644 

NATURE 
[AUGUST 12, 1915 


LETTERS TO THE EDITOR. 
[The Editor does not hold himself responsible for 
opinions expressed by his correspondents. Neither 
can he undertake to return, or to correspond with 
the writers of, rejected manuscripts intended for 
this or any other part of Nature. No notice is 
taken of cnonymous communications. ] 
The Principle of Similitude. 
Dr. RiapoucHInsky directs attention concisely in 
Nature (July 29, p. 591) to an important point, which 
must have arrested the notice of readers of Lord Ray- 
leigh’s weighty exposition and illustration of the scope 
of the method of dimensions, as an instrument of 
precision in the analysis of physical problems. The 
example under consideration was the cooling of a hot 
wire by a stream of air passing across it. The point 
is that temperature, although in ultimate analysis it 
must be expressible in terms of the three fundamental 
dynamical entities—mass, space, and time—can yet 
be in that problem considered effectively as a fourth 
independent entity, thus vastly increasing the informa- 
tion derivable from comparison of dimensions. In the 
formal analysis of mere diffusion or conduction this 
is clearly yalid, for the dynamical aspect of tempera- 
ture is not involved. - 
In so far as thermodynamic considerations, such as 
work of expansion, etc., are of secondary importance 
in the analysis of the convection from a hot wire 
(as is the case so long as only an adhering surface 
layer of the gas need be taken as operative '), the same 
principle will apply approximately there. But in a 
problem the content of which is mainly thermo- 
dynamic, the relations will be far more complex. In 
fact there is nothing transcendental about dimensions; 
the ultimate principle is precisely expressible (in New- 
ton’s terminology) as one of similitude, exact or ap- 
proximate, to be tested by the rule that mere change 
in the magnitudes of the ordered scheme of units of 
measurement that is employed must not affect sensibly 
the forms of the equations that are the adequate 
expression of the underlying relations of the problem. 
Cambridge, July 31. 

Tue question raised by Dr. Riabouchinsky (Nature, 
July 29, p. 591) belongs rather to the logic than to the 
use of the principle of similitude, with which I was 
mainly concerned (NaturE, March 18, p-. 66). It 
would be well worthy of further discussion. The con- 
clusion that I gave follows on the basis of the usual 
Fourier equations for conduction of heat, in which 
heat and temperature are regarded as sui generis. It 
would indeed be a paradox if the further knowledge of 
the nature of heat afforded by molecular theory put 
us in a worse position than before in dealing with a 
particular problem. The solution would seem to be 
that the Fourier equations embody something as to 
the nature of heat and temperature which is ignored 
in the alternative argument of Dr. Riabouchinsky. 
August 2. RAYLEIGH. 

The Probable Error of the Amplitudes in a Fourier Series 
obtained from a Given Set of Observations. 
To express a periodic variation by one or more 
terms of a Fourier series is such a common and 
convenient method that it is of importance to know 
the extent to which the constants are trustworthy. 
This is particularly the case when the quantity is 
1 Cf also L. V. King, on hot-wire anemomerry, PA7d. Trans , A214(1914), 
P- 373+ 
NO. 2389, VOL. 95] 


subject to a large casual and therefore non-periodic 
variation as well. 
Suppose there are n evenly spaced observations, 
Yor Vay Yo»+++¥n—-y then confining attention to one 
term only, y may be expressed by the relation 
y=asin(x—a). This may be written 
1 - j 7 
y=p sinx+9 sin( +7) 
and the amplitude a is given by a= /p?+q?. 
The standard error of a therefore depends on that 
of p and q. To obtain p we write— 
P= Sin o+)) sin Se sin aE: ete aea SI "ton 
whence, if o is the standard error of the y’s, it is easy 
to prove that o/2//n is the standard error of the p 
or q. This is also the standard error for any higher 
order term depending on the n observations. 
From the way in which p and q are cbtained it is 
obvious they may be positive or negative; the sign 
depends, in fact, upon the origin chosen, and their 
mean value obtained from many samples would be 
zero. But even in the case of there being no period 
at all, and the y’s being purely casual, it is very 
improbable that both p and q should be zero. In this 
special case the amplitude a is really zero, but it 
is almost certain, since it equals “7+ 7, not to be 
given as zero. Its mean square is plainly equal to 
twice the mean square of p or q, but this is not the 
square of the standard error, because it is not taken 
about the mean value. 
To ascertain the general magnitude of the error the 
following plan has been adopted. [T'rom a sort of 
roulette board of roo compartments, 500 pairs of 
numbers have been taken, the numbers on the board 
being arranged to have a standard deviation of 10, and 
a distribution as nearly normal as the limit of 100 
numbers permits. From these pairs, representing the 
p and q of the sine curve, 500 amplitudes have been 
found. The result gives a mean amplitude of 12-5, a 
standard deviation of 6:5, and a distribution such that 
4 per cent. exceed the value 25. 
Similar results from 500 other pairs, but with a 
genuine sine curve of amplitude 10 superadded, give a 
mean amplitude of 16:5, a standard deviation of 8-0, 
and a distribution such that 8 per cent. are below 5, 
and 5 per cent. above 30. 
From theoretical considerations it is apparent that 
as the periodic part of the variation becomes large 
compared with the casual part, the mean amplitude 
will still exceed, but will approximate to, the genuine 
amplitude, and the standard error will approximate 
towards that of p or q. 
Suppose, then, that we have a set of n well-distri- 
buted observations, and that nothing is known about 
them in regard to their periodic variation. Let their 
standard deviation be o. Now let the first three terms 
of the Fourier series be found in the usual way, and 
let the amplitudes be a,, a,, and a,, the question that 
arises is, to what extent do a,, a, and a, represent 
genuine periodic variations? The standard deviation 
o may be due entirely to the periodic variations, or 
may be purely casual. In the former case it is easily 
proved that c= /a,?+a,?+a,?//2, and if this relation- 
ship is approximately satisfied, the larger ampli- 
tudes may be accepted as correct. But if the 
deviation o of the observations is distinctly larger 
than that produced by the periodic part, the 
casual part of it will lead to error in the a, 
a,, and a,, and unless’ an amplitude when com- 
pared with oV2//n is well outside the limits for the 
special values given above, it need not be significant. 


