this formula, with the tentative values of (ky÷k) and λ obtained for each metal in my paper on the Thomson Effect and Thermal Conduction,1 agree with those found by Bridgman from his experiments. Close agreement is, of course, not to be expected, even if my theory is everywhere correct, for the values of (ky ÷ k) and λ given in the paper just mentioned were obtained without any regard whatever for the Peltier effect, they being merely certain values, not the only values, that will, approximately at least, meet the requirements of the Thomson effect and thermal conductivity. Moreover, the experimental data used in arriving at these values are doubtless faulty in many particulars; for example, in a number of cases I had no observed value of the thermal conductivity at 100° C., and was obliged to make use of a value calculated, not very safely, by means of the Wiedemann-Franz law.

Departing from the common practice, I have substituted bismuth for lead as the reference metal, taking as my definition of the Peltier heat the amount of heat (ergs) absorbed in the passage of (1÷e) electrons from any given metal to bismuth. All the values in the table below are to be multiplied by 104. Con. is constantan and Man. is manzanin.

by means of equation (10), from the values of (ky÷k) and λ, for 0° C., given in my already printed paper on the Thomson Effect and Thermal Conduction. Comparison of these two columns shows that the calculated values are three or four times as great as the observed values, but this disparity, before any adjustments have been attempted, is not discouraging. Column V gives the observed values for 100° C., while VI gives those found by means of equation (10) from the values of (ky÷k) and X, for 100° C., given in the paper just mentioned. Here again the calculated values are some-fold larger than the observed values.

Columns III and VII, which are to be compared with I and V, respectively, are obtained from columns II and VI, respectively, merely by using new values of (ky÷k) and X, for bismuth, these new values serving quite as well as the former ones, on the whole, for the Thomson effect and the thermal conductivity of bismuth.

Columns IV and VIII, which are to be compared with I and V, respectively, are obtained from III and VII, respectively, by using new values. of (ky÷k) and λ for each of the other metals than bismuth, due regard being paid to the Thomson effect and the thermal conductivity in the selection of these new values.

The agreement between columns I and IV is as good as need be. The agreement between columns V and VIII is not so good, the values in V being in every case the larger, as column IX shows. A new readjustment in the case of bismuth, making [(kk) X100 for this metal about 14% greater than the last re-adjustment left it, would reduce most of the differences shown in column IX to a negligible size. But such a re-adjustment cannot be made without introducing disagreements at other points. The disparities between columns V and VIII, together with equally serious disparities, at 100° C., between thermal conductivities given by my formulae and those found by other means, must stand for the present.

A plausible explanation of these discrepancies, where they are not to be acounted for by mere imperfection of experimental data, may be found in the crudity of my assumption that the number, n, of free electrons per cu. cm. of a metal, can be expressed by the formula

n = zT

where z and q are constants. In making this assumption I did not expect it to hold for so great a range of temperature as that between 0° and 100° C. In fact, I was greatly surprised to find, a year or so ago, that I could make the jump from 0° to 100° with any measure of success. It is interesting to inquire how the theory of the Peltier effect set forth in this paper can deal with Bridgman's observations on the magnitude and sign of this effect between compressed and uncompressed pieces of the same metal. Testing twenty metals, including two alloys, he measured the heat absorbed by unit quantity of "positive electricity in passing from uncompressed metal to [the same] metal compressed" or, as I prefer to

put it, by (1÷e) electrons in passing from the compressed to the uncompressed 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 (ky÷k)λ under pressure, and a negative effect by an increase. It seems probable that (kƒ ÷ k) is generally decreased by pressure, and that the increase of (kk)λ indicated for certain cases by Bridgman's experiments is to be attributed to an increase of A sufficient to overbalance the decrease of (kƒ ÷ k). A priori one might expect λ to decrease with increase of pressure, causing reduction of volume, since, according to the formula 'λc' + sRT, it increases with rise of temperature, causing expansion. But it is unsafe to assume that a contraction caused by pressure will have the same effect on the properties of a substance as a contraction caused by fall of temperature. Thus Bridgman says: "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 resistance is only a few per cent less than under normal conditions."

A change of about 8% in the value of (kƒ ÷ k)λA 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 Jn (2) 0 are known to be all real if n<-1. The methods of Sturm when applied to the function J,(z) show that the roots are increasing functions of n if n>0, that is to say, denoting by (n) the kth positive root (n')>(n) if n'>n>0. In the following it will be shown that (n')>(n) if n'>n>-1 so that the inequality holds as well when −1<n<0.

Putting f„(2) = Σr(n+r+ 1)г(† + 1)

the polynomials g(2) are to be considered. They satisfy the recurrence

formula g,+ 1

=

o

=

(n + v + 1)g, + zg,-1 with initial values g-1 go = 1 and are connected with f, by the relation f2 = gyfn + + 28, – 1fn + + 1. Putting A, + 1 − g + 1 gy' where differentiation is with respect to z it is known (Hurwitz, Math. Ann., 33, 1889, p. 246) that A, +2 = (n + v + 2)g2, + 1 + z2 A,, A1 = n + 1, ▲2 = A2

=

(n + 1)2(n + 2) 8 +1 is an in

creasing function of z if n + 1>0.

If differentiation is taken with respect to n and D,

Do

signed to z D,>0 and 8+1 is an increasing function of n, with the con

From these properties of &+1 it follows if z1 is a root of g, = 0 that

gy – 1(21)>0 or <0 according as g, changes from negative to positive or the reverse when x increases through 21. Denoting the dependence of g, on n by g" it follows if n' is slightly greater than n that g(1)>0

in the first case and <0 in the second because in both cases

n

gr (21)

g. – 1(21) from z1 it follows in both cases that 21'>21. are known to be the limits of the roots of g" (z) if denote the kth root of fn Pk O and Pk The equality can easily be excluded. This is the theorem stated at the beginning. It has been assumed that ≈<0 and if n + 1>0, all the roots 21 are <0 in fact.

-1.

A formula of another kind may be obtained as follows: Differentiating the equation fn = &v fn + + 28v - 1 fn + v + 1 and equating the result 8v v to fn + 1 expressed in terms of fn + » + 1, fn + + 2, and comparing coefficients it is found that g = g" + (n + v + 1)(g′′)' + (zg” – 1)'

An application of the second equation may be made to the calculation

of the roots of fr

of g and g

=

=

=

0. Assuming n + 1>0 as before, so that the roots O are all real and negative, let Z1, Z2, Z3 be the kth roots of g +1 g +1 = 0, then z1>22>23 and the functions g all change in the same sense from negative to positive or the reverse when z passes through the corresponding root. If z' is the kth root of (g" + 1)' 0, from the above equation g" + 1 (z′) g (z') and it can be seen that (g+1(2))'>0 or <0, according as g+1 changes from negative to positive or the reverse when z passes through 21. This allows a series of approximations one, as follows:

to a root of f

1

=

n

=

0, say the first

g2= 0; form the quantities 23 = Z2

Let 22 be the root of

83 (Z2) 83' (22)

81 (23) g1'(23)

etc. These

z's converge to the first root of fn = 0. In the case of the first root the

approximations are rational functions of n.