PROFESSOR TAIT ON ORTHOGONAL ISOTHERMAL SURFACES. 107 
d. Now if = c be one of a system of surfaces isothermal as well as orthogonal, we must 
have, by the above equation, 

Diu, &) = 0, 
But the orthogonality gives. 
D@, €) =9, 
Dé) =90, 
and the elimination of £ among these three equations gives 
Alu, ; 6) = 0 ) 
ae. by the property of functional determinants, w is a function of » and Galone. ‘Thus we 
have 2 
Ey = So 9) 
a well-known relation, &c. 
1. Consider the equation 
T.(@+/M)y be =1, 
S.-(6 +f(h)) - =—-1 Ci. inca Pls 
where ¢ is any self-conjugate linear and vector function, of which 2, 7, & are the 
principal vector directions. We assume that the roots of Hamiiron’s equation 
M,=0 
are finite and different from one another, so that cylinders, surfaces of revolu- 
tion, &c., are excluded from (1). 
For any assigned value of — , (1) gives in general three values of /(h) and 
therefore of 4. Omitting for the present the consideration that each value 
of /(h) may give more than one value of h, these values may be any assigned 
functions of the position of a point in space; because, when they and the 
function / are assigned, the squares of the constituents of ~ (or, what comes to 
the same thing, the values of -’, S~¢0, S-¢’-) can at once be found in terms 
of them, by a system of three /inear equations. 
Tn this first part I confine myself to cases in which each of these squares is 
positive, so as to avoid for the present the use of biquaternions. 
2. For any assigned constant value of /, (1) represents in general a surface 
whose normal vector, Vh, is given by 
Sc oJ’ (W)Vh = 23(i8. yc) a 
or, as it may be written, 
Wy being written for convenience in place of ¢ + / (A). 
Now if h,, h, be the other two values of h given by (1) for a particular value 
of ~, the conditions of orthogonality of the surfaces h, h,, h., are of the form 
0 =S.VAVh, = 3.8.2 ye8. Lye fae ee) 

