of a New Class of Infinite Series. 265 
OS, + 15.,,4+258,15435,,,4+45,,,+ 55,64 
@ 
6 S46 +7 S47 ae S.i8 see S49? 
will represent the figure which occurs in the unit’s place of 
any number 2: substituting w, instead of x, we have 
1 
ee 18 We 2 es _ as Ee tei IST F9% 535 (8). 
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 () is 
u=9z+ceS +65 ..4+4S8 5+ 2 a baleteus 
where the constants must be determined from the conditions. 
In the further pursuit of any inquiries in this direction, much 
assistance may be derived by consulting a paper of Mr. Her- 
schel’s in the Philosophical Transactions for 1818, ** On cir- 
culating functions.” 
Amongst the conditions for determining the general terms 
of series by some relation amongst particular figures, there 
occurs a curious class, in which, if we consider only whole 
numbers, the series becomes impossible after a certain num- 
ber of terms. 
Let the equation determining u_ be 
Au, =1(UFu._,+ UF w,, ,). 
Then the following series conform to this law, 
Series. Diff. Series. Diff. Series. Diff. 
1 3 4 6 ] 9 
4 5 10 4 TORS 
9 14 4 11 1 
18 12 3 
15 
If the law is restricted to whole numbers, none of these series 
admit of any prolongation; nor have I, with that restriction, 
been able to discover any series of the kind possessing more 
than five terms. 
Devonshire Street, Portland Place, C. BABBAGE. 
March 29, 1824, 
Vol. 67. No. 336. April 1826. 2 L XLI. On 
