312 
MATHEMATICS: T. H. GRONWALL Proc. N. A. S. 
of rotation and the distances from the plate to I, S, and the disk D. Plot- 
ting wave radii as ordinates and the corresponding time intervals a^ 
abscissas, the tangent to the curve at any point gives the instantaneous 
velocity of sound at that distance from the source. The accompanying 
curve shows that the velocity varies from about 660 meters per sec. at a 
distance of 3.2 mm. from the sound source to 380 meters per second at a 
distance of 1.8 cm. Results have been obtained since the above curve 
was plotted for points much closer to the source and for distances up to a 
half meter. 
The writer designates as "weak sparks" those produced when each 
condenser, L, consists of two Leyden jars; as "strong sparks," seven jars. 
The curve appears to show but little difference in the velocity of the re- 
sulting sound waves. However, considering the shortness of the sound 
pulses produced by such electric sparks, all are really intense waves at 
points near the source. 
This investigation, made under a grant from the American Association 
for the Advancement of Science, will be published in full in the Physical 
Review. 
CONFORMAL MAPPING OF A FAMILY OF REAL CONICS ON 
ANOTHER 
By T. H. Gronwall 
Technical Staff, Office of the Chief of Ordnance, Washington, D. C. 
Communicated by E. H. Moore, April 27, 1920 
Note IV On Conformal Mapping Under Aid of Grant No. 207 From the 
Bache Fund 
Let z = X -\- yi and w = u -\- vi two complex variables, and let 
the analytic function w = w{z) define a conformal map of the 0-plane 
upon the w-plane. It is the purpose of the present note to determine all 
functions w{z) such that there exists a family (containing at least one real 
parameter) of real conies in the ^-plane which is mapped upon a family 
(obviously with an equal number of parameters) of real conies in the w- 
plane. The particular case when the conies in the w-plane are straight 
lines parallel to the real axis has been investigated by Von der MiihlP 
and Meyer. 2 
It is convenient to use the isometric coordinate z = x -\- yi and z = % — 
yi, and we begin by establishing the following general results: 
When0 = f{z, t), where / is analytic in both its arguments and t is a 
real parameter, represents a family of real curves in the ;s-plane, and simi- 
larly w = F{w, t) is a family of real curves in the z£;-plane, the necessary 
and sufficient condition that w = w{z) shall map these two families upon 
each other is that the relation 
