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MATHEMATICS: D. N. LEHMER 
the placenta. The two sexes are nearly equally apt to be affected — of 243 
affected persons 56.8% are males. The disease tends to recur without a 
break in the generations and is equally apt to come down the male and the 
female line. Consequently it looks as though the hereditary factor in neuro- 
fibromatosis is a dominant one. In each affected fraternity, indeed, about 
50% of the individuals are affected, as is expected if it is a dominant trait. 
Actually 43.5% were found affected. In some cases, however, a generation is 
skipped — a result that can be explained on the hypothesis of occasional failure 
of dominance. 
The symptoms of neurofibromatosis are very diverse. But inside of one 
family they are apt to be alike. This speaks strongly for the hypothesis of an 
inheritance factor. Similarly the location of the principal tumors is apt to be 
the same in one family, although it shows the greatest diversity in different 
families. Other multiple tumors are inherited in the same way as neuro- 
fibroma ta. Thus the tendency to form vascular tumors of the skin and 
mucous membranes has been shown by Osier (1901) and others since to be a 
dominant one. Polyadenomata are inherited similarly. Likewise the tend- 
ency to form pigmented patches in the skin (ephilides) was shown by Ham- 
mer to be a dominant trait. To this same group of heredity belong epi- 
dermolysis bullosa, angioneurotic oedema, and persistent hereditary oedema, 
also such skin diseases as psoriasis, porokeratosis and ichthyosis. 
In not a few cases the removal of neurofibromata has been followed by 
malignant growths, at the same spot. It is plain that neurofibromata are 
in some way related to cancerous growths. The fact that neurofibromata 
have an inheritable basis strengthens the view that cancers in general have 
such a basis. 
The complete paper will be published jointly with Dr. S. A. Preiser. 
ARITHMETICAL THEORY OF CERTAIN IIURW1TZIAN 
CONTINUED FRACTIONS 
By D. N. Lehmer 
Department of Mathematics, University of California 
Communicated by E. H. Moore, April 2, 1918 
The following research is the outcome of the discovery, made some three 
years ago, that the denominators of the convergents of order 3n, 3n — 2, and 
3n — 6, as well as the numerator of the convergent of order 3n — 3 in the regular 
continued fraction which represents the base of Naperian logarithms, are all 
divisible by n. The convergents recur modulo n with a period of 6n when n 
is odd, and with a period of 3n when n is even. 
