Notices respecting New Books. 391 



pendicular to it. But, although Mr. Childe has thus limited the 

 subject, he nevertheless introduces so many illustrative propositions 

 of a novel and interesting kind, that it cannot fail to attract the 

 attention of all who desire to see the higher methods of analysis 

 skilfully applied to the solution of optical questions. The polar equa- 

 tions investigated in the Appendix, although they are unusually 

 simple and easy of application, afford the means of resolving inverse 

 problems by integration, and lead to some useful diff^erential equa- 

 tions. In discussing the theory of plane reflected caustics, Mr. 

 Childe has, very projierly, omitted the consideration, as special ex- 

 amples, of the cycloid, circle, logarithmic spiral and other well- 

 known curves. The examples which he adduces have, therefore, 

 with some few exceptions, a marked character of novelty. 



One of the most interesting illustrations in the book is that with 

 reference to the rings of the planet Saturn. After showing that if 

 rays of light from a luminous spheroid are reflected from a circular 

 ring symmetrically placed with respect to the luminous surface, the 

 envelope of the reflected ray- surfaces will be two right cones of 

 which the axes pass through the centre of the ring perpendicular to 

 its plane, he says : — 



" In the complex system of the planet Saturn with his attendant 

 rings there seems to be presented to us a physical illustration of the 

 preceding investigation. If, for the sake of illustration, we should 

 conceive the planetary rings to be composed of a congeries of circular 

 rings of infinitesimal thickness, concentric with the planet, and 

 capable of reflecting light, a certain portion of each component 

 ring will reflect all the incident rays which emanate from a definable 

 surface of the planet. Each luminous point in this surface will gene- 

 rate a ray-surface containing all the rays which, issuing from that 

 point, have been reflected from any one of the elementary rings ; 

 and the intersection of all such ray-surfaces, reflected from that ele- 

 mentary ring, will be in the conical envelope which we have here 

 considered. Now if we conceive these concentric reflecting circles 

 to be multiplied within the known limits, so as to generate the satel- 

 lite rings of Saturn, the ray-surfaces reflected from each component 

 ring will be multiplied in like manner, generating an infinite number 

 of conical surfaces of light. The absolute thickness of any section 

 in the stream of light (so to speak) which is thus thrown off from 

 the rings, is a function of the distance between the interior limit of 

 the inner ring and the exterior limit of the outer ring, an interval of 

 the known magnitude 69079 miles ; and it can hardly be doubted, 

 that in this manner an important element in the optical ceconomy of 

 the planet is provided. There will be in this system three separate 

 classes of ray-surfaces generated from the rings. Considering the 

 rings to be circular, the rays incident directly from the sun will be 

 reflected from each elementary circle in a cylindrical surface deter- 

 mined by the equations in page 17, chapter 1. Again, from the 

 same elementary circle, the rays which are incident upon it after 

 reflexion from the planet will generate a conical surface, of whicli 

 the general equation is that obtained in page G5. In addition to 



