Mr. Babbage on a New Class of Infinite Series. 259 
No. 1. From Preston quarry in the Cleveland dyke. Mi- 
mosite, fine grained, imperfectly porpheroidal from the salient 
crystals of pyroxene. It is a basalt of the ancient mineralo- 
gists. The specimen contains a great abundance of dark- 
greenish gray felspar, mixed with a very small quantity of 
pyroxene and titaniferous iron. Some points of pyrites are 
to be seen. The paste also envelops laminar crystals of fel- 
spar, having a considerable lustre, which give the paste a scaly 
appearance which distinguishes it from basalt. 
No. 2. From Coaly Hill dyke near Newcastle. Mimosite, 
small grained, passing into zerasite. Many of the cavities 
contain green-earth. It is imperfectly porpheroidal. The 
crystals of felspar very brilliant. 
No. 3. From Walbottle Dean dyke. This has a more de- 
cided character of a dolerite, very fine grained, the felspar 
whiter than in the others. 
As these distinctive terms are not generally adopted by 
English mineralogists ; it may be proper to state that mzmosite 
and dolerite are granular rocks. Xerasite and basalt are com- 
posed of the same elements, but microscopic, and having the 
appearance of a paste. 
XL. -On the Determination of the General Term of a New Class 
of Infinite Series. By CHARLES BaBBaGE, Esq. M.A. Fellow 
of the Royal Societies of London and Edinburgh, and of the 
Cambridge Philosophical Society*. 
(THE subject of investigation on which I have entered in 
the following paper, had its origin in a circumstance 
which is, I believe, as yet singular in the history of mathema- 
tical science, although there exists considerable probability, 
that it will not long remain an isolated example of analytical in- 
quiries, suggested and rendered necessary by the progress of 
machinery adapted to numerical computation. Some time has 
elapsed since [ was examining a small machine I had con- 
structed, by which a table, having its second difference con- 
stant, might be computed by mechanical means. In consi- 
dering the various Bp ps which might be made in the ar- 
rangement of its parts, I observed an alteration, by which the 
calculated series would always have its second difference equal 
to the unit’s figure of the last computed term of the series ; 
other forms of the machine would make the first or the third, 
or generally any given difference equal to the unit’s figure of 
* From the Cambridge Philosophical Transactions, vol. ii. Part I. 
9K 9 the 
