218 Mr. Baspace on the General Term 
the digit fixed upon might be that occurring in the ten’s place, 
or generally in the »" place. I did not, at that time, possess 
the means of making these alterations which I had contemplated, 
but I immediately proceeded to write down one of the series 
which would have been calculated by the machine thus altered; 
and commencing with one of the most simple, I formed the 
series. 
Series. Diff. 
el 
D> 
ornvnnaor 
If uw, represent any term of this series, then the equation which 
determines w, 1s 
Au, = unit’s figure of «., 
an equation of differences of a nature not hitherto considered, 
nor am I aware that any method has been pointed out for the 
determining u. in functions of = from such laws. I shall now lay 
before the Society, the ‘steps which I took for ascertaining the 
general terms of such series, and of integrating the equations to 
which they lead. I shall not, however, commence with the 
general investigation ofthe subject, but shall simply point out 
the paths through which I was led to their solution, conceiving 
this course to be much more conducive to the progress of analysis, 
although not so much in unison with the taste which at present 
prevails in that science. 
If we examine the series, and its first differences, it will be 
