533] 
FACTORIALS, AND DERIVATIONS. 
471 
this is evolved from the top term (a, a) by a process given implicitly in Arbogast’s 
Calculus of Derivations, and which may be termed the rule of the last and the last 
but one. Let the direction “operate on any letter,” be understood to mean that the 
letter in question is to be changed into that which immediately follows it, but in 
such wise that when the letter occurs more than once, e.g. as in a, a the operation 
affects only the letter in the right-hand place. Then operate on the a, a in regard 
to a, we obtain a, ¡3; operate on this in regard to /3, we obtain a, 7; and in regard 
to a, we obtain /3, /3. Again we operate on a, 7 in regard to 7, and obtain a, 8; we 
do not operate on it in regard to a for the reason that a is not the letter immediately 
preceding 7. Operate on ¡3, 0 in regard to /3, we obtain /3, 7. The next step, if the 
series extended to e would be to operate on a, 8 in regard to 8, giving a, e; do not operate 
on it in regard to a, for the reason that a is not the letter immediately preceding 8. 
But in the example, since the series does not extend to e, there is no operation on 
a, 8. Passing then to the next term /3, 7, we operate in regard to 7, obtaining /3, 8, 
and since ¡3 is the letter immediately preceding 7, we also operate in regard to ¡3, 
obtaining 7, 7. Similarly if e were admissible, /3, 8 would give /3, e, but it in fact 
gives nothing; 7, 7 gives 7, 8; thus if e were admissible would give 7, e and 8, 8, 
but it in fact gives only 8, 8, and, e being inadmissible, the process is here concluded. 
The rule is, operate on the last letter, and when the last but one letter is that 
which, in alphabetical order, immediately precedes the last letter (but in this case 
only) operate on the last but one letter. 
Taking another example, but with numbers instead of letters, and supposing the 
highest admissible number to be 5, then from 111 we derive as follows: 
113 
114 
115 
125 
135 
145 
155 
255 
122 
123 
124 
134 
144 
235 
245 
345 
222 
133 
224 
225 
244 
335 
444 
223 
233 
234 
334 
344 
333, 
too OOO 
the original single column being here for greater convenience broken up into distinct 
columns ; but the order of the terms, when the columns are taken one after the other 
in order, each being read from the top to the bottom, being the same as before; it 
will be noticed that the successive divisions are the partitions into 3 parts (no part 
exceeding 5) of the numbers 3, 4, ..., 15 respectively; the partitions being in each 
case obtained without repetition, and those of the same number being given, say in 
their numerical order (corresponding with the alphabetical order when letters are 
employed). It is necessary to show that the partitions will be obtained without 
repetitions; and that all the partitions will be obtained; for this purpose consider, for 
example, the partitions of 9 ; any one of these is either a partition 135 where the last 
number 5 is not a repeated number; and in this case there is a partition of 8, viz. 
134, from which operating on the last we obtain 135 ; but there is no other partition 
of 8 which would give 135, the only such partition would be 125, but here, as 2 is 
not the number which immediately precedes 5, there is no operation on the last but 
one, and we do not from it obtain 135. Or else a partition of 9 is of the form 144