Mr. A. Cayley on a Problem in the Partition of Numbers. 345 



nance of soda, sulphuric acid, and chlorine, which distinguishes 

 it from the Ottawa. It is an interesting geographical feature of 

 these two rivers, that they each pass through a series of great 

 lakes in which the waters are enabled to deposit their mechanical 

 impurities, and thus are rendered remarkably clear and trans- 

 parent. 



The presence of large amounts of silica in river-waters is a 

 fact but recently established. Until the late analyses by H. 

 St.-Claire Deville of the rivers of France*, the silica in water 

 had generally been overlooked wholly or in great part ; and, as 

 he suggests, had, from the mode of analysis, been confounded 

 with gypsum. (The purity of the silica in all my determinations 

 was established by the blowpipe.) The importance in an agri- 

 cultural point of view of this large amount of dissolved silica, 

 where river- waters are employed for the irrigation of the land, is 

 very great : and geologicallj', the fact is not less significant, as 

 it marks a decomposition of the siliceous rocks by the action of 

 waters holding in solution carbonic acid, and the organic acids 

 arising from the decay of vegetable matters, which, dissolving the 

 alkalies, the lime and magnesia, from the native siUcates, liberate 

 the silicic acid in a soluble form. Silica is never wanting in 

 natural waters, whether neutral or alkaline, although proportion- 

 ally less abundant in neutral waters which contain large amounts 

 of earthy ingredients. The alumina, whose presence is not less 

 constant, although in much smaller quantity, appears equally to 

 belong to the soluble constituents of the waters. The amount 

 of dissolved silica annually carried to the sea by the rivers must 

 be very great ; yet sea-water, according to Forchammer, does 

 not contain any considerable quantity in solution ; it doubtless 

 goes to form the shields of Infusoria, and may play an important 

 part in the consolidation of the ocean sediments and the silitica- 

 tion of organic remains. 



Montreal, March 1, 1857. 



XXXVII. On a Problem in the Partition of Numbers. 

 By A. Cayley, Esq.-\ 



IT is required to find the number of partitions into a given 

 number of parts, such that the first part is unity, and that 

 no i)art is greater than twice the preceding part. 



Commencing to form the partitions in question, these are 



&c.: 



* Annales de Chimle el de Physique, 1848, vol. xxiii. p. 32. 

 t Coinmunicatcd by the Author. 



