of Edinburgh, Session 1870 - 71 . 505 
where <p and {[/ are scalar functions of T to be found. This gives 
(Ta)°f = (?—+<d(?>F' + *f) 
= <p 2 F" + (<p<p' + if/<p + <pf) F'+ (<pE + 4 /2 )F 
Comparing, we have 
<P a = 1, 
, 2 
<pi]/' + i// 2 = 0 . 
From the first, 
whence the second gives 
<P =± 1, 
^ - ± 7p- > 
and the third shows that the upper sign must be taken. That is 
TA = A + I . 
cIT T 
Also, an easy induction shows that 
v n / d \ n n / d \ n_1 
Hence we have at once 
aTA / d 
6 ~ 1 + a (^T 
+ T 
t) 
+ 
d 
a dT 
= € 
+ 
d 
a a err 
T e 
so that 
aTA 
€ FT<7 
rr d v 1 1 
+ &c. 
Ter 4- ci 
~Ta^~ 
F(T<j~ + a ) . 
In using such a formula we must carefully remark that F is 
defined as a function of a tensor , Ae., of a quantity essentially 
positive , so that should a be negative and of greater magnitude 
than To- the quantity of which F is a function becomes a - To-. 
