PARTICULARLY THOSE OF TWO VARIABLES. 
855 
(Since it has been assumed that 
A*+A'* -J- A" t + A'" t =even 
for all values of t which occur, no imaginary quantities are introduced). 
Thus, as a term on the right-hand side, there will be 
_^_] ^[SAi+2A m +SA„+2A P ] 
where 
Ai, A 25 Ag-(-l, . .. , A k + 1, A ? +1, A w +1, A„+ lj A p , A 2 ,. 
, . . ., ]ST k , 1ST; +1, N OT 4-1> X„ 4- 1, 1, N 2 ,. 
Nj, X 3? X 3 
^ A 1 —A7+ A! 1 -J- A" 1 - j- A' 
• 5 AA-y ST 
X, 
The coefficient of ^ in the index of —1 is SA^-j-SA^+XA^+SA^, by Eule II., since 
the numbers N^+l, N w ,+ 1 , N#+ 1 , N /; +l are all that differ by unity from those in 
the first term, and the sum of the numbers which correspond to N/, . . . , N*, . . . , 
N w , . . . , N ;j , . . . , in the four functions in that term is 2 A^+ 2 A*+ 2 )A ?i + tA p ; and a 
— sign is prefixed, by Eule I., because there is an odd number of pairs of corresponding 
numbers—!’ 
members of the similarly situated pahs in the first term. So another term will be 
/_jU{2Ai+2A,„+2A, l +2A,}TT < j ) f /A 1; A 3 , A 3 -fl, . . ., A* 4-1, A/+1, 
V ' l\X x ,X 3 ,X 3 ,...,X* ,N l+ l, 
A/* 4-1, A n ,, Af) 4-1, Aq ,... , AAy y y 1 
N.4- 1 ,^+ 1 ,...,^, ,X 2 4-1,.. .,nJ A i ’ ’ a '-j ; 
and the sign and coefficient of any term may be written down from an inspection of 
its characteristic. 
64. As it has been proved that every number in the characteristic is either zero or 
unity, and the assumption has been made that the sum of any four similarly situated 
numbers in the characteristics of the four functions is even, the general product 
theorem comprises (4') 3 , i.e., 2 Gr , particular cases, the variables being still left perfectly 
general. 
65. In a manner similar to that adopted in Section I. the following formulae are 
obtainable :— 
A»4-l, 
X w 4-1, 
A„4-l 5 
X 
” ’ each member of which differs by unity from the 
n 4" I, 
5 R 
MDCCC LXXXII. 
