44 Lovering and Bond on Magnetic Observations at Cambridge. 
intricate path by which the heat deposited at one moment in the 
centre of our system arrives at its final result of causing a deviation 
in the direction of the magnetic meridian. And while this is the case, 
it will be impossible to enter upon the mathematical analysis of the 
problem and deduce formule which can be used for detecting the 
errors of theory or correcting or supplying the deficiencies of obser- 
vation according to the well-known relation subsisting between these 
different methods of investigation. But the artifices of analysis will 
frequently take hold of cases which cannot be approached by any 
direct process. The observations allow us to proceed upon the 
ground that the declination or the ordinate of the diurnal curve 
of declination is a function of the solar day. It may, then, like 
any other periodic function be supposed to be expressed in a se- 
ries of terms arranged according to the sines and cosines of the 
time and its integral multiples.* Thus if 
t = the time expressed in parts of a day as its unit, 
d — the ordinate of the diurnal curve for the time f, 
a = the ratio of the circumference to the diameter, 
nm = any integer whatever ; 
and if § denote the sum of the terms which correspond to the dif- 
ferent values of n, we have for the general form ; 
D=A+S.C, sin. 2 rn (t+ ,). 
The values of 4, C, and c, are readily determined by the follow- 
ing formule. Let observations be taken at equal intervals for several 
whole days and let 
h — time of observation counted from the beginning of each Magnetic day in 
parts of a day as unity : 
* It was according to this mathematical developement that Professor Peirce 
calculated the empirical curves. 
