresponding normals from 0. Then 
OI/=«OL J 
0N'=/3 0X; 
whence 
t,.^ OL' a OL ct __,. 
tanPX= 0¥ / = ^0N = ^ tanPX - • ' ' C 1 ) 
Similarly, if OQ', OQ be the normals to a second plane at 
these two temperatures, 
tan Q'X= |tan QX ; (2) 
whence, if the angle P'Q' = PQ, or the planes P, Q be isotropic 
for the two temperatures t 1} t 2 , we have 
tan (Q'X-P'X) = tan (QX-PX), 
^tanQX-tanPX) tanQS _ tmps 
1 + g tan QX tan PX 1 + tan Q X tan PX ' 
whence 
tan QX= =tan P'Z from (1), . (3) 
?tanPX 
QX=P / Z = 90°-P / X, (4) 
a simple relation giving the position of a plane Q isotropic 
for this pair of temperatures to a given plane P. 
It is now necessary to find the angle F f/ Q f/ between these 
same planes P, Q at a third temperature t 3 . 
If the unit lengths along OX, OZ at t x become a and b 
respectively at t 3 , we find, just as before, that 
tanP"X=^tanPX, 
tanQ"X=^tanQX, 
= j — cot PX, from (3) ; 
