Ill 



from a point, and the caustic by refraction of a circle for rays pro- 

 ceeding from a point ; the result in the last case is not worked out, 

 but it is shown how the equation in rectangular coordinates is to be 

 obtained by equating to zero the discriminant of a rational and inte- 

 gral function of the sixth degree. The memoir treats also of the 

 secondary caustic or orthogonal trajectory of the reflected or refracted 

 rays in the general case of a reflecting or refracting circle and rays 

 proceeding from a point ; the curve in question, or rather a secondary 

 caustic, is, as is well known, the Oval of Descartes or ' Cartesian : ' 

 the equation is discussed by a method which gives rise to some forms 

 of the curve which appear to have escaped the notice of geometers. 

 By considering the caustic as the evolute of the secondary caustic, it 

 is shown that the caustic in the general case of a reflecting or refract- 

 ing circle and rays proceeding from a point is a curve of the sixth 

 class only. The concluding part of the memoir treats of the curve 

 which, when the incident rays are parallel, must be taken for the 

 secondary caustic in the place of the Cartesian, which, for the parti- 

 cular case in question, passes off to infinity. In the course of the 

 memoir, the author reproduces a theorem first given, he believes, 

 by himself in the Philosophical Magazine, viz. that there are six 

 different systems of a radiant point and refracting circle which give 

 rise to identically the same caustic. The memoir is divided into 

 sections, each of which is to a considerable extent intelligible by 

 itself, and the subject of each section is for the most part explained 

 by the introductory paragraph or paragraphs. 



III. " On the Figure, Dimensions, and Mean Specific Gravity of 

 the Earth, as derived from the Ordnance Trigonometrical 

 Survey of Great Britain and Ireland." Communicated by 

 Lieut-Colonel JAMES, R.E., F.R.S., &c., Superintendent of 

 the Ordnance Survey. Received April 30, 1856. 



(Abstract.) 



The Trigonometrical Survey of the United Kingdom commenced 

 in the year 1 784, under the immediate auspices of the Royal Society ; 

 the first base was traced by General Roy on the 16th of April of 



