Mr. J. W. L. Glaisher on Arithmetical Irrationality >. 193 



refer to the way in which e is usually shown to be irrational, 



n 

 viz. that if e — — (m and n integers), then 

 m v o / 



m_ 1 _J^_ 1 _ 1 



»"1,« 1.2.3'- 1.2...n + 1.2...(a + l) i ~'"' 



so that 



m(l . 2...?i — 1)= integer +(— pr- 7 — -777 — ToT + ••■•)' 



v ' ° \rc + l (ro + l)(rc + 2) / 



or integer = integer + fraction ; 



so that m and n cannot be finite integers. 



This method can only be applied when the denominators in- 

 volve all numbers (or multiples of them), and when each deno- 

 minator includes all the preceding ones, the numerators being- 

 constant. 



There cannot be much doubt that in an ordinary natural 

 canon sin 30° is the only rational sine, though I believe this has 



not been proved; (sin- and cos- are easily seen to be irrational; 

 oc ss 



for co an integer, the unit being the arc equal to radius, by the 

 sort of reasoning applied above to e -1 ;) and many other con- 

 stants are in the same state, as, e. g., log7r, e 71 ", &c, or Euler's 

 constant "577215 .... '. 



Lambert's principle will be found to be not often applicable, 

 as the conditions requisite are but seldom fulfilled; it is far 



8 8 

 more common for ' 2 * ' ' to be infinite than zero. 

 *fit . . . 



The method of proving the irrationality of certain quantities 

 which I now proceed to explain, although very simple, does not 

 seem to have been noticed ; at all events Eisenstein, who was 

 occupied with the irrationality of some of the quantities to which 

 it is directly applicable, certainly did not perceive it. An ex- 

 ample will make the principle of the method clear. Consider 

 the series 1 + q + q A -\- q 9 + q 16 ; + ... which occurs in the theory of 

 Elliptic Functions; it follows at once that the value of this 

 series is irrational whenever q is the reciprocal of an integer 



greater than unity; for if q= —, then in the scale of radix r the 



value of the series would be written 



1-1001000010000001000000001 . . . , 



which, as the intervals between the l's consist of 2, 4, 6, 8, 10 . . . 

 ciphers, does not circulate. In the scale of radix r, therefore, 



M 



the series cannot be expressed in the form ^, M and N being 



Phil. Mag. S. 4. Vol. 45. No. 299. March 1873. O 



