516 Mr. J. W. L. Glaisher on the [June 15, 



which is greater or less than unity according as / is less or greater than y 

 The ratio of the (m+2)th to the (m+ l)th term 



= H- 2 0-A 



m\4 / 



which is always greater than unity. 



Thus the mth or (m-j-l)th term is the least according as / is less or 



greater than -L 



When m is large the value of the mth term is very nearly 



1 1.2.3.... 2 m 



on ' {2ivxf m 



which, by use of the theorem 



1.2.3 . ? . cv= V(27r4^--^1+ I L\ 



becomes 



Vm " (27rJ) 2m \ + 24m) 



«vi-(m>+m) » 



The number which forms the negative characteristic of the common 

 logarithm of this quantity (the mantissa being made positive), reduced by 

 unity, denotes the number of ciphers which precede the first significant 

 figure in its value ; so that if the mantissa be not made positive, the cha- 

 racteristic indicates the number of decimals directly obtainable from the 

 series. 



The logarithm of (ii) with its sign changed, after some expansions and 

 reductions, becomes 



2^4-|log 10 a- Iog 10 2-JJ- T +tL+ _LA 

 2 \2xk a tt 2AccttJ 



f.i being the modulus 43429448 .... Neglecting the last three terms, 

 which can only very rarely lead to an error of a unit, we obtain as a result 

 that the number of decimal places which the formula (i) will afford directly 

 for a given value of x is equal to the greatest integer contained in 



2/x7r#+-log 10 tfr- log 10 2 (iii) 



The corresponding value of the least term is, of course, easily obtained 

 fiom (iii), or it follows at once from (ii) ; for the least term 



