PARTICULARLY THOSE OF TWO VARIABLES. 
857 
M> 
• • • > M-W* 1 ’ 
a 3 , . . 
• • j A s 4-1, . . . , A;, 
... ,v r J \v v 
»• 
., V s , ... ,V t , 
-rraA 1 ’ 512 . 
A s, • ■ 
. . > Ajf+1, . . . > A,.\ 
\v v v 2 , ... , 
V s , ■ • 
. , v t , . . . . V r J 
+ 8^ ** ' • 
* j Aj-f-’lj . . , , \f, . . 
1 = 1 \v-t_, V 2 , . . 
• j Vi 
, ... , V s , Vt, . . 
+nof Xl ’ X3 ’' ‘ ’ 
A,+l, A*+l, . . ., A,.\ 
W V v • • • ’ 
v s 
, Vt , . . . , V r j 
-sW x ^’-- 
1=1 \v v v v . . 
• , Vl 
, . . . , v s , . . . , : 
— Sim/* 1 ’ Xs ’ ' • 
.» Az+1, . . . , A^ + 1, . . 
Z = 1 W Vet, * • 
• j vi 
Vs , . . 
. . , A; 
. . , v. 
, A r 
, V 
Vt , 
• » A;, 
■ , ^ 
> A,. 
. v r 
, A 
. v v 
, Aj. 
, V r 
+ s^sno^ 1 ’ Xs ’ * ’ x?+1, • • • ’ Xwi+1, • * * ’ ■ ■ ■»^ 
| 5 +1? A 3 +1, ... ,\i, ..., A Wi , ..., A+1,...,A+1,...,A+1\ 
Z =1 m= 1 W > ?/ 2 , . . . , Vl, ... , V m , . . • , v s , . , . , Vt , . . . , Vf. J 
-sW Xi+1,X3+1 ’ 
?=)• \^i ) V 3 
W 
Z=1 
/Aj +1, A 3 + l, . 
\*i » ^3 * ■ 
+n<i> 
+ i sW Xi+1 ’ Xs+1 
Z=1 
—m> 
—n<t> 
1 > y 3 
A 1 + l, Acj +1? • 
V-, , Vc, 
+m> 
• • » A; ? 
• • , v h . 
• • 5 A u 
. •, v b 
1 As,. . 
, Vs, • • 
• • ) Aft 
• • * Vh 
, A» . • 
, V s , . . 
Vr 
. , A ot A; + l, . . . , Af + 1 
■ , v„ Vt , . . . , v r 
• ) Aj-) - 1, . 1 .) . . . 1 A,. +1\ 
. , v s , ... ,v t , 
A;, . . . , A,.+ l 
Vt, ... . v r 
■ , A # 4 -1, . . . , A^ +1, 
• t ^ , • • * , Vt , 
, A,.+ l\ 
(237) 
, A,.+ l 
, V r 
Az+1, 
Vt , 
/ 
, A^ + 1, 
? v, » 
, A, + l, 
, V.o 
. , A;, 
• > v h 
, X^ + l\ 
» v,. / 
1 ’ "3 
^A x +1, A 3 +1. . 
, v 3 , . 
'Aj + l, A 3 +l, . 
h -Vet , . 
= (—1 )k^+^+x , '»+a" / i+a.+x , .+x".+x w . ) ^ Same expression with A written for A and N 
for v throughout, except for v s , v t for 
which write respectively N g +1, N f +1; 
the variables being as in (235)] 
. , A^+l, . .. , A,.+l\ 
. , Vt , .... v r 
} 
in which s, t are to have in succession the values 1, 2, 3, . . . , r, (but never the same 
l=r 
value together); and S implies summation for every value of l in 1, 2, 3, . . . , r, 
except l—s , and l=t. Thus the equation comprises \r{r— 1) cases. 
5 R 2 
