325 



equation of the curve formed by the perpetual intersection of rays, 

 which, diverging from a luminous point, are reflected by a polished 

 surface of a given curvature. Curves of this description have been 

 denominated caustics ; and the method usually employed to discover 

 their polar equations, or the relation between the radius vector of any 

 point of the curve and the tangent at that point, is both long and in- 

 elegant, and is considered by the author as involving considerable 

 inaccuracy of reasoning. He proposes, therefore, to substitute a new 

 method of investigation, by taking the polar equation of one of the 

 reflected rays, and differentiating this equation with respect to the 

 arbitrary quantities solely which determine its position, and thus ob- 

 taining the polar co-ordinates of the point of intersection of two con- 

 secutive lines ; and finally, by elimination, the equation of the curve 

 in which all such points are found. He is thus led to results remark- 

 able for their simplicity, elegance, and generality : and he gives par- 

 ticular applications of his method, exemplifying the facility with which 

 it effects the solution of problems extremely difficult of management 

 by the ordinary methods hitherto employed. His method is also ap- 

 plicable to the determination of the equations of the evolutes of curves, 

 and to various other problems of a similar nature. 



A paper was also read, entitled, " Discovery of the Metamorphoses 

 in the second Type of the Cirripedes, viz. the Lepades, completing 

 the Natural History of these singular Animals, and confirming their 

 affinity with the Crustacea." By J. V. Thompson, Esq., F.L.S., 

 Deputy Inspector General of Hospitals. Communicated by Sir James 

 Macgrigor, Bart., M.D., F.R.S. 



The discoveries madeby the authorof the remarkable metamorphoses 

 which the animals composing the first family of the Cirripedes, or Ba- 

 lani, undergo in the progress of their developement, and which he has 

 published in the third number of his Zoological Researches (p. 76), are 

 in the present paper, which is intended as a prize Essay for one of the 

 Royal Medals, followed up by the report of his discovery of similar 

 changes exhibited by three species of two other genera of the second 

 tribe of this family, namely, the Lepades. The larvae of this tribe, like 

 those of the Balani, have the external appearance of bivalve Monoculi, 

 furnished with locomotive organs, in the form of three pairs of members, 

 the most anterior of which are simple and the other bifid. The back of 

 the animal is covered by an ample shield, terminating anteriorly in two 

 extended horns, and posteriorly in a single elongated spinous process. 

 Thus they possess considerable powers of locomotion, which, with the 

 assistance of an organ of vision, enable them to seek their future per- 

 manent place of residence. The author is led from his researches to 

 the conclusion that the Cirripedes do not constitute, as modern na- 

 turalists have considered them, a distinct class of animals, but that 

 they occupy a place intermediate between the Crustacea decapoda, 

 with which the Balani have a marked affinity, and the Crustacea en- 

 tomostraca, to which the Lepades are allied ; and that they have no 

 natural affinity with the Testaceous Mollusca, as was supposed b,v 

 Linnaeus, and all the older systematic writers on Zoology. 



