PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 111 
From these we have 
— 3.7Vv.i 4 — 68. vq + 2V9-4.g = 9, 

or’ 
Vu et . aa ty 
a oeeMe Ger Vg .g = 
Hence 
DAVer g = 0 
and é 
Vu i = Vq-! G : (12), 
Gn ia ari Sa ae 
or, finally, 
Weng =] : : (13). 
10. But this is not the only relation between wand g. For by (12) we may 
write (11) in the form 
— V.(Va"gq*ag)q~* — 28. (g~*agv)g~* =9 . (14). 
It is obvious that, by adding the three equations of this form, each 
multiplied by a proper scalar, we may derive from them three equivalent ones 
of the form 
NIVG=-9?) ¢= "4 250V)q-" = 0. . (15). 
This may be written by the help of (12) in the form | 
ses 
2 i= V ANG = Viwtav. wV log. u (16), 
and we thus see that the constancy of the tensor of qg is recognised. 
Differentiating again after multiplying into g—', we have 
dg —* 
27a = V(eV log. u) ne + Viv log. u.q-* 
= (V.iV log. u)’q7* + Viv loge... 
Adding the three equations of this form, we have 
aONE == (Valoga ig" 3 (7); 
for obviously V? log. w is a scalar. 
But we have also 
Vega. = 0 (18), 
which gives 
V log. u.g—* + Vq-? = 0 
and 
V’ log. u.g— + 2.1V log. yo = ore Va =0), 

