118 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
de d2s- do~ da du 
de dal "de ded, ae 
which give 
parc dud-~  dud-~ dude 
= ge (ee ee 
ae du do du do dud-  dud-~ 
“te de + “\ de da * dy dy a dz dz 

From the three equations of this form we obtain, first, 
duds  dud-~ dude 
2072 = m\ e 
UN — UN ee Sade ded): 

and, secondly, three equations of the form 

= (as _  lfdude- , dud~ | dude 
da\a® day = | ub dx dx + dy dy + he di 
These are summed up in 
u’daz)]~ dy 

@ f1 de- d (ldc- d (\de- 1 
al (aq) = rAG&s a ; (36), 
which express some of the conditions of orthogonality of the three series of sur- 
faces given by equation (1). 
21. To obtain the others, remark that by (30) we have 
= 7 ; 
Ngai: Le RS ae rea = 2V.jVg + im 
ut dxdy dy u dx’ 
or 
dl Sie 7 du 
at Gea V.iVjV. log. u += = V.jViV. log. u + 2 gee 
and that each of the two latter expressions may be written 

idu , j du 
u dy + ou dx’ 
Hence 
d? o- 1 (do- du do- du 
dxdy — w\dx dy ae (37), 
and there are, of course, other two vector equations of a similar form. 
From these we have nine equations of the form 
PE 1(dédu , dé du 
dady ~ u\dady * dy dx ior 7 ae 
Now. 

d Lary (pate @E 2 dudé 
dy \u? da u? dady~— u® dy dx 
