362 MESSES. C. AND E. CHAMBEES ON THE MATHEMATICAL EXPEESSION 
3. There is a yet simpler case, the importance of which possibly did not press itself on 
Bessel’s attention, but which the present writers (having occasion to describe in con 
nexion with actual observations) find it convenient thus to introduce. It is that of 
two or more combined phenomena, each of which separately recurs after a certain 
period, which is of different duration for each phenomenon ; and the object of this 
inquiry will be to determine under what circumstances, and with what degree of 
accuracy, may the coefficients of the expression, according to Bessel’s form, of each 
separate phenomenon be found from a series of observed values of the combined 
phenomena. 
4. As the result will be equally applicable to any number of combined phenomena, 
we will consider the case of only two, whose periods are respectively z and z'. 
x f . 
Let ~i=gif and g being the least integral numbers that will satisfy this condition*; 
and x being the interval in time (supposed to be constant) between every two conse- 
cutive observations, let the series of observations extend over the time gz or _/V, of 
which x is a measure, then will the number (r) of observations be ^ , and the angles 
2^ o^r 
corresponding to the time x under the respective periods — x=z (say) and y x—-z; 
further, the angles corresponding to the time rx or gz will be 2^or and 2 fir respectively. 
If ot m represent the observed value of the combined phenomena at the time mx , and 
0 m and y m be the separate phenomena of which it is composed, 0 m recurring after the 
period z, and y m after the period z', we shall have 
0 m =i> 0 +p 1 cos mz-\-q l svn.mz-\-]) 2 cos 2 mz-\-q 2 sin 2 mz-\- &c., (1) 
y m =.Y 0 -\-¥ x cos^-mz + Qj sin^ , m 2 +P 2 co s 2 |^;s+Q 2 sin 2 ^m.s+ &c., . . (2) 
• (3) 
The observations will furnish r equations of the following form : — 
{ +Po+ih cosmz-\-q l sin mz -\-]) 2 cos ‘^mz J r g 2 sin 2to2+&c. 
f f f f 
+ P 0 -f cos - mz + Qj sin - mz + P 2 cos 2 - mz + Q 2 sin 2 - mz + &c. 
9 9 9 9 
and the most probable values of^ 0 ,^„ &c., P 0 , P„ Q n &c. will be those which give 
a minimum value to 
V f +p 0 +Pi c °s mz-\-q x sin mz-{-][) 2 cos 2 mz-\-q 2 sin 2mz-\- &c. 
% —«„,=< f f f f 
m=0 l^ + Po+P, cos-m^+Q, sin - mz +P 2 cos 2 mz + Q 2 sin 2 J -mz+ &c., 
the sum being taken between the limits m=. 0 and m=r—l for integral values of m. 
This will have such a value when the differential coefficients of it with respect to 
* It will be shown later that it will suffice that / and g very nearly satisfy this condition ; hut it is conve- 
nient in what immediately follows to regard them as doing so rigidly. 
