Mr. J. W. L. Glaisher on Partitions. 307 



moves silently on, and at length asserts herself, with perfect 

 indifference to the fame of her votaries, and the final verdict is 

 in favour of truth. Many a fact wanders about the world as a 

 scientific waif, not finding rest in any sufficient theory, until at 

 length it falls into its proper place and becomes associated with 

 a host of other facts, outcasts like itself, or interlopers in theo- 

 retical lodging-houses ; but now, in its right place, it performs 

 valuable and unexpected work. The motions of camphor &c. 

 on water were known during upwards of a century and a half 

 before they found their true resting-place in the surface-tension 

 of liquids ; and it is quite possible that the varied phenomena 

 connected with supersaturation may, at some future time, be 

 embraced by some general law, or be held together by some 

 satisfactory theory, when the labours of those who have been 

 working on the subject from the commencement of this century 

 to the present time will have been forgotten, or referred to by 

 the curious in journals that will then have become old, with a 

 wondering smile that men could have been so blind to the 

 obvious teaching of the facts. 

 Highgate, N. March 9, 1875. 



XXXIII. Note on Partitions. By J. W. L. Glaisher, M.A * 



DENOTING by P(«, b, c. . . q)x the number of ways of 

 forming x by addition of the elements a } b, c . . . q, each 

 element being repeatable any number of times, I propose to 

 consider the value of P(l, 3, 5 . . . ) x y viz. the number of ways 

 of partitioning x into parts all of which shall be uneven. 



To fix the ideas, suppose #=10, and consider any one parti- 

 tion, say 1 + 1 + 3 + 5 ; write this in the form 



i, i, i, i 



2, 4, 

 the top line consisting of units only (as many as the partition 

 contains parts), and the second line containing only even num- 

 bers ; the other partitions of 10 into four parts, viz. 1 + 1 + 1 + 7, 

 1+3 + 3 + 3 are to be written 



1, 1, 1, 1 1, 1, 1> 1 



6, 2, 2, 2 ; 



so that the number of partitions of 10 into four uneven parts is 

 equal to the number of partitions of 10 — 4 into even parts not 

 exceeding four in number. It is at once evident that this pro- 

 cess is general, and that the number of partitions of 2x into 2r 

 uneven parts is equal to the number of ways of partitioning 

 * Communicated bv the Author. 



