NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 4'21 
which give 
Pi ^i+205 Pi+\0 {^i~ ^i+3o) “1“ (^i + 10 ®! + 2o) 
^i+10 ^i + 30 ^j+20- 
These being calculated, the function 0.60,+9.60^_i4-&c. reduces itself to the fol- 
lowing, which seems the simplest form it admits : 
320(30^ 30^_2-1-30^_3 — 30^_5-l-30^_g — 30^_8-l-30^_9-l-30^_io— 30^_ii-j-30^_i3 
~30^_i4-l-30^_ie— 30^_,7-1-30 ^_i9} 
-l-360{20._.o+20,_„-f20,_42+20._.3+20,_44+ 20._.9} 
-f{320.10,-l-9.10,_i-216.10,_2-31.10,_3-576.10,_4+585.10,_3-l-104.10,_g 
- 351 . 10,_7-576. 10,_8-1-329. 10,_9 (53.) 
(41.) The problem, “In how many ways can a given number be constructed,” is 
reduced by the author of a short but interesting paper in the Cambridge Mathema- 
tical Journal, iv. p. 87*, to the integration of the equations of differences 
'^x,y '^x—y,x+y and 'l^x,y '^x—y,y '^x—\,y—li 
which last equation corresponds to the case where it is required to find in how many 
ways X can be composed of numbers none greater and not all less than y. The ana- 
logy of this problem with that here treated is obvious, the function ^ being in effect 
y 
identical with that which in the above notation would be expressed by Yl{x),y corre- 
sponding to our s. Accordingly, as far as i/ = 4, to which limit only the inquiry is there 
extended, the results are identical (the mode of expression excepted) with those of our 
equations (49.), (50.), (51.)'|'. The method there pursued (by the successive integra- 
tion of equations of differences) would of course continue to afford similar results, but 
without some systematic processes of notation, transformation and reduction, such as 
those delivered in the foregoing pages, would speedily become too complicated to be 
followed out, though the sort of form which would ultimately be assumed by the result 
seems to have been clearly apprehended. Observing that in the cases oiy—2, 3, 4, 
the results express in fact the nearest integers to certain rational fractions, such as 
0^ «y>3 I ^ /yi2 /yi3 I "D O /yt 
Y 2 in the case of y=S, — ^ — {x even) and ^ odd) when y=4, it is sug- 
gested that “probably this simple species of description might be continued.” This, 
on examination of the value above given, when y or s=5, appears to be the case, but 
for higher values it will be neeessary to enlarge the terms of the description, so as 
to take in circulating functions of higher orders, and with more complicated coeffi- 
cients. To make this apparent, suppose .s=6. Then, without going into the whole 
calculation (which however would not be materially more complieated than for s=5, 
and would lead, as in that case, to a final period of 60, only not reducible to the sum 
* It bears no name, but I have reason to believe it to be the production of Professor DeMoegan. 
t Mr. Waebtjeton has also obtained expressions for the number of partitions as far as 4, and his results, 
mulatis mutandis, agree with the above. 
