491 



A=l, 

 B=-> 



r _2 2 



\j = i> = - . 

 5 3.5 



whence substituting in (4) and (3), integrating, and observing tbat 

 c=0 when t=0 or a?= oo, we have the series (2). On substituting 



now in succession tan" 1 ^ and tan~ ] -= for c, and therefore 10 and 50 

 o 7 



for x, and in the latter case multiplying the numerators and denomi- 

 nators of the successive terms by successive powers of 2, we obtain 

 the series (1). 



These series, which the author believes to be new, follow a simple 

 law, and converge with great rapidity* But their distinctive feature, 

 compared with other series which have been given for the same object, 

 consists in the fact that the denominators involve the successive 

 powers of 10, the division by which is effected at once. 



II. " On the Production of Vibrations and Sounds by Electro- 

 lysis." By GEORGE GOHE, Esq. Communicated by Pro- 

 fessor TYNDALL. Received November 12, 1861. 

 (Abstract.) 



In this communication, which is a continuation in subject (but 

 different in title) of a previous investigation " On the Movements of 

 Liquid Metals and Electrolytes in the Voltaic Circuit," the author has 

 described the most convenient and effective method of obtaining 

 vibrations and sounds by electrolysis. 



The paper contains a full account of the influence of various cir- 

 cumstances upon the vibrations and sounds : viz., of the electrodes, 

 the electrolyte, mechanical circumstances and temperature, the 

 electric current, size and number of voltaic elements, quantity of 

 the current, coils of wire in the circuit, induction coils and iron 

 cores, electrolytes in the circuit, and of magnetism : also the in- 

 fluence of the vibrating medium itself upon the electric current. 



The best liquid for producing the vibrations and sounds consists of 



