82 
MATHEMATICS: J. LIPKA 
transformation {T) on the / family, (4). (r^), (r^), and (T^) lead 
respectively to 
v" = { [(a - w - log Vx)« -{a- log VX)r] + [(a - CO - log Vx), + 
(a -log Vy^Uv'} [\+v''], (16) 
z;''={[(«-log\/x)«-coj + [(a-log \/x). + coJ?;'} [\-^v'^] (17) 
v" = (cou + co.z;0 (1 + v'^). (18) 
Now equations (2) and (4) are special forms of a more general equation 
= {rP - <i>v') (1 + V'^), (19) 
which reduces to type I or t3^e iV according to the restriction 
+ = 0 or ^„ - 0„ = 0 (20) 
respectively. Applying the criteria (20) to equations (16), (17), and 
(18), we may draw the following conclusions: 
Given any I family, (T) always transforms this into an N family, and 
(T^) always gives the orginal I family. In general (T^) and (T^) give neither 
an I nor an N family; but if the auxiliary arbitrary function, a, is so chosen 
that the system v' = tan {a — log \/ X) is an isothermal system,^ or if 
our surface is developable^ and the system v' = tan a is isothermal, 
then (r^) gives an / family and {T^) gives an N family. 
Given any N —I Jamily, i.e., the isogonals of an isothermal system {cf. §4), 
(r) always transforms this into an N family, (T^) gives neither an I family 
nor an N family, in general, (T^) always gives an I family, and (T^) always 
gives the original N — I family. If the auxiliary arbitrary function is 
so chosen that the system v' = tan {a — log \/ X) is isothermal, or if 
our surface is developable and the system v' = tan a is isothermal, then 
(r), (T^), and (T^) give N—I families (not the original family). 
4. If the N family 
= [(log F V\\ - (log F V\)u v'] [1 + v''] 
and the / family 
v" = (co„ + CO. v') (1 + v'^) 
coincide, then we must have 
(log F '\/\)v = ccu and (log F \/x)w = — co, '] 
or _ K21) 
COuu + COr, = 0 and (log F \/\)uu + (log F V X)ri, = 0 J 
Therefore the curves v' = tan co and v' = tan (log F -y/x) are isother- 
mal systems, and the functions co and log F \/x are conjugate harmonic. 
Thus the base system of our isogonals is isothermal, and if H is con- 
