PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 109 
But, in this case, by (2) 
VA \\2, &e., 
and we have series of rectangular planes. 
6. Hence there must exist a scalar function w, and a quaternion q (which 
may obviously be taken as a mere versor), such that 
dco I 
de = Utd, &e., 
or, im one expression, 
d=- = ugdpqé' (7). 
Thus it appears that, in order that (1) (with the limitations above imposed) 
may represent a triple series of orthogonal surfaces, ~ must be such a function 
of p that, if the extremities of a set of values of p form the corners of an 
indefinitely small cube, those of the corresponding values of ~ (drawn from a 
common origin) form the corners of another such cube ; and that, therefore, the 
passage from p to ~ is that from one mode of dividing space into indefinitely 
small cubes to another. 
Whatever, therefore, may be thought of the logic of the investigation 
above, it is worth while to pursue the inquiry thus suggested, by developing 
the consequences of the equation (7) to which it has led us. 
7. From the equations just written we see that if 
c=e+ m+ hkl, : : : : (8), 
the direction cosines of giq—* are 
1dé 1dn 1dg- 
ude udx wdz 

From these, and other six of similar form, we see that the direction cosines of 
2 referred to giq—*, qjq—', gkq7—' are 
Ldg 1de Lae 
wdx’ udy’ udz’ 

and similarly for those of 7 and &. 
Hence it follows that VE, Vy, V¢ form a set of mutually perpendicular 
vectors whose common tensor is 2. 
The same result may be obtained as follows :— 
a o ad ; 
Vé=—(iz, TI dy aig b.) Sie 
=— u(tS.igigq~' + J8.igjq—* + &S.igkq-?) 
=— Uu(iS.ig—*ig + JS.jq7-' tg + &S.kq-? ig) 
=a ak ae : ; ; : é ; (9). 
VOL. XXVII. PART I. 2 

