PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 123 
requires that all three sets of vectors y, y,, y, shall be null, and thus gives no 
solution. A similar nugatory result is obtained from the second. 
The third thus shows that the only solution is 
il 
Tle &e. 
Hence 

and, finally, 
ct ie (iSy,p +957; a KS7;Pp) : 
where y,,7,, 7; form a rectangular system with equal tensors, whose common 
value must obviously be the reciprocal of w. But we have seen that 
do~ dow do 
aa? dy” ‘dz 
also form a rectangular set of vectors with equal tensors. Hence 
(Siy,)’ + (Six) + Siri) = Syn) + 877) + Si7i) = 
0=Siy,Sjy, + Sty Sir, + Sty] S777, &e. 


These equations also are satisfied identically, and we therefore have, as 
before, 
= Ugpg > 
where w and g are each constant. 

