of a New Class of Infinite Series. 225 
where the constants must be determined from the conditions. 
In the further pursuit of any enquiries in this direction, much 
assistance may be derived by consulting a Paper of Mr. Herschel’s 
in the Philosophical Transactions for 1818, ‘On Circulating 
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 number of terms. 
Let the equation determining uw. be 
Au. = B (UFu,_, + UF u,,1). 
Then the following series conform to this law, 
Series. Diff. Series. Diff. Series. Diff. 
1 3 4 6 1 9 
4 5 10 4 10 1 
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. 
C. BABBAGE. 
Devonshire Street, Portland Place, 
March 29, 1824. 
Fr 
