348 Prof. Young oji the Stimmation of 



rise to new decompositions, and in that way only, causing ob- 

 struction to the passage of the electric current, I was freed 

 from the necessity of considering the peculiar effects described 

 by that philosopher. I was the more willing to avoid for the 

 present touching upon these, as I must at the same time have 

 entered into the views of Sir Humphry Davy upon the same 

 subject*, and also those of Marianinif and Ritter:f, which are 

 connected with it. 



[To be concluded in the next number.] 



LVII. On the Summation of slowly converging and diverging 

 Infinite Series. By J. R. Young, Professor of Mathematics 

 in Belfast College, 



/COMMODIOUS methods for approximating to the sum 

 ^-^ of a slowly converging infinite series are very valuable in 

 many departments of physical science. Philosophical in- 

 quiries of the highest interest and importance frequently ter- 

 minate in series of this kind, which would be practically use- 

 less, on account of the impossibility of the actual summa- 

 tion, did we not possess the means of transforming them to 

 others of such rapid convergency that the sum of a moderate 

 number of the leading terms may in each case afford a near 

 approximation to that of the entire series. Of all such me- 

 thods of transformation that furnished by the well-known 

 Differential Theorem is, perhaps, the most extensively appli- 

 cable ; and it is, therefore, in one form or other, generally 

 employed for this purpose. In the application, however, of 

 this theorem, as well as in that of all other practical formulae 

 intended to abridge numerical labour, there is room for the 

 exercise of some ingenuity as to the most advantageous ar- 

 rangement of the arithmetical process ; for if this arrangement 

 be not such as to render the amount of calculation by the pro- 

 posed formulae a minimum, it is plain that we do not derive 

 from that formula all the advantage, as a facilitating principle, 

 which it is capable of affording. 



In the present paper it is my wish, first to give a short and 

 easy investigation of the differential theorem, and, by deducing 

 it in a somewhat more complete form than that in which it 

 usually appears, to show that it is capable of furnishing, not 

 only a near approximation, but also very close superior and 

 inferior limits, to the sum of a slowly converging or diverging 



♦ Philosophical Transactions, 1826, p. 413. [or Phil. Mag. and Annals, 

 N.S., vol. i. p. 193.— Edit.] 



t Annalex de Chimie [et de Physique], torn, xxxiii. pp. 117, 119> &c. 

 X Journal de Physique, torn. Ivii. pp. 349, 350. 



