Mr. A. Cayley on a Problem in the Partition of Numbers. 247 

 A' + B'x + C'x^ + Sic. =(l+2a; + 2a;2+...)(A + Ba?2 + C^'* + &c.) 



1—x ' 



And if we form in a similar manner A", W, C", D", &c. from 

 A', B', C, D'j &c. and so on, we have 



A" + B"a? + C V + &c. = \^ (A' + B'a;2 + C'.^'* + &c.) 



=i3fr3|I(A+B^''+c^«+&c.), 



and so on. Write A = 1, and suppose that the process is repeated 

 an indefinite number of times, we have 



1 +38a; + Ca72 + lia?3 + &c. = ^ +^- 1 +^^- 1+^^- &c- ^ 

 \ — x. 1— «^. v—ar . &c. 



And the coefficients 1, 33, C, B, &c. are precisely those of the 

 infinite series 1, 2, 4, 6, &c. We have more simply 



l + iix + €x'' + mx^+&cc.= .- ^^- 5-1^ 5— — -, 



{l—x)^{l—x^){l—x^){l—x^)kc.' 



which gives rise to the following very simple algorithm for the 

 calculation of the coefficients : — 



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 



0, 0; 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56 



1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72 



0, 0, 0, 0; 1, 2, 4, 6, 10, 14, 20, 26, 35, 44, 56, 68 



1, 2, 4, 6, 10, 14, 20, 26, 35, 44, 56, 68, 84, 100, 120, 140 

 0, 0, 0, 0, 0, 0, 0, 0; 1, 2, 4, 6, 10, 14, 20, 26 



1 I 2 I 4, 6 1 10, 14, 20, 26 1 36, 46, 60, 74, 94, 114, 140, 166| 



&c. 



The last line is marked off into periods of (reckoning from the 

 beginning) 1, 2, 4, 8, &c. ; and by what has preceded, the series 

 which gives the number of 1-partitions, 2-partitious, 3-partitions, 

 &c. is found by summing to the end of each period and doubling 

 the results; wc thus, in fact, obtain (1), 2, 6, 26, 166, 1626, 

 &c. : and the same scries is also given by means of the last terms 

 of the several periods. 



The preceding expression for 1 + J3a; + C.r^+ &c. shows that 

 J3, C, &c. are the number of partitions of 1, 2, 3, 4, 5, 6, &c. 

 respectively into the parts 1, 1', 2, 4, 8, &c. : and we are thus 

 led to — 



Theorem. The number of ^-partitions (first part unity, no part 



