224 Mr. BaspeaGe on the General Term 
whenever those two figures are the same, a similar period must 
re-appear: now as there are only two figures concerned, they can 
only admit of 100 permutations, consequently, this is the greatest 
limit of the periods in such species of series.—In the one in ques- 
tion the period is comprized in ten terms. This reasoning may 
be extended to other forms of series in which higher differences 
are given in terms of the digits occurring in the unit’s, ten’s, or 
other places of wu. or u.,; or elsewhere, but I am aware that it 
does not in its present form present that degree of generality 
which ought to be expected on such a subject: probably the 
attempt to solve directly that class of equations to which these 
and similar enquiries lead, may be attended with more valuable 
results. 
As the term ‘“ wnit’s figure of” occurs frequently, it will be 
convenient to designate it by an abbreviation; that which I shall 
propese is the combination of the two initials, and I shall write 
the above equation of differences thus 
INTE CMON NTRS on parca gl) 
This may be reduced to a more usual form by the following 
method. If S, represent the sum of the a" powers of unity, 
divided by ten; then 
OS,+1S,, +28, 424+38,43+48, 44+ 5 S,45+68,46+7 8, 47+85,+t9S,45; 
will represent the figure which occurs in the unit’s place of any 
number «: substituting uv. mstead of x, we have 
~Au.=0S,_+ DS gi ee a t= oo eer aeete amet Oe 
an equation in which w. enters as an exponent. 
From the previous knowledge of the form of the general terms 
of the series we are considering, it would appear that the general 
solution of the equations (a) and (4) is 
u=9s+ cS, + 4 S,., + CoS.42'+ 10-00. C5945: 
