78 
MATHEMATICS: J. LIPKA 
groove. This fold is well shown in Keibel's"^ figures of embryos of the 
pig, and must represent the supramaxillary fold of the Teleostomi, the 
lachrymal groove representing a part of the supramaxillary furrow of 
those fishes. The supramaxillary fold is apparently not continued on- 
ward anterior to this point, as it is in Chimaera and Ceratodus, and the 
Schnauzenfalte of His's^ descriptions of human embryos, notwithstand- 
ing that it strikingly resembles the median portion of the supramaxil- 
lary fold of Chimaera and Ceratodus, is probably not a part of that 
fold. The lips and nasal apertures of the Mammalia could, accordingly, 
not be derived from those in Ceratodus without marked reversions, but 
they could readily be derived from those in Amia or Polypterus by the 
simple shifting of the secondary upper lip from a position oral to the 
nasal apertures to one between those apertures. 
In the Amphibia the formation of the nasal appertures, as described 
by authors, is markedly different from that above set forth, but this 
is certainly due simply to condensations and abbreviations of the nor- 
mal developmental processes, for the posterior nasal apertures of the 
adults of these vertebrates lie, as they do in the Amnio ta, between the 
primary and secondary dental arcades, and the nasal apertures of either 
side are, in embryos of certain of these vertebrates, connected by an 
epithelial cord (Gymnophiona) or line (Urodela) derived from the ex- 
ternal epidermis; this cord or line certainly indicating the line where 
nasal processes have fused with each other above the nasal groove to 
form a normal nasal bridge. 
iMiiller, J., und Henle, J., Systemaiische beschreibung der Plagiosiomen, 1841, Berlin, 
xxii + 200 pp., 60 Taf. 
2 Allis, E. P., Jr., Q. J. Microsc.Sci., London, N. S. 45, 1901, (87-236), pi. 10-12. 
3 Peter, K., Handbuch vergl. exper. Eniwickelungslehre d. Wirbeltiere von O. Hertwig, 
Bd. 2, Teil 2, 1906, (1-82). 
^Keibel, Fr., Anat. Anz., Jena, 8, 1893, (473-487). 
6His, W., Arch. Anat. Physiol., Anat. Abth., Leipzig Jahrg. 1892, (384-424). 
NATURAL AND ISOGONAL FAMILIES OF CURVES ON A SURFACE 
By Joseph Lipka 
DEPARTMENT OF MATHEMATICS. MASSACHUSETTS INSTITUTE OF TECHNOLOGY 
Communicated by E. H. Moore, December 13, 1916 
1. If F is a function of the coordinates of a point and ds is the ele- 
ment of arc length in any space, the curves along which S Fds is a 
minimum are said to form a natural family of curves. Such families 
include many interesting special cases. Thus if W is the negative po- 
tential function and is a given constant of energy in a conservative 
