PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 121 
If, in these, we write 
ia + jb +ke=y 
(1G, + II, + Ms= Vy 
ip! + 7p" + kp = V> : 
we obtain, by multiplication by 7, 7, & respectively and addition, 
1 / 
pe Pres ee ee Se (46), 
which is equivalent to the three equations (46), and may be put in the form 
iL Wf 
a Vee yp typ by Nae s |  (46"). 
23. It was shown above, § (7), that 
VE, Vn, VE 
form a rectangular system of vectors whose common tensor is w. Hence, by 
(46’’) we have three equations of the form 
1 
we 
—— = (¢e—28yp)'p" + 7p + + Srp — P've 
+2p’Syp(e¢—2Syp) +28 y,p(@—2Syp) + 2p’Syy, + 2p'S.yy,p+28.y, y2p, 
expressing the equality of the tensors; and three others of the form 
0= (e¢—2S yp) (¢’ — 28 7p) p? + p*(e’ —28y'p)Syp + (¢ — 287’) Sy, 
+ (¢—2S8 yp) p’Sy/p + p*Syy’ + p’Sy'y, + p'S.7' 720 
+ (¢—25 yp)Sy.p + pSyy + Sry, + 8.772" 
+ P°S.7 70 +8.7,%9 + 887.0 —Syy7op"- 
Here the constants in Vn; VZ are expressed by the application of one, and 
two, dashes respectively to those of Vé 
In the first set of three, the terms in the various powers of Tp must be equal. 
This gives the following sets 
y’ = y” = y” 
S(Vyy,— ey)p—S(V77,—¢y )p = 
Ciao ego ap toy poy p=” 
S(V¥,7. a ey,)p SS eon ay Tetulce Uc 
112 
Deh A (OH 
» | val x V1 - 
VOL. XXVII. PART I. OT 

