116 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
27 = 3s" (idh(—H+ FOE 
of which the integral, by (23), is 
C’— 2 log. uw = [2 log. (kh) — log. (A + /(h)) (B+ 7())(C + (2) ], 
or, if we write, 

2k BATA erty tt bee pa 
th J (A + fh) (B+ fH) (C + £0) FQ) OS OSS ee 
en. 
oaayrajEay* )\ ae 
17. The following is the first quaternion method that occurred tome. I 
give it here, though it is considerably more prolix than the preceding, because 
it exhibits, incidentally, many curious properties of the system -, uw, g above 
defined. 
Starting again with equation (7), we see that it gives 
—-V?- = qVuq —2uq> (V.ivg'F ae 
or, 
he a 
to which value, as we shall see immediately, it may be reduced. ; 
18. From (7) we obtain at once 
1 do _ dg, 1 1dqg,-! , 1 dude 
u dedy — dy 2 oe i on u* dy dx’ 
Si dg = 1du 
aa — gig” dy! ‘+= = qi iq ae 
1 d?c- . ad 
7a ap “ig” —gjg 2g” + - aig. 

Comparing the last two values, we have 
eet Melero ey NT EO ee VO ot “7 -ldq , j du 
a il 2V.i1Vq a ae 2V.7Vq aa? cae (30). 
Operating by S.4, we have 
bi -—ldg _ . —ldq 
S.99 re ee = 
From this, and other equations similar to it, it is obvious that 
- —1d - —1ld —1d 
S.2g “= 8-39 “em Sa ‘J=0 oo ye oan 
