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



finite integers. And as M and N are not integers in the scale 

 of radix r, neither are they integers in the scale of radix 10. as 

 a number expressible as an integer in one scale must clearly also 

 be so expressible in any other scale, both the radixes being 

 integers. 



It is evident that the same kind of reasoning applies to all 

 series of the form 1 4- qQb) 4. g<P(2) 4. q$ (») -[-... } where <j>(n) is a 

 rational non-linear function of n such that, when n is an integer, 

 <j>{n) is so too; so that all such series are irrational when g is 

 the reciprocal of an integer greater than unity. We see also 

 that the method is susceptible of being still further generalized, 

 and gives a result in a great number of cases where the coeffi- 

 cients are not unity nor all positive ; thus 



l-g + q 4 -q 9 + ...= -900099999000000099 . . . 

 (q being, as throughout the rest of this paper, -, and the sym- 

 bol 9 denoting the digit r — 1) cannot circulate; and the same 

 is the case with i +- q + 2q 4 + 3q 9 + . . ., 1 +- q + 4q 4 +- 9q 9 + ..., 

 &c, and generally with 1 + ^{\)q + ty\2)q 4 '+^{3)q 9 + . . . , 

 where ^r(n) is such that the number of figures constituting it in 

 the scale of radix r always bears a ratio less than unity to 2n—l 

 (which is the difference between the number of ciphers preceding 

 the first significant figures in q( n ~ 1 ) 2 and q n2 ), — as- then the 

 numbers of figures in the groups of ? s or 9's continually 

 increase, so that the decimal cannot circulate. 



The number of figures that any number u will occupy 

 when expressed in the scale of radix r is equal to the integer 

 next above log r u; so that if we apply the above reasoning 

 to the series 1 + xq + x q q 4 ± x s q 9 + . . . , we see that so long as x 

 is an integer such that log ? .a: ?i+1 < 2n (that is, if x be an integer 

 less than r 2 ), its value is irrational. 



That the number of zeros in a group is in this case ultimately 

 infinite is apparent, as the number in question when x—r'*— 1 

 is approximately 2n — log r (r*— 1)* +1 , which 



and is infinite with n\ that the series is irrational when w=r* 

 is easily seen otherwise. 



The most general case to which the method is applicable is 

 that of the series 1+^(1)^(04.^(2)^)4.^(3)^(3)4.,,^ 

 which is irrational if log r ^r(n) +-1 <<j){n) —<j){n—\) } so that 

 when n is infinite, tf>(n) — $(ra— 1)— ty(ri) is infinite too,— $(») 

 being as above, and ^ (n) a rational function of n, which is an 

 integer when n is an integer, 



