OF OBSERVATIONS OF COMPLEX PERIODICAL PHENOMENA. 
367 
of the r times of observation, they would all generally differ as to its value at any time 
intermediate between any consecutive two of the r observations. In the first of the 
equations (7), if we were to attribute the whole of a 0 to p 0 or P 0 , it would imply that 
the phenomenon a 0 occurred at all times irrespective of any periodicity ; but if we attri- 
bute it all to (say) p it would imply that the phenomenon a 0 occurred only at the times 
of observation, whilst at intermediate times the corresponding phenomenon would be 
represented by a 0 cos 7 - mz, which passes through a complete cycle of change during the 
interval between every two consecutive observations, or whilst m passes from one integral 
value to the next ; and combined with this there may be a phenomenon represented by 
q 1 - sin^ mz of any arbitrary range. Similarly, the distinction between the different 
terms of the other equations is that they go through a full cycle of change in different 
periods ; and graphically each term would be represented by a complete wave whose 
length corresponded to the period of that term. 
6. As the mathematical theory of this process affords no criterion for selection, we 
ought to find reasons apart from it for preferring particular appropriations of a 0 , a t , 
&c. to the several component parts of their equalities ; otherwise it is clear from what 
has been said that no useful result will be attained. It may be remarked that an 
ambiguity, similar to the one under consideration, attaches to Bessel’s treatment of a 
single periodical phenomenon, the values corresponding to our a 0 , « 15 b x , &c. being given 
at the foot of page 26,* Section III. of Bessel’s paper. Bessel remarks that if we 
compare a mathematical theory of any periodical phenomenon, based on physical prin- 
ciples, with the observations, his expression for the values of the phenomenon is more 
convenient for the purpose than the observations themselves — the reason of this being 
that, as the expression given by the mathematical theory is developed in the form in 
which the observations have been expressed, the two expressions may be compared term 
by term, or by equal subordinate periods. This is probably the most important use of 
the method ; and as the most striking features of a variation will generally be those of 
long period, they may be examined apart from the others. The next most important 
use of this method is probably that which has for its object the elimination of casual 
irregularities from the observations ; but this is served only when the subordinate varia- 
tions of short period are rejected ; and after such rejection, it must always be borne in 
mind that the remaining expression is incomplete : this does not, however, interfere with 
the comparison of the subordinate variations retained with other phenomena of nature 
involving variations of the same subordinate periods ; indeed by indicating the periods 
followed by the subordinate variations which are of largest amount, it suggests a means 
of distinguishing other phenomena that on examination may be found to be related to 
the one which is the subject of the observations. The reason assigned by Bessel for 
giving preference to the terms of long period, viz. that “the development of the 
