Mr. J. W. L. Glaisher on Arithmetical Irrationality. 19 L 



tion between water and gas) a sufficient number of examples 

 may be found. In an experiment with a strong solution of am- 

 moniacal gas, in which a silver wire and a strip of platinum were 

 immersed, no bubbles appeared on either surface, although in- 

 numerable bubbles were set free by the action of heat. Strong 

 hydrochloric acid behaved in the same way with platinum ; only 

 on heating it more strongly fewer bubbles were produced, showing 

 the much greater attraction between water and hydrochloric acid 

 gas. By long exposure to the air both solutions became much 

 weaker ; and on gently warming them, the immersed wires became 

 covered with bubbles, arising doubtless from the liquid having 

 absorbed atmospheric air. 



In conclusion, some experiments on fatty bodies maybe referred 

 to. Large drops of olive-oil, oil of almonds, and linseed-oil on the 

 surface of water well impregnated with carbonic acid produced no 

 separation of bubbles. The attraction of these oils for water is 

 therefore not sufficiently energetic for the purpose; and it remains 

 to be seen whether they absorb the gas to a slight extent. Stearic 

 acid melted at the bottom of a small glass cylinder, and after 

 becoming solid covered with soda-water, gave off an extraor- 

 dinary display of bubbles. After some time the stearine loosened 

 its hold and rose to the surface, separating from the fluid fat, 

 a chemical action having taken place. With water previously 

 boiled no bubbles were produced. 



Freiburg, im Breisgau, January 1871. 



XXIV. On Arithmetical Irrationality. 

 By J.W. L. Glaisher, Fellow of Trinity College, Cambridge*. 



IT is rather curious that, although very many of the numerical 

 quantities with which the mathematician is constantly con- 

 cerned are generally believed to be irrational (i. e. not to termi- 

 nate or circulate*}- when expressed as decimals), yet the fact of 

 such irrationality has been demonstrated in only a few cases ; 

 and it is still more remarkable that so little attention seems to 

 have been given to the matter at all. To take an example, I 

 suppose no one has any doubt (using the words in the sense 

 that there is no one who would not be very much surprised if 

 the contrary were proved) that all the sines in an ordinary Brig- 

 gian or hyperbolic logarithmic canon, in which the arguments 



* Communicated by the Author. 



f In the rest of this paper the word circulate will be supposed to include 

 the case where the decimal terminates, as is indeed the fact ; for a termina- 

 ting decimal is merely one in which the circulating period consists entirely 

 of zeros. Thus every numerical quantity either circulates and is rational, 

 or does not circulate and is irrational. 



