122 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
In the other set of three, we have by the same process 
0 = Sy7= Sy7y= Sy"y 
0 =SpiVy 7. Vay ey 27 | — 
O = p? fee’ +Sy’7, + Syy¥—S7.75} —28 y'pSy,p—2S ypSyip + Sy, pSy,e= 
0 = SoiVa.9, Hvis Sept er} Ser 4 ee Ale 
0=S8y%=Sn771 =SiMnH- 
We might easily have obtained this last set of equations from that which pre- 
ceded, by a species of differentiation, p being constant, and dy = 7’, dy’=y”, &c. 
24, From these we conclude that, if they exist at all, y, 7’, y”, and y,, y, 
y,, form rectangular systems with equal tensors. In terms of them we obtain 
—%= «7,—Cy, +e, = cyte’ —e'y” 
sae y= e”y, =. KY; —ey, — — ely =- cy + ey” 
—y, =—ey, +ey, tKylL = ey—ey’ +cey’, 
where « and © are scalar constants to be determined. 
Expressing, from these, y, in terms of y, 7’, y’”, we have 
N=—rt {0c + @)y + (ee + Ke”) y+ (ce” —Ke')y’t, 
where 
io er e! 
if Ne e", «—e@|=K{e+e+e7%4+ 6%}. 
—@é, @ kK 

Now, the above expressions for y,, &c., show that 
Ty, =Ty. &¢., 
hence by expanding and simplifying 
he ote (2 ay (2 kay te). 

D K 
This admits of no values but 
and 
The first of these three values of x gives 
Y aperemme a &e., 
and thence, by the equations at the end of § 23, leads to an impossibility, which 
