112 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
or 
V’ log. u.g—* + $2.7V log. u V.7V log.u.g7' + V?q-' = 0, 
which may be simplified into 
V* log. u.g—* + (V log. u)’g~* + V’q~* = 0 (19): 
Together, (18) and (19) give 
Vilea¢7, =V 9g = 0. « } 
J 2 an 
2V’ log. u + (V log. u)? = 0 ( 
The latter of these equations may be written 
Vu 
us}? 

V(w V log. vu) = 0 = v( 
or finally 
Wi) 20°. YS: ey oe 
11. Hence w is the square of the potential of some distribution of matter, 
none of which is contained in the space occupied by the surfaces. 
Hence the only strict solution, 7.¢., the only one which holds at every point 
of infinite space, is 
uw = constant, 
and, of course, 
q = constant. 
From this we have 
VE = uq- gq = ulia, + Jb, + key) 
E=e + Uma t+ by + G2). 
Thus the constituents of —, separately equated to constants, give the equations 
of three series of mutually perpendicular planes cutting space into cubes, for « 
is the same for all. When we turn the axes so as to be perpendicular to these 
planes respectively, and adopt a suitable origin, we have 
&= 1a jp, CHaz, 
whence 
Or = Up, 
and thus equation (1) gives in this case the confocal surfaces of the second 
order. 
12, We omit for the present, in consequence of the remark at the beginning 
of last section, other obvious solutions. of (21), such as 
1 al ( oe 
i = apron we a eae, 
