of EcUnhurgh, Session 1883-84. 
603 
and so on. 
and so on. 
There remains for us the summation of these series of inverse 
powers. M. Callet says — “ Sommant enfin toutes ces series, nous 
aurons, en nous hornant a vingt decimales ” (lastly, summing all 
these series, confining ourselves to twenty places, we shall have), 
&c. , leaving us to understand that the results are obtained by actual 
summation. Let us proceed to verify the first coefficient by this 
method : — we have 
l-^= 1.00000 00000 00000 00000 
3 -^= mil mil 11111 mil 
5-2 4000 00000 00000 00000 
7-2= 2040 81632 65306 12245 
9-2= 1234 56790 12345 67901 
11-2= 826 44628 09917 35537 
101-2= 9 80296 04940 69289 
1001-2= 9980 02996 00499 
1003-2= 9940 26892 40355 
Here the progression is so slow that, after five hundred divisions, 
which bring us to the divisor 1001, we have got only to the sixth 
place. The convergence is becoming rapidly less, and we should 
need to use five thousand million of terms ere our quotient he 
brought below the twentieth decimal place. Worse than that : — 
the succeeding terms diminish so very slowly that their sum may 
