12 Drs. Tilley and Maclagan on the Conversion of Cane-sugar 



In the same way that it has now been proved thatT 1 — j- ) 

 is always equal to * 3 ... , whatever be k, above 3, may it be 

 further shown that (l+-r-) is always equal to 2* ... ; and 



/ 1 \Aoo/ [ # t 



thence that \l+-j-) is necessarily infinite when k is : so 



that it is indisputably true that the extreme of the convergent 

 cases of the above series S, usually written in the form 



1 — 1 + 1—1+1 — 14- &c. 



is — , and that the extreme of the divergent cases, usually 



written in the same form, is really infinite, as stated in my 

 former paper; which last conclusion could never have been 

 anticipated from the theory hitherto prevalent. The views 

 now developed are only the continuation and completion of 

 those exhibited in my paper on Series submitted to the British 

 Association in June 1845. If I have been anticipated in any 

 of these views, which are doubtless calculated to produce a 

 reform in the existing theory, I hope to be informed of the 

 circumstance through the medium of this Journal. I have 

 only further to add, that when an expression for the conver- 

 gent cases of a series is found — as it often may be by aid of 

 the differential theorem — then the general equivalent of the 

 series may afterwards be ascertained by developing this ex- 

 pression sufficiently far to unfold to us the general form of the 

 remainder. The expression for the convergent cases of the 

 general series, discussed at page 439 of the last volume, may 

 in this manner be determined ; and the development of this 

 expression by common division, as there proposed, furnishes 

 the formula by which that expression must be corrected, in 

 order that the algebraical equivalent of the series may be ex- 

 hibited in its utmost generality. 

 Belfast, November 21, 1845. 



IV. On the Conversion of Cane-sugar into a substance isomeric 

 •with Cellulose and lnuline. By Thomas Tilley, Esq., 

 Ph.D., and Douglas Maclagan, M.D., F.R.S. Edin.* 



WHEN the juice of beetroot undergoes fermentation at 

 temperatures varying from 30° to 40° C, the cane- 

 sugar which it contains is at first converted into sugar of 

 grapes, and after some time into mannite, lactic acid and a 



* Communicated by the Chemical Society; having been read April 21, 

 1845. 



