1893.] On Operators in Physical Mathematics. 117 



37. Now passing to the case of a bigger w, viz., ^, we may remark 

 that this differs from the known good case n = -J by an integral 

 differentiation, so we may expect good results again. We have 



x / x I x f 5x / 7# / " 



W=1+ 2 ( l -S 1- 6 ( l -V ( l -n 1- - (56) 



...... (57) 



Taking x = 1 first, giving 



=i + i(l-iKl-i(l-A(i- ..... - (58) 



' (59) 



2"5625 

 we find w = 1*4464, v = ^z = 1:4462,, 



by not counting the last convergent, that is r th smallest term in the 

 v series. Its inclusion makes v appreciably- too big, viz. 1'46. 

 Next take x = 2. Then 



3'2124 



giving tt = 1-81275, v = = T812, 



/ 1& 



again not counting the smallest term. 

 Lastly, with x = 3, we have 



giving u = 2-12.60, v 2'1256, 



again neglecting the smallest term in v, though it is of little moment 

 in this example. The tendency for v to be too big when the smallest, 

 term is fully counted should be noted. 



38. A further increase of n to % gives good results, and likewise 

 -iV Thus, for T 9 o we have 



u = i + ^ x(l-^x(l-^x(l-^x(l-^x(l-. . . . , (64) 



