108 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
where 
fi = ¢ + f (hy) . 
3. The three equations (3) may be put in the new form 
= Gory te = 
Sy «2(Z8. eh -)=0, &, 
whence 
do- de = = =i 
3(Z8.q¥ ‘-) [ Viioiokh «. ae 
4. But, by the nature of self-conjugate linear and vector functions, 
—1,-1 
Sock c=Sieb th - 
—1 
Aas’ 2 la i 
= jay) oe 2 ee 
with two other equations of the same kind. These give (when the values of 4 
are different) three equations of the form 
fe Va ede 
where, of course, we may dispense with the V. 
5. By (4) and (6) we see that we have three equations of the form 
—1 do fos esl 
ye [2(GS.gN c), 
and these show at once that 
do- do- do 
dz’? dy’ dz 
are rectangular vectors whose tensors are equal. For 
TS.aBy =aS.By7 + BS.yat + yS.aBr 
is the only decomposition of 7 parallel to a, B, y respectively ; and we have 
here the equation 
T ||aSaz + BSBr + ySyr, 
holding good for the three non-coplanar vectors W's, {,~'<, Ww '~, and there- 
fore true for all vectors. Hence we must have . 
a|| VBy, B\| Vyo, yl| VaB, 
of which any two include the third as a necessary consequence, and in all three 
of which the coefficient of proportionality is evidently the same. The only 
exception to this is when 
de 
dx 
| wo, &e. 

