370 MESSES. C. AND E. CHAMBEES ON THE MATHEMATICAL EXPEESSION 
the first set of expressions of (6) may be put into the following typical form, 
2 [_p s cos smz-\-q s sin smz] cos tmz 
. ( 10 ) 
which, in the case before us, is 
2 cos smz-\-q s sin smz ] cos tmz 
( 11 ) 
and since Uz=2ct, and neglecting, with Bessel, the terms of (5 m and y m for which s is 
not small, and also dividing through by this becomes 
But after each successive period the quantity 
passes again, in each of its terms, through the same identical values ; it is therefore a 
proper periodical function, and passes at the same phase of each period fx' through 
some maximum value, which cannot ever be of magnitude so great as the sum of all 
the P’s and Q’s disregarding their signs ; much less can 
ever reach that sum ; hence the last term of (12) can never he so great as twice the 
sum of all the P’s and Q’s regardless of signs. Suppose this to be its value at some 
time during the first period fx', then at no time in the second period fx' can it exceed 
the half of this, since B will have been at least doubled, whilst the part under the sign 
of summation cannot have increased; similarly, at no time during the nth period fx' can 
2 
its value be of greater magnitude than -ths of the sum of its P’s and Q’s regardless of 
signs. Hence if n be made large enough, i. e. if the observations be sufficiently extended, 
this quantity can always be reduced till its effect upon the value of jp t is inappreciable. 
11. Now it has been shown in the preceding investigation that, as B increases and 
fx! <2fxJ 3 fx! dfx' 
passes successively through the values &c., the quantity 
