106 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 


Then, if 
DE 1) =D@, ))=DEH=?9, 
we have 
as ag a 
dz dy dz 
dy dn) \dn dn| ~ [dy dy 
dy’ dz dz” dx dx’ dy 
a at) jae at] jae ak 
dy” dz' dz’ da dix’ dy 
. dé d& d& 
“4 Tihs Phe dy = Piso oe suppose 
a it ope ee 
with two similar sets in 7 and €¢. 
b. Since 
4% dQ, d& dN, dg dA 
#8)” 8G) * 8G) 
A=uwDé=r70Dn= wD. 
we have at once 


Hence 
adA 
dé 
uUu = (ae = A 
dé yes 
dx 
or 

: dé a ua) 
3 4() Gi) = 
log. A=C +4 log. OE. 
which gives by integration 
Thus, finally, 
A=V fone. 
ae cake. De 
c. Thus we have 

But, identically, 


Ce ae ae 
s () (Gy ta 
dar . \dy, (<:) 
ult use dé\i 
e “dx +5, (“a Tg 7 (ug ee 
uVv7é — Du, &) =9, 
with similar equations in v, 7 and w, ¢. 
or 

or 

