204 

material might be formed at such a temperature if 
some helium were present. 
But of course the heat used up in forming these 
substances would cool the rest of the mass : any energy 
gained in radio-active form would be lost in the form 
of heat. It could never avail to explain a solar con- 
stant such as has been measured for longer than 
Kelvin’s 20 million years. In other words, radio- 
active substances produced would act only as accum- 
ulators of energy, not as primary batteries. 
To recapitulate: As Kelvin showed, gravitational 
energy can only account for 18-3 million years of 
sunshine at the present rate. Invoking radio-activity 
as a source of energy implies the assumption that 
unknown radio-active materials liberating considerably 
more energy than uranium were created by some 
unknown agency within a measurable period of time, 
and that these are now breaking up. This assump- 
tion is not necessary to account for the existence of 
uranium, as it is quite conceivable that a certain 
amount of radio-active matter might be produced 
afresh during every stellar collision. The energy of 
substances formed in this way would not be avail- 
able to explain a greater amount of energy on 
the sun as their energy is abstracted from the 
gravitational energy, and has already been taken into 
account. 
F. A.; LINDEMANN. 
Sidholme, Sidmouth, April 5. 
Harmonic Analysis. 
IN a paper which I read to the Physical Society 
last January (see Nature, February 11, p. 662) I 
suggested that the best way of analysing a wave, the 
graph of which was given, was to apply the rules for 
the mechanical quadrature of integrals which are 
given in treatises in the calculus of finite differences. 
I am convinced that these methods when. applied in- 
telligently are much simpler and ever so much more 
accurate than most, if not all, of the methods in 
everyday use. 
In the paper referred to above I applied a well- 
known method of mechanical quadrature (Weddle’s 
rule) to the case of a semicircular alternating wave, the 
equation to the positive half of which is y=/x—x?. 
I chose this wave because I found that the evaluation of 
the Fourier integrals for it by analysis was laborious. 
Prof. A. E. Kennelly, of Harvard University, has 
kindly written to me to point out that the equation 
to the curve can be readily put in the form— 
y =J,(z/2) sin xx —(1/3)J,(37/2) singzx+ 
(1/5)J.(57/2) sin 57x — 
where J,(x) is the Bessel’s function of the first order. 
Hence from tables of these functions we get :— 
y¥=0°567 sin 7X +0-0939 Sin 37x. 
+0:0422 Sin 57x +0:0252 Sin 77x. 
+O-OI71 singnmx+ ..- , 
Very close approximations to these numbers can be 
obtained very simply by Weddle’s rule. For example, 
if b, denote the amplitude of the first harmonic, we 
have : 
iz 
100, = 5Viig + % 3Va/3t OV i05 
where y,=/n—n?, and hence b,=0-568. 
To get an accuracy of the same order for the third, 
NATURE 
| horizon, whereas the sight is somewhat rarer. 

fifth, seventh, and ninth harmonics we must calculate 
NO; 2373, VOL? OS) 
[APRIL 22, 1915 

or measure the lengths of 8, 13, 18, and 23 ordinates 
respectively. Doing this, we find that b;=0-0942, 
b;=0-0423, and that b; and b, are given correctly. 
It will be seen that from the practical point of view 
the simplicity and accuracy of the method in this case 
leave little to be desired. It has the great advantage 
that the amplitude of each harmonic can be computed 
independently of the others. 
When the wave passes smoothly through the ex- 
tremities of the ordinates we measure, we can apply 
the rule with confidence. Jagged or very distorted 
waves must be treated more carefully. For example, 
if we apply the rule to a rectangular alternating wave 
of height unity we find from the formula given above 
that 10b,=11+¥73, and so b,=1-27321 approx. The 
true value is 4/7, i.e., 1-27324...., and hence the 
error is less than 1 in 40,000. For a triangular alter- 
nating wave of height unity, however, if we apply 
the rule intelligently we get bj=o-88 .. . instead 
of 081057... The error in this case arises from 
applying Weddle’s rule through a point of discon- 
tinuity. If we apply it over one-quarter of the wave, 
it being necessary to measure six ordinates instead of 
three, we find that b,=o0-81056.. . 
ALEXANDER RUSSELL. 
Faraday House, Southampton Row, W.C., 
April 12. 

A Mistaken Butterfly. 
REFERRING to Prof. Barnard’s letter so titled in 
Nature of April 15,. which describes the apparent 
mistake of a butterfly in visiting a peacock’s feather 
as if expecting to ‘‘extract food,’’ I think it probable 
that there are no animals that do not make mistakes 
at times. I observed an analogous mistake made by 
a species of Pierida—Appias nero—in Sumatra, as I 
have recorded in ‘“‘A Naturalist’s Wanderings,’” 
p- 130:—‘‘In the open paths I netted scarlet Pieridae 
. often flying in flocks of over a score, exactly 
matching in colour the fallen [withered] leaves, which 
it was amusing to observe how often they mistook for 
one of their own fellows at rest, and to watch the 
futile attentions of an amorous male towards such a 
leaf moving in the wind.” 
Henry O. Fores. 
Redcliffe, Beaconsfield, Bucks, 
April 17. 

The ‘‘ Green Ray’”’ at Sunset. 
Pror.. A. W: Porrer, in. Nature of February 
18- (vol. xciv., p. 672), seems to think “that 
the “‘green ray” is more of a subjective pheno- 
menon than anything else, or at least often 
is so; but the fact that it is seen at sunrise 
also shows that in this case at least it is not a result 
of complementary colours. Besides, if it were a 
subjective phenomenon, one would expect to see it on 
every occasion when the sun set behind a clear 
I once 
saw a lovely blue flash, and I read a description 
recently of a sunset in’ Palestine where the writer 
speaks of the sun vanishing like a blue spark. If 
you -hold a lens almost edgeways on between your 
eye and a light and move it until it is quite edgeways 
on a few discs of light will be seen, and at last 
these vanish in a green or blue flash, the effect of 
dispersion. ; 
35 Roeland Street, Cape Town, 
d March 17. 
