110 PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 
Hence we have 
—d& = uS.g—'igdp , 
of which the condition of integrability is 
V.V.ug t¢-= 0. 
Thus w and g must be determined go as to satisfy the equation 
V.V.uq—laq = 0 : - HED), 
whatever constant vector be represented by a. 
We may state our present conclusions in the following simple form. In 
order that (1) may represent a triple series of orthogonal surfaces, it is necessary 
and sufficient that the constituents of - satisfy three equations of the form 
(9) ; 2.¢., that, when severally equated to constants, they represent three series 
of surfaces which together cut space into cubes. 
8. As a verification of (9) we see that it and the similar expressions for 
V7 and VE give 
—d-=u(iSq—igdp +. ; . .) 
| =— ugdpq~* , 
which is equation (7). 
9. Performing the operations indicated in (10), it becomes 
V.V.ug~*aq + WV. 2 (tga ih — ig? qa) = 
or 
Vu | : de _ dg 
V.7 9 (aq + 22V.iV.9—agq “Gq = 0 
3 
(this simplification being permitted because (S 6) the tensor of g may be 
regarded as unity) or, finally, 
Vu - d ‘ dq 
Vig Gag SS o9 pag Sige oe + 22. (8:9 'agi)g-* == 
which may be written 
V. ve tag + 2q~tagS.Vq-"¢ — 23.(S.q7aqi) Eas =2 1s 
or 
V. te + 29g~"agS.Vq-'¢ — 28S. (g7agV)q-1.g = 0. 
Here a has the values 7, 7, 4, so that if we write 
G7 0g a0, er 
we have three equations of the type 
—V.i 2 + 208.Vg'y—28(7V)g--g = 0. (At), 

