314 MATHEMATICS: T. H. GRONWALL Proc. N. A. S. 
with center at the origin go into the straight Hnes 
Two parameters, denoted by r, B or p, B (four types) : 
= z-^-l', (IV) 
any equilateral hyperbola 
e-2ei^2 + e^''z~^ = r2 
with center at the origin and foci at Zi = re^\ Zi — ~Z\ is transformed 
into the equilateral hyperbola 
^-2»^-^2 _^ ^2^i-2 = ^2_2 COS 2B 
with center at the origin and foci at 
Wx = ^lr'^ — 2 COS 2d e^\ = —Wi. 
In the special case f ^ = 2 cos 2B, we have a pair of perpendicular straight 
lines through the origin in the T£;-plane. 
any straight line 
is transformed into the parabola 
e'^'''^u>-e^'^'''w = Hp 
with focus at the origin and at a distance of ^/2p from the directrix, B being 
the angle between the axis of the parabola and the real axis in the if-plane. 
w = {z'^ — Ciy, where Ci = 0 or 1; (VI) 
any equilateral hyperbola 
e'/^^i -z^-e~ "''''z" = ii^Jp + 2ci sin V26I) 
with center at the origin and foci at 
= ^Vu^-)'(v^ + 2ci sin 72(9)'''', Z2 = -Zi 
is transformed into the parabola 
e^'-''^w-e-'^'''^Jw = Hp 
with focus at the origin. 
w = Hz-iy; (VII) 
any parabola 
e'^'''^Jz-e-'^'''^Jz = iHp + 2 sin 1/2^) 
with focus at the origin is transformed into the parabola 
e'^^''^jw-e-'^^''^w = H} 
with focus at the origin. 
One parameter, denoted by r, B or p (nine principal types, the linear 
composition of each of these with itself or with another giving a total of 
forty-five types). There are tabulated below the nine functions which 
map the one parameter families of conies specified on the parallels to the 
