Vol.. 6, 1920 MATHEMATICS: T. H. GRONWALL 315 
real axis in the Z -plane. When <pi{z) and (p2{z) are any two of these func- 
tions (identical or different), all maps transforming one parameter families 
of conies into others are found by making 
(P2{w) = A(pi{z) + B, 
where A and B are constants and A real, and then making the linear trans- 
formations on z and w indicated at the beginning of this summary. 
Z = z; (VIII) 
all parallels to the real axis* (special case of I). 
Z = log z; (IX) 
all straight lines through z = 0 (the circles with center at ^ = 0 go into 
parallels to the imaginary Z-axis) . 
Z = i log z; (X) 
all circles with center at ^ = 0 (the straight lines through 2; = 0 go into 
parallels to the imaginary Z-axis) . 
Z = ^2. (XI) 
all equilateral hyperbolas z"^ z'^ = r"^ with center at the origin and foci 
on the real axis (special case of III). , 
Z = log(^2_i). (XII) 
all equilateral hyperbolas 
^-26i^2 ^ ^2er^2 ^ 2 cos 2B 
witn center at the origin and foci =t V2 cos 2d (which are the end-points 
of the diameter with angle of slope % of the lemniscate with foci at + 1 
and —1). These hyperbolas all pass through the foci of the lemniscate. 
Z = log {z + V^"^) or ^ = V2(^^ + e-^)\ (XIII) 
all hyperbolas with foci at -j- 1 and — 1 (all ellipses with these foci go into 
parallels to the imaginary Z-axis) . 
Z = lA log {z -h V02-I) or 0 = cos Z; (XIV) 
all ellipses with foci at +1 and —1 (all hyperbolas with these foci go into 
the parallels to the imaginary Z-axis) . 
Z = V0; (XV) 
all parabolas V^— = i^p with focus at the origin and the real axis 
as axis (special case of V). 
Z - log (Vi-1); (XVI) 
all parabolas 
with focus at the origin and passing through the point z = I. 
1 Von der Miihll, /. Math., Berlin, 69, 1868 (264). 
2 Meyer, "Ueber die. . . Isothermen," Dissertation, Gottingen, 1879. 
