158 Cambridge Philosophical Society. 



Wenlock shale. But in Banisdale there are old slate quarries in 

 the group which, without sectional difficulty or ambiguity, does over 

 lie the Coniston grits. For the future, therefore, the author pro- 

 poses to use the term Banisdale slates for the lower part of the 

 group which overlies the Coniston grits. 



Having approximately laid down the /««//* above mentioned, there 

 was still something wanting to complete the evidence ; for the 

 slate rocks between Duddon Sands and Leven Sands, in spite of 

 their contortions, form an ascending section of great thickness. If, 

 therefore, these slates be a repetition of the Coniston flags, the 

 Coniston grits (typical section No. 4) might be looked for somewhere 

 towards the south-east. To put this to the test, the author (accom- 

 panied by his two friends) made a complete traverse from Broughton 

 to the upper part of Leven Sands ; and they found, as they were 

 finishing their traverse, that the ridges which skirt the estuary of 

 the Leven, below Penny Bridge, were composed of the Coniston grits 

 in their characteristic form. The evidence was then complete, and 

 they next day left the country. 



Also a paper was read by Professor De Morgan, " On the Beats of 

 Imperfect Consonances." 



This subject has been left in great obscurity by Dr. Smith, and 

 subsequent writers have either neglected it, or misunderstood it, or 

 obtained results by methods which miss the principal simplification 

 of which the theory appears susceptible. Omitting historical matter, 

 Mr. De Morgan's method may be described as follows : — 



The grave harmonic of Tartini, formed by sounding two notes of 

 which the vibrations take n and m equal parts of time (m : n being 

 in its lowest terms), has a vibration which lasts through mn of those 

 times. This is called Tartini' s beat, whether it produce a sound, 

 or whether it only produce what Dr. Smith calls a fluttering. This 

 beat is most perfect when the consonance is in perfect tune. If the 

 consonance be a little out of tune, Tartini's beats are not destroyed, 

 but do not succeed each other with perfect reiteration of circum-, 

 stances, owing to the gradual advance or regression of the )iosition 

 in one vibration of the commencement of the other. A cycle of 

 disturbances is the result, which cycle is repeated, or repeated quam 

 proxime ; and the ear recognizes this recurrence in Smith's beats, 

 which are entirely due to the imperfection of the consonance. The 

 connexion has a close resemblance to that of the instantaneous ellipse 

 of a planet and its disturbed orbit. 



The simplest connexion of beats and vibrations is as follows : — 

 The smaller of the two numbers, n and m, being n, every vibration 

 by which the upper note is tuned wrong gives n beats per second. 

 Thus, the consonance being a fifth (2 : 3), every vibration by which 

 the upper note is too flat or too sharp gives two beats per second. 



In an appendix, Mr. De Morgan gives some tables of beats, repeats 

 some theorems on temperament from the Penny Cyclopaedia, and 

 recommends and argues in favour of tuning being performed by a 

 whole octave of tuning forks, adjusted by beats to the system em- 

 ployed. 



