PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 115 
I’) 
So + 2a = BS. View a (24): 
The left hand member, if multiplied by /’ (2) dh, is the differential of a function 
of h only. If, as in § 11 we have 
oc —up, w= constant, 
the right hand side vanishes, and integration gives 


f'(h) dh 
Cas eS +f(h)) (B+ 4) (C+) = 08 : 
If ~ have the value given in § 12, equation (24) is obviously not satisfied. 
Thus confocal quadrics are the only isothermal orthogonal surfaces included in 
equation (1) with our present limitations. 
16. It is interesting in itself, and will be useful for the second part of this 
paper, to eliminate ~ from (24) by the help of our previous equations. For this 
purpose we may write (2) in the form 
Tye f/(b)Vh = 23 (68.5 ye) 
= 2ud(iS.ig7 “ww '-q) 
=—Qug bog 400) gests. (25); 
the tensor of which is 
Ty of (h)TVh = Qu. 
But it is shown in (29) below that : 
Vu=q Voq, 
so that (25) gives 
Tp J’ (h)S.VuVh = 2uS. Vey a , 
or, by (24), 
= 2u(—H +2 Taye ees: 
The three equations of this form give 
>.T'y bof’ (h) VAS. ViVu = 24? >. Vh( — Hi, +: 2 rasp ap ): 
or, by (25) and (3), 
—4u°Vu = 2u°S. f(b) Vh (- sr 2 Uy ): 
Operating by S.dp, this gives 

