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Vol.. 6. 1920 MATHEMATICS: T. H. GRONWALL 313 
dF ^ 
^ dw dw 
'dt di 
shall become an identity in z and / upon making w = w{z). 
In order that the z- and w-planes may be mapped upon an auxiliary 
Z-plane in such a manner that our two families of curves are mapped into 
the parallels to the real Z-axis, it is necessary and sufficient that the right- 
hand member in the relation above equals the product of a function of 
^ by a function of t. When this is not the case, and our two families of 
curves are both algebraic, then w{z) is necessarily an algebraic function 
of z. 
Applying these general propositions to the various kinds of real conies, 
we obtain all the conformal maps having the required property, and re- 
placing z and w by az b and aw + /3, respectively, or by (az + 6)/ 
(cz + d) and {az + l3)/{yz + 5) when the conies involved are circles, 
and interchanging z and w when necessary, we may reduce our conformal 
maps to the following types, arranged according to the number of param- 
eters in the families of conies : 
Five parameters (one type) : 
w = z; (I) 
any conic in the 2;-plane is transformed into the same conic in the w-plane. 
Three parameters (two types) : 
w = l/z; (II) 
an}^ circle or straight line is transformed into a circle or straight line. 
w = z\- (III) 
denoting the parameters by k, r and 6, any ellipse (k > 1) or hyperbola 
{l>k>-l) 
Zi Zi 
with center at the origin and foci at 
Zi = re^\ Z2 = - Zi 
is transformed into the ellipse {k>l) or hyperbola (1 >^> — 1) 
w = K w -i- ^ -f — V(i^2 _ _ ^ ) 
Wi Z Wi 
with K = 2k^-l and foci at 
Wi = Zi = r^e^^\ 7^2 = 0. 
In the special case k = 0, the equilateral hyperbolas 
e-'^'z-' + e^'^'z' = r' 
