206 Mr. W. Steadman Aldis. 



From Wolfram's table, taking thirty-six places, 



log 2 = 0-693 147 180 559 945 309 417 232 121 458 176 568. 

 The value of ^(0) is given in a note by Gauss as 



^(0) = -0-577 215 664 901 532 860 606 512 090 082 402 431. 

 The algebraical sum of these is 



0-115 931 515 658 412 448 810 720 031 375 774 137, 



which is, therefore, the value of E to many more places than will be 

 required. 



The quantity - ^ (0) is, of course, Euler's constant, and the above 

 value is also to be derived from a paper by the late Professor J. C. 

 Adams in the ' Proceedings of the Eoyal Society.' 



6. The calculations of I (^), Ii(^) 5 KO( X )' ^i( x ) are ^ 3es ^ carried on in 

 connection with one another. We have 



The first process is to find the values of I (a;) and l^x). 



If a series of quantities, /? , /3 V /3 2 , ...... , f$-2 r , /?2r+i ...... be deter- 



mined by the successive relations 



!, 1^ 



far+I = ^y-fe ftr+2 = ^7p$ 



coupled with the condition /3 = 1, it is easily seen that 



r=0 



Thus the successive terms of I (x) and I^x) are obtained by multiplying 

 by a series of factors of the form \x\r + 1 ; the alternate terms when 

 obtained are written down underneath one another, the odd ones in 

 one column, the even ones in another, and by addition of each column 

 the values of I () and I^rc) are obtained. 



In working out the values of l Q (x) and I^x) given in Table I, all the 



