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NOTICES AND ABSTRACTS: 
OF 
MISCELLANEOUS COMMUNICATIONS TO THE SECTIONS. 
MATHEMATICS AND PHYSICS. 
On the Mean Results of Observations. By the Rey.H.Luovp, D.D., M.R.LA. 
Iv is well known that the mean value of any magnetical or meteorological element, 
for any day, may be had approximately, by taking the arithmetical mean of any num- 
ber of observed values obtained at equal intervals throughout the twenty-four hours ; 
the degree of approximation, of course, increasing with the number. It is important 
to ascertain the law which governs this approximation. 
Any periodical function, u, of the variable v, being represented by the formula 
u=ay) + a, sin (v + 2) + a, sin (2v + a) + &e., 
in which a, is the true mean, or 
1 tell F 
FS Uav 
0 Oe —@ 2 
if %, Uz, Us, &c., U», denote the values of the function u, corresponding to those of 
the variable 
Qa Ag 2(1—1)x 
Vs; arn: O+ A op 
it may be shown that their arithmetical mean is equal to 
A+Gnsin (nv + e,) + do, Sin (2 nv + Han) + &e., 
whatever be the value of v. Hence, as the original series is always convergent, we 
have, when the number 2 is sufficiently great, 
1 
My = (th + ta + tis + &e. + u,)5 
nearly; the limit of error being a,, nearly. Hence, when the period in question is a 
day, we learn that the daily mean value of the observed element will be given by the 
mean of ¢wo equidistant observations, nearly, when a, and the higher coefficients are 
negligible ; by the mean of fhree, when a; and the higher coefficients are negligible ; 
_ and so on. 
The coefficient a3 is small in the case of the temperature ; the curve which repre- 
sents the course of the diurnal changes of temperature being, nearly, the curve of 
sines. In this case, then, thé mean of the temperatures at any two homonymous 
hours is, nearly, the mean temperature of the day. This fact has been long known 
to meteorologists. 
The Coefficient a; is small in all the periodical functions with which we are con- 
cerned in magnetism and meteorology: and therefore the daily mean values of these 
functions will be given, very nearly, by the mean of any three equidistant observed 
values. _ The truth of this was shown by the author in the case of the magnetic 
declination, the atmospheric pressure, and temperature, as observed at the Magne- 
tical Observatory of Dublin. 
In choosing the particular hours for a continuous system of observations, we 
should select those which correspond nearly tothe maxima and minima of the observed 
res so as to obtain also the daily range. This condition is fulfilled, in the 
° B 
