66 
MATHEMATICS: J. H. McDONALD 
Proc. N. A. S. 
put it, by (It?) electrons in passing from the compressed to the un- 
compressed metal. Working from 1 atm. to 2000 kgm./cm. 2 and from 
0° C. to 100° C, he found this absorption to be positive throughout in 
fourteen of the metals; negative throughout in three, cobalt, magnesium, 
and manganin; mixed, sometimes positive and sometimes negative, in 
three, aluminium, iron, and tin. 
In terms of my theory a positive effect here can be accounted for by a 
decrease of (kf 4- k) X under pressure, and a negative effect by an increase. 
It seems probable that (k f k) is generally decreased by pressure, and 
that the increase of (k f — k)\ indicated for certain cases by Bridgman's 
experiments is to be attributed to an increase of X sufficient to overbalance 
the decrease of (kf 4- k). A priori one might expect X to decrease with 
increase of pressure, causing reduction of volume, since, according to the 
formula X' = X/ + sRT, it increases with rise of temperature, causing 
expansion. But it is unsafe to assume that a contraction caused by pres- 
sure will have the same effect on the properties of a substance as a con- 
traction caused by fall of temperature. Thus Bridgman says: 2 "The 
volume of many metals at 0° C. and 12,000 kg. [per cm. 2 ] is less than the 
volume at atmospheric pressure at 0° Abs. The resistance of most metals 
tends towards zero at 0° Abs., but at 0° C. at the same volume the re- 
sistance is only a few per cent less than under normal conditions." 
A change of about 8% in the value of (kf -5- k) X would account for the 
maximum Peltier effect between compressed and uncompressed bismuth, 
as observed by Bridgman, and a still smaller per cent change, in most cases 
much less than one per cent, would serve for the other metals dealt with 
in this paper. 
1 These Proceedings, October, 1920, p. 613. 
2 Proc. Amer. Acad. Arts & Sci., 52, No. 9 (638). 
ON THE ROOTS OF BESSEL'S FUNCTIONS 
By J. H. McDonald 
Department of Mathematics, University of California 
Communicated by E. H. Moore, December 13, 1920 
The roots of the equation J„(z) = 0 are known to be all real if n< — 1. 
The methods of Sturm when applied to the function J n (z) show that the 
roots are increasing functions of n if n> 0, that is to say, denoting by ^(w) 
the k th positive root ^.(w') >^fe(w) if n'>n>0. In the following it will 
be shown that \l/ k (n f )>if/ k (n) ii n'>n>—l so that the inequality holds 
as well when — 1< n < 0. 
Putting/ M (z) = "S? so that J n (z) = (* )/,/--) 
hJ ^T(n + r + l)r(r+ 1) W \ 4/ 
the polynomials g*( z ) are to be considered. They satisfy the recurrence 
