402 SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
mon measure, and the functions to be combined be 
'P^=a^.s^+b^.s^_,+ &c. 
Q^=A^.4+B®-^.r-i+ &c., 
we have by equation (22.), 
.... .... &c., 
and consequently 
/{P^, Q^}=/{«« AJ Bjm^_i+ &c 
For example, 
• 2^ “b & . 2^_ 1 ) ( A . 3^ -|- P • 1 + C . 3 j_ 2) = (2 A . 6^ -)- &B . 6j,_ 1 + aC . 6^_2 + ^ A . 6 j_3-1‘ <^B . 6^_4 4" . ( 
(11.) The readiest way in practice and the surest way to avoid mistakes, which in 
complex cases are very likely to occur, is to proceed by a much easier process, as 
in the following- example. Suppose we would express the product of the three 
periodic functions 
P^=24:+2 .2^_i, Q4.=34,4-2.3 ^_i + 3.3^_2, B4:=4^+2.4j,_i-[-3.44,_2+4.4^_3, 
the product of 2, 3, 4 divided by the greatest common measure of 2 and 4 is 12, 
which will therefore be the period of the product. Write then the several coefficients 
in order as follows : for 
p. 
h 
2; 
1, 
2 • 
1, 
2 • 
1, 
2 • 1 
2 • 
1 , 
2 • 
&c. 
Q. 
1, 
2, 
3; 
1, 
2 
3; 
1, 
2, 3; 
1, 
2 
^9 
3; 
&c. 
K 
1, 
2, 
3, 
4; 
1, 
2, 
3, 
4; 1, 
2 
^9 
3, 
4; 
&c. 
1, 
8, 
9, 
8, 
2, 
12, 
3, 
16, 3, 
4, 
6, 
24, 
and we shall have for the product 
12,+ 8.12._,4-9.12,_2+8.12,_3-{-2.12,_4+ 24.12,_„. 
If the signs of the coefficients, or any of them, differ, they must be of course annexed 
to each, and the proper sign affixed to each product. 
X 
(12.) If we denote by = the integer part of the quotient of any number x divided 
by another s, then, universally. 
X X j X — 1 j ^ — 5+1 
“ 5 * s 1 r • • • ^ s+ 1? 
1 •'^^•-1+2 .^^-2+ 1 .-fx-s+ij 
and therefore the remainder, in the same division, is expressed by 
(23.) 
O.Jf^+1 .54 ,_i4-2.6“,,_2+ s— 1 ..S,;_s+1 
(24.) 
(13.) If, again, we would express the integer part of the quotient in the division of 
