368 MESSES. C. AND F. CHAMBERS ON THE MATHEMATICAL EXPRESSION 
expression which represents the given values of y will in general only be interesting 
when it converges so rapidly that only a few of the first terms have appreciable values,” 
had reference doubtless to the incompleteness of the partial expression — this being of 
no consequence when the rejected part, the absence of which makes the expression 
incomplete, is of inconsiderable amount. We may, however, be guided as to the validity 
of this reason by noting well whether the values of a t , b„ A t , B„ &c. do themselves 
become inappreciable whilst t is still small. 
7. Now in many special inquiries f, g , and r will have such values that (s + t)g, 
(sf+tg), (s+t)f, (sg + tf), &c. will first become a multiple of r only when s or t has 
ceased to be small ; in which case, following Bessel, we may neglect as inappreciable 
all the terms on the right-hand side of equation (7), except and P 0 in the first 
equation and the first term of each of the others ; we then have 
J p 0 -t-p 0 =« 0 =^i; 
q^b—% 
r m=0 
■p,—a 2 =~ 2 
^K=- r m £ 
&c. &c. 
9 m=r 
Jp t =a t =-X^ 
St=b t =~% 
p I =A I =!r 
Q I= B 1= ^ 
9 m—r— 
P,=A,=- 2 
[«J» 
[a m cos m 2 ], . 
[<x m sin m 2 ], 
[<z m cos 2 m 2 ], 
[ a m sin 2 mz], 
&c., 
[a m cos tmz], 
[a m sin tmz] ; 
cos ^ mz J , 
a m sin - mz 
9 
[«m sin ^ m 2 ], 
j ~a m cos 2^mz~J ; 
Q a =B 2 =^S_ [a m sin 2 ~ m 2 ] . 
&c. 
&c. 
9 
&c, 
P«=A 
[a* cos m 2 ], 
Qt= B t=~ r \ [a* sin t~ g mz\^\ 
which are the same values as those 
that would be found by applying 
Bessel’s method to the r observa- 
tions, on the supposition that they 
are unaffected by the phenomenon 
whose period is %! . 
which are the same values as those 
that would be found by applying 
Bessel’s method to the r observa- 
tions, on the supposition that they 
are unaffected by the phenomenon 
whose period is js. 
J 
( 9 ) 
