BOTANY: C. DRECHSLER 
221 
imaginary double points is clearly zero, which is another verification of pre- 
vious results. (16) is also true when mi is even. Polar and cartesian equa- 
tions for mi even may be established in a similar manner as in (14) and (15). 
3. Examples. — When mi = 3, 2 k = 4, the polar and rational parametric 
equations of the curve 3 are 
p 4 - Rp 3 cos 3d - ?L R 2 P 2 + ^-R* = 0 
?-Ut 2 +3 2t(3-5t 2 ) , D n 
4 a + t 2 ) 2 4 (i + t 2 ) 2 
When mi = 4, 2 k = 6, cartesian and polar equations of the curve are 
S( X 2 -|_ 3,2)3 _ 2 4(x 4 + y 4 ) - 32 x 2 y 2 + 39(x 2 + y 2 ) - 18 = 0, 
3 p^ - (2 cos 49 + 22)p 4 .+ 39p ? - 18 = 0. 
As m and & increase, the construction of the equations with numerical co- 
efficients becomes increasingly difficult. 4 
1 Greenhill, Les Fonctions Elliptiques et leurs Applications, chap. III. 
2 Tannery et Molk, Elements de la Theorie des Fonctions Elliptiques, 4, pp. 176-192; Appell: 
Train de Mecanigue Rationnelle, 1, p. 494. 
3 This curve is known and was investigated by G. de Longchamps: /. Math. Elementaires, 
4, 1885, 269-277; also by H. Brocard: /. Math. Speciales, (Ser. 3), 5, 1891, 56-64. 
4 A curve of this type may be symbolically denoted by C^J. Tabulations of all curves, 
with their real and imaginary double points as far as Cll have been made, also actual graphs 
of Cl, Cl, C\. C\- 
THE TAXONOMIC POSITION OF THE GENUS ACTINOMYCES 
By Charles Drechsler 
Cryptogamic Laboratories, Harvard University 
Communicated by R. Thaxter, May 14, 1918 
To the genus Actinomyces are usually assigned a variety of peculiar and 
very minute filamentous organisms, widely distributed in nature, concerning 
the taxonomic relationship of which divergent views have been held. Most 
of the earlier medical writers, whose attention was centered on forms associ- 
ated with human and animal diseases, placed the genus with the pleomorphic 
bacteria. Others recognized conidia in the clavate elements produced by one 
of these pathogenic forms, Act. bovis, within the animal body, and referred this 
parasite to the Fungi. These elements were later shown to represent de- 
generative structures; but as subsequent investigations on a number of species, 
including various saprophytic forms, as well as the common potato scab 
