472 
OX THE BINOMIAL THEOREM, 
[533 
where the last letter is repeated; there exists in this case no partition of 8, such 
that operating on the last we obtain from it 144, but there does exist a partition 
of 8, viz. 134, such that operating on the last but one we obtain from it 144. That 
is, for any partition whatever of 9 there exists one (and only one) partition of 8, 
such that operating on the last or the last but one, we obtain from it the partition 
of 9; that is, taking the entire system of the partitions of 8, and operating on the 
last and the last but one, we obtain, and that without repetitions, the entire series 
of the partitions of 9; and so in general. 
Translating the example into letters, but using for greater convenience a 2 , &c. 
instead of a, a, &c. the process 
a, b, c, d, e, we have 
i will 
be 
precisely 
the 
same ; 
taking the 
letters to be 
a 3 orb a 2 c 
a 2 d 
a 2 e 
abe 
ace ade 
ae 2 
be 2 
ce 2 de 2 e 3 
ab 2 
abc 
abd 
acd 
ad 2 bee 
bde 
ede 
d : e 
b s 
ac 2 
b 2 d 
b 2 e bd 2 
c 2 e 
d 3 
b 2 c 
be 2 
bed, c 2 d 
cd 2 
Attributing weights to the 
seven 
il letters, 
viz. to a, 
b, c. 
, d, e 
the weights 
1, 2, 3, 4, 5 
respectively, the several columns show the terms of the weights 3, 4, ...15 respectively. 
I have said that the foregoing rule is given implicitly in Arbogasts Calculus of 
Derivations; this calculus includes in fact a process for the expansion of a function 
(fr (a + bx + cx 2 + dx 3 + &c.) 
in powers of x; the expansion in question may be obtained by means of Taylor’s 
theorem, viz. 
(fr (a + bx + cx- + dec 3 4- &c.) 
= <fra 
+ ip (bx + cx 2 + dx 3 4- &c.) 
<b"(l 
+ (bx + cx 2 4- dx 3 4- &c.) 2 
4- ^ ^2 ^3 CX ~ + ^ c -) 3 > 
viz. expanding the several powers of the polynomial increment, and arranging in powers 
of x, this is 
= cfra 
+ x {(fr'a. b) 
4- x 2 (^fr'a . c 4- <fr' a . 
4- x? ($>'a . d 4- (fr"a . be + (fr'"a. pj 
4- x 4 1 (fr'a . e 4- <f>"a. (bd 4- %c 2 ) 4- <f>"'a. ~ 4- (fr""a .