234 Cambridge Philosophical Society. 



nature and origin to those substances, and he considers that when 

 indigo is met with in urine in considerable amount, it forms a vehicle 

 for the elimination of any excess of carbon contained in the system. 

 This view is borne out by the important fact, that the greater 

 number of cases in which indigo has been observed to be developed 

 in the urine in large amount have been cases of extensive tubercular 

 disease of the lungs, and in which the decarbonizing functions of 

 those organs are greatly impaired. 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 



[Continued from vol. vii. p. 458.] 

 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. — 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 isa series of the form i2«+i_2'^'*'^V + 3^""^^ . f^-kc, 

 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- repeating 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 /, then the denominator also is a 

 recurrent function of /. 



♦ Sec Memoirs of the Academy of Sciences, 1777- 



