Royal Society. 149 



The fifth set includes equations whose solutions, whether in as- 

 cending or descending series, are always necessarily divergent. 



The distinguishing mark of this class is, that it transgresses both 

 the restrictions to one or other of which the last set is subjected. In 

 this case the divergency is infinite, and appears to be of an unma- 

 nageable character. 



The analogy of the process leads to a presumption, that in all 

 cases of divergency, above referred to, the corresponding convergent 

 solutions are in series infinite in both the ascending and descending 

 directions. 



The author observes in conclusion, that the inverse calculus of 

 the process here developed may be employed for the discovery of 

 the generating functions of series whose laws of relation are given. 



April 26. — "A Report upon further Observations of the Tides of 

 the English Channel made by order of the Lords Commissioners 

 of the Admiralty in 1848, with remarks upon the Laws by which 

 the Tidal Streams of the English Channel and German Ocean appear 

 to be governed." By Captain F. W. Beediey, R.N., F.R.S. Com- 

 municated by the Lords Commissioners of the Admiralty. 



The author commences this report by observing, that the result 

 of the observations upon the tides in the English Channel, made in 

 the course of the summer of 1848, had confirmed in a satisfactory 

 manner the view he had taken of the tidal phenomena of the chan- 

 nel, in the report communicated to the Royal Society last year, and 

 printed in the Philosophical Transactions (Part L 1848), namely, 

 that there is a meeting and a separation of the streams between 

 Alderney and the Start : that the whole space between the Start 

 and SciHy is under the joint influence of the channel and offing 

 streams : that from the vicinity of the Start to the vicinity of Hast- 

 ings the stream runs true up and down the channel ; and moreover 

 that this stream throughout turns nearly simultaneously with the 

 time of high and low water on the shore at the virtual head of the 

 tide, which he places in the vicinity of Dover ; and lastly, that the 

 streams which meet off the Start are turned down into the Gulf of 

 St. Malo, and vice versa. 



He then takes a comprehensive view of the tidal system of the 

 English Channel and German Ocean together, and considering them 

 as one great canal open at both extremities to the free admission of 

 a great tidal wave, which might be supposed to meet and form a 

 combined or stationary wave (art. 187, Encyclopedia Metropolitana), 

 he infers that in such a case, there ought to be in the eastern half 

 of such a canal, a recurrence of the phenomena which had been 

 found to exist in the western half. He proceeds to explain that, 

 from a valuable series of observations in the German Ocean by Cap- 

 tain Washington, R.N., and other authorities, it does appear that, 

 inverting the direction of the stream, there is a correspondence of 

 phenomena in almost every respect: that the offing and channel 

 streams meet off Lynn, as off the Gulf of St. Malo, at the same hours, 

 and at the same distance nearly from the virtual head of the tide : 

 that the phase of the tide at Lynn corresponds with the phase of the 



