114 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
=a dh do-~ .—1 
Set er oe 
This gives 
28. EV oS (l) GE .eb (0) (GB) + 8.0 “of (GZ) 
= d*h d? =e d -—1dc- 1 = 
+ S.cp aft) a= 28.55y 9 +28. oy 2-28, Sy rE 
dae netetes OUD ; 
Eliminating 7 from these equations, we have 
do- —2 do- =i) d- er de a 
43:5 cS.a oes y Sie mag o- ee oo ae say 3 
Sic Jo .o- o- Seah oe co "(Fy ae 
ee a 
as a*h @2 ae yea Sel Peng se a 
+ Sh “eS Wig 28.Ga 0 e+ 28. BU GE AS EW a 
Now, whatever vector « may be, we have by § (5), (6) 
dentine leis (team 
d-, do 
2 ==. Gs!29 pe piel Set eee 3 eee 
— Ws = GS. Get GZS. qe dz S. az”? 
so that, if w be any other vector, 
: ; : 2.eutee ar it 
Adding, then, the three equations, of which that containing 7 is given above, 
we find 
_ 4p Sheen sutS.cu- Spey ie ae SW) Sp eye 
age bd 
Sip? 


Sty o ry (F(R)? Sop oe 
& ee SG —2 —1 
=—28. Vey +238. Soy S + 4 eae 

where the term in V7 of course vanishes by (22). This is seen at a glance to 
be equivalent to 
fh) -1 do- , —1.de- 
oe 2 =—28.Vewb oo + 225.70 ie 
The last term here is seen at once, by HamitTon’s beautiful theory of linear and 
vector functions, to be equivalent to 
aie 5 1 it 1 
2V2S.if 1=—2u = + By7qHy t <= : (23), 
if A, B, C be the constants of ¢. Calling the expression in brackets for the 
present H, we have 
