638 
MATHEMATICS: L. P. EI SEN HART 
dx' , dx' ^bx 
— = h — J — = / — 5 (2; 
bu bu bv bv 
provided that h and / are functions of u and v subject to the conditions 
^= a{l-k),^-=b{h-l). (3) 
bv bu 
Moreover, each pair of solutions I of these equations leads by quad- 
ratures to a net N' , such that the tangents at corresponding points M 
and M' to the curves of the net are parallel. All nets parallel to N are 
obtained in this way. 
If 0 is a solution of equation (1), and 6' is the corresponding function 
given by 
be' J bd be' , be ... 
— = h — , — = / — , (4) 
bu bu bv bv 
then the functions Xi, yi, Z\ defined by equations of the form 
X\ = X — — x' (5) 
e' 
are the coordinates of a net TVi, so related to TV that the lines joining 
corresponding points M and Mi of these nets form a congruence whose 
deveJopables meet the surfaces on which these nets he in the curves 
of the nets. We say that the nets so related geometrically are in the 
relation of a transformation T. Parallel nets are in such relation. We 
have shown^ that any transformation T oi N into a non-parallel net Ni 
is given by equations of the form (5). Hence any transformation T of 
N is determined by a parallel net and by a solution of the point equa- 
tion of the net. 
When two surfaces are applicable to one another, in the correspond- 
ence thus established there is a unique conjugate system of curves on 
one surface corresponding to a conjugate system on the other. We 
say that these nets are applicable. This paper is concerned with the 
transformations of applicable nets into applicable nets. 
If we have two applicable nets N and N with the respective point 
coordinates x, y, z and x, y, z, the analytical condition of their appli- 
cability is 
f^Y, S f s (^X (6) 
\bu/ bu bv bu bv \bv/ \bv / 
the sign 2 indicating the summation of terms in x, y and z. Since 
the coefficients a and b of equation (1) are functions of the left-hand 
