702 
MATHEMATICS: J. H. MCDONALD Proc. N. A. S. 
Spectrum was the same at a given temperature, showing no effect of the 
reduced possibiHty of conduction by the vapor. With the first method, 
an extra protection of the tube from loss of heat permitted the observa- 
tion of the test spectrum with a potential difference of less than 0 . 6 volt 
per centimeter length of the tube. 
Two experiments in which there was no potential difference on the tube 
were next tried. In one, the tube was heated, the current broken, and 
the exposure made during an interval while the tube cooled with no cur- 
rent. The spectrum was found to be the same as when the furnace was 
operated in the regular way at the mean temperature of the no-current 
experiment. In the other test, a tube containing iron was heated by a 
high-current arc between horizontal electrodes supported beneath the tube. 
The vapor in the tube used as a crucible in this way emitted a spectrum in 
which the test lines appeared as in the resistance tube carrying a current. 
The conclusion from this series of experiments is that for the tem- 
peratures required in the line classification which the writer has carried 
out for a number of metallic spectra, the potential difference acting on 
the tube is not effective in modifying the spectrum, the lines sensitive 
to arc conditions appearing with equal ease whether a voltage is acting 
on the tube or not. Higher temperatures, accompanied by increase of 
both ionization and potential drop, should be checked when possible as 
to the effect of conduction by the vapor. Lines brought out only at these 
higher temperatures are, however, usually so faint that they are of little 
importance in the furnace spectrum. For a study involving these lines, 
the arc would usually be employed. 
AN APPLICATION OF THE PORISM OF FOUR TANGENTS OF A 
TWISTED CUBIC 
By J. H. McDonald 
Department of Mathematics, University of Cai^ifornia 
Communicated by E. H. Moore, October 25, 1920 
The problem of determining the involutions of degree n of a given 
discriminant has not been solved except when n = 2,3,4. 
Special forms of discriminant occur in Jacobi's transformation of elliptic 
integrals and in researches on reducible integrals. Geometrically the 
problem is to construct a rational curve in w-space of degree n to touch 
2{n — 1) hyperplanes of a pencil. For n = 3 the solution depends on 
a certain porism;^ that of four tangents of a twisted cubic. There is 
in fact the theorem: No proper twisted cubic can be drawn to touch 
four lines unless they satisfy a condition in which case an infinity of cubics 
^ Dixon, A. C, on twisted cubics which fulfil certain conditions, Quar. J. Math., 
23, 1889 (343-357). 
