4 REPORT 1864. 



the imaginaiy circular points at infinity, and we have the ordinary method of inver- 

 sion as first proposed, in 1836, by Professor Bellavitis, of Padua, and now universally 

 employed. When the fundamental curve is an equilateral h^^erbola, -^^dth its centre 

 at the origin, we have the hyperholic transformation of Professor Schiaparelli, of 

 MUan. 



On an easy Mode of Measurinc/ Helglits. By M. Moggeidge. 



On Syniholical Expansions. By "W. H. L. Eussell, A.B., F.E.S. 

 In this paper the author pointed out the connexion between his own binomial 

 theorems and the general theorems of symbolical development, given by Professor 

 Boole a long time ago. He also made some remarks upon the application of the 

 calculus of symbols to the integration of linear diflerential equations. It would 

 be interesting to know the real extent of the power of the calculus of symbols as 

 applied to the latter subject. It is certain that many differential equations known 

 to be integrable by other methods are solved with gi-eat facility b}' the calculus of 

 sjTnbols. The author particularly specified Laplace's equation, the well-known 

 equation occuning in investigations respecting the figure of the earth, and a dif- 

 ferential equation, integrated by Professor Stokes, relating to the pressure on rail- 

 way bridges. The most remarkable instance, however, was the equation known 

 as Pfaff 's equation, which was of considerable generality. In this case, as had 

 been ascertained by Professor Boole, the calcidus of symbols gave the same con- 

 ditions of integrability as had been previously ascertained by Pfaffj by an entirely 

 different process. 



ASTEONOMY. 



On Methods of Detecting Clianges on the Moon's Surface. 

 By W. E. BiET, F.li.A.S. 



The author commenced by alluding to the theories that had been submitted for 

 explaining the appearances on the moon's surface. One, he said, had reference to 

 the fixity of those appearances ; so that how long soever observations might be 

 continued, no changes would be detected, all volcanic action on the surface (which 

 appears to be admitted on aU hands) having long ago ceased — the largest lunar 

 forms having been the result of the most violent outbreaks, the smaller mountains, 

 especially in the larger craters, indicating the last expiring efibi-ts of this action. 

 This theory, the author said, would not satisfy all minds ; and accordingly astro- 

 nomers were not wanting who leaned to the hypothesis that eruptive action still 

 exists, although in a subdued form. 



In order ultimately to set at rest the question as to whether the surface of the 

 moon is in a state of quiescence or activity, the author recommended the fonua- 

 tion of a catalogue of lunar objects, remarking that our existing records were 

 inadequate to determine the question. He gave instances of lunar craters figured 

 as far back as 1792, by the astronomer Schroter, which have been entirely over- 

 looked by Beer and Madler, and consequently omitted in their map, which is the 

 acknowledged authority in lunar matters. These craters, the author said, he had 

 himself seen very recently ; and in order to assist in detecting futiu-e changes, should 

 there be any, he had abeady commenced a catalogue in which 386 objects were 

 entered, many of them very small. 



I 



On the present Aspect of the Discussion respecting the Telescopic Appearance 

 of the Solar Fhotosp>here. By the Eev. "W. E. Dawes. 

 After reviewing the statements of Sir W. Herschel, ]\Ir. Nasmyth, Mr. Stone of 

 the Greenwich Observatory, and others, the author considered the discussion to 

 be reduced to these alternatives : — Either, first, the objects described by Mr. Stone 

 as like "rice-grains," are not identical with those JMr, Nasmyth has compared to 



