1] 
three Acts, an Interact and an Exodion 
25 
where 
and in general 
S x = x 4- # 2 4- x 3 + x* + a? + — 
$ 2 = x? 4 a? + x 7 + x 9 4- x 11 + ... = 
5 3 — x 6 + of 4- x 12 + a? 5 4- x w 4-... = 
5 4 = x 10 + x u + a? 18 4- x 22 + x 26 4-. •. = 
S r> = x 15 4- x 20 4 x 25 + x*° 4- x? 5 4-.. • = 
_ £.1+2+3+...+r _j_ £,2+3+4+...+(r+1) ___ _ 
X 
1 — X 
X 3 
1 — X 2 
X 6 
1 — a? 
X 10 
1 — x* 
X 15 
I — x 6 ’ 
x hr(r +1) 
1 — x r 
So much of Prof. Sylvester’s theorem as relates to a single sequence 
follows from inspection of the above scheme. For Si = X ; adding to S 3 
JL — X 
X^ 
the first term of S 2 , we get -—— ; adding to S 5 the first term of 84 and the 
X^ 
second term of S 2 , we get -—— ; adding to $ 2OT+1 the first term of S 2m , the 
second term of $ 2(m _d, the third term of $ 2(m _ 2) ,..., and the mth term of $ 1( 
£.2m+i 
we get - 2m+i I the proposition is proved. The fact is made more 
evident to the eye if we write the scheme as follows: 
Sx = x + x 2 4- x? 4- x 4 4- x? 4-... S 2 = a? 4- x 5 4- x 7 4- x 9 4- x 11 + ... 
S 3 = x 6 4- x° 4- x 12 4- x™ + # 18 + ... S 4 = 
S 5 = x 15 4- x 20 4- x 25 4- x 30 4- X s5 4 ... S 6 = 
S 7 = x 2S 4- x 35 4- x i2 4- ¿e 49 4- x 56 + ... S 8 = 
S 9 = of 5 4- ic 54 4- x 63 4- x 7 ' 2 + x si 4-... £10 = 
at 3 
x 10 4- ¿c 14 4- x K + x 22 4- ... 
x 21 4- x 27 4- x 33 4- •. • 
se s * + x M + ... 
# 55 4- ... 
Here 
, for instance, is obtained by adding the fourth column on the 
1-x 9 
right to the fifth row on the left. 
It may be noted that we have thus found that 
x oc 3 a? x‘ 
a? 
1 — x 1— x 3 ^ 1 — x 5 
r 2m+i 
+ ...+ 
X £.‘im+l 
X X? 
+... 
4- 
1 — X 1 — X‘ 
+ ■ 
x \n (n + 1) 
+ ...4--i