OF OBSERVATIONS OE COMPLEX PERIODICAL PHENOMENA. 
371 
vanishes at each passage ; when, therefore, the series of observations is not sufficiently 
extensive to obliterate the effect of the last term of (12), it may be worth while, in the 
first place, to calculate approximately the values of P„ Q,, P 2 , Q 2 , &c., choosing for the 
purpose a number of observations R/ which very nearly completes an integral number 
of periods fx!, and thence the value of 
% £P S cos s ~mz-\-Q s sin s ^ mz^ cos tmz 
for the fractional part of a period fyJ which is in excess of the last completed 
period. 
Similar reasoning, with a similar result, may be applied to each of the expressions 
of (6), of which (10) is a type. 
III. 
12. The variations in a series of n observations (equidistant in time) are by hypothesis 
due to a periodical phenomenon whose true expression is 
a ™=Po+Pi cosw.s+2'i sinm;z+^ 2 cos2TO2+2' 2 sin2m2+&c., . . . (13) 
2c7t • 
in relation to which 2 =—, c=a constant integer not small, «=the period of the 
phenomenon, #=;s^=the interval of time corresponding to the angle z, £=the time 
reckoned from the commencement of the observations, and m=^. Let the interval 
between successive observations be (x-\-Ax), so that the n observations will extend over 
a period n[x-\- Ax) =c(k-{- An). The angle {z-\-i) which corresponds to the interval of 
time (x-\-Ax) will be equal to z^^~=z-\-z~- or i=z^-=z let this be so small 
that mi is also a small angle, s being the suffix of a or q. Under these conditions, to 
find the coefficients ^> 1? See. Let it first be observed that the condition that sni is a 
small angle, implies that n has been so chosen that (z-\- Ax) approximates as closely as 
possible to the known or assumed value of %. The phenomenon a m occurring at the 
time mx, let that which occurs at the time m(x-\- Ax) be called a m , ; then we shall have 
a m’ =jPo +i> i cos m(z + i) -j- q 1 sin m(z + 7) -\-p 2 cos 2 m(z + i) + q 2 sin 2m(z + i) + &c . , (14) 
the general term of which is cos sm(z -f i) + q s sin sm(z + 7) } , where s represents the 
positive integral suffix of a qp or q; and we may, for shortness, write 
«m'=[i>i c ossrn( 2 !+^)+g' s sin sm(s-H’)]> (15) 
the square brackets indicating that the general term within them is to represent the 
sum of its series of values when for s is put 0, 1, 2, 3, &c successively, whence 
a m'=[j? s ( cossm 2 cossmi— sinsmz sinm^+^sinsms: cossrni+cossmz sinsjm)] ; (16) 
