120 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
hx’z + Qlaz + nx? + Qhe + a =— hy’z — 2myz — ny’ — 2ay — ts 
hay’ + 2may + ly? + Qey + ae = — haz — l? — Inzxe — %z — ie (43). 
hy2 + Qnyz + me + 2az +- = — hax’y — 2lay — max? — 2ex — wi 

The elimination of /, from the two first, by differentiation, gives 
d? d? 
h(a? — y?) + Qle — my + ies = ma + A(z? — y?) — 2my + 2Qnz, 

and the third gives 


df df ; 
Z a2 5 1 2 
hz? + Qnz + dady ~~ dyda — hx? —2z, 
so that we have 
: “hs Peay ele A ee 2 ey. 
ha? + Qle + oS hy” + 2my + Tede = 2” + Qnz + dade te =0. (4), 
which proves that h, 7, m, n are separately zero, and that each of the J’s is the 
sum of two separate functions, each containing one of the constituent variables 
only, 7.¢., 
Si(Y, 2) =Y,+2Z, 
His ee 2) = 7, + x | : ; é (45). 
J; (@, Y) = Xs + Y; 
But by (43) and (45), we have 
det ly =k 
2b + dy = 2ey ae > we 
whence 
dY 
aE = — pf" — 2ay 
aX 
ai = p’”’— 2a, &e., 
giving 
Y,=-a’+C —— iY 
X= — ba? + C+ pz, &e., 
so that, finally, 
lad 
= = = a(x? — y? — 2”) + Way + Qeze + ce + 9.— py + pz 
1d 
aE ie = 2ary + b(y? — ae z*) + Qeyz + ey + In — Put px (46). 
a, 
So = 2aze + 2byz + e(2?— a? —y?) + ez 4 Js—p'u+t ply J 

