145 



May 1, 1854. 



A paper was read by Professor De Morgan on the Convergency of 

 Maclaurin's Series, being an Appendix to a paper on some Points in 

 the theory of differential equations. See the abstract of the former 

 paper, Phil. Mag. vol. vii. p. 450. 



Mr. Kingsley made an oral communication on the Chemical 

 Nature of Photographic Processes. 



May 15, 1854. 



A paper was read by Mr. Warburton on Self-repeating Series. 



In computing Bernoulli numbers by the formula of Laplace*, the 

 author of this paper was led to notice, that in the fraction whose 

 development is a series of the form 1 2w + 1 _2 2w+ V + 3 2ra+1 . * 2 -&c, 

 the numerator of that fraction is a recurrent function of t. This led 

 him to investigate the question, what are the conditions which the 

 denominator of the generating fraction, and the terms of the series 

 generated, must satisfy, in order that the numerator of such a frac- 

 tion may be a recurrent function of t. The paper contains the result 

 of that investigation. 



The author calls those series " self -repeating ," which, when ex- 

 tended without limit in opposite directions, admit of separation into 

 two similar arms, each arm beginning with a finite term of the same 

 magnitude. Between this pair of finite terms, either no zero-term, 

 or one or more zero-terms, may intervene. One arm repeats, and 

 contains arranged in reverse order, the terms of the other arm, either 

 all, or none, of the terms having their signs changed. The different 

 positive integer powers of the natural numbers, of the odd numbers, 

 and of the figurate numbers of the several orders, present familiar 

 examples of self- repeating recurring series. 



The author demonstrates the following three theorems respecting 

 self- rep eating recurring series : — 



I. If the series arising from the development of a proper fraction 

 is the right arm of a self-repeating recurring series, and if the deno- 

 minator of such a fraction is a recurrent function of t, then the nu- 

 merator also is a recurrent function of t. 



II. Other things remaining the same, if the numerator of the 

 fraction is a recurrent function of t, then the denominator also is a 

 recurrent function of t. 



III. If the numerator and the denominator of a proper fraction are 

 each a recurrent function of t, then the series, arising from the deve- 

 lopment of the fraction according to the positive integer powers of t, 

 will be the right arm of a self-repeating recurring series. 



By way of example, the author applies his first theorem to the 

 summation of the infinite series I 9 — -2 y + 3 9 — &c, and compares his 

 process with the corresponding processes of Laplace and of Sir John 

 Herschel. The sum in question is given by Sir John Herschel (see 

 Jameson's Journal, January 1820) in terms of the differences of the 

 powers of 0, extending from A'O 9 to A 9 9 . In the author's process, 



* See Memoirs of the Academy of Sciences, 1777- 



