PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 119 
1 (dEdu 4aé =) 
~ u® \dy da dx dy 
Symmetry shows from this that 
a (ld di ilde 
dy \ve da = alas : : : (39), 
which is one of another set of nine equations, three each for €, n, ¢. 
22. Now, by (36), it is obvious that we may write, w, beimg a new 
variable, 


lide “d's, Ide do, dt dia, 40 
ude dydz’? uw? dy” dedx’? utdz — dady ° ; (40), 
and thus (39) becomes 
oa, oa, . 
Pie dee + A oye ay AD, 
with two others in a, and three each in o,, a,. 
Putting 
Ey 
1 = dadydz? 
these give by differentiation 
dio, _ _s Vay 
dy Mae 
Ey Gee 
Cae dy? 
CY aa ies 
Ce Gea? 
so that all three quantities vanish. Hence we have 
8 
wo, = vate = 2hayz + 2lyz + 2mzxr + 2nay + Zax + 2Zy + 2z +e, 
where h, /, m, n, a, b, c, ¢ are absolutely constant. From this, and (40), we have 
1 
aE = haya + Qlays + mea? + na’y + aa? + Lay + Qezw + ex + K(Y,2); 
u* dx 
ld 
AP = =hayez + ly’z + 2mayz + nay? + Zany + by’? + 2yz + ey + /(Z, 2), ¢ (42). 
1 
u - = hayz + ly? + mea + Wnayz + 2aze + Qyz + 2 + ez + f, (2, y), 
Applying (26) to these, we obtain 

