TRANSACTIONS OF THE SECTIONS. 7 



On a Generalization of Poncelot's TJieorems for the Linear Representation 

 of Quadratic Radicals. By Professor Sylvester, M.A., F.R.S. 



The author explained the application of Poncelet's theorems, to practical ques- 

 tions of mechanics in the case of forces acting in a single plane as in the theory of 

 bridges. 



He next referred to the mode of extension of this theorem, suggested by Poncelet, 

 applicable to the case of forces in space, and pointed out its insufficiency, and, in a 

 certain sense, its incorrectness. 



The essential preliminary question to be resolved in the first instance (after which 

 the matter became one of easy calculation), was shown to be that of cutting off by 

 a plane the smallest possible segment of a sphere that should contain the whole of a 

 given set of points lying on the sphere's surface. Some years ago Prof. Sylvester 

 had proposed in the * Quarterly Mathematical Journal,' without any suspicion of 

 its having any practical applications, the following question : — " Given a set of points 

 in a plane to draw the smallest possible circle that should contain them all." By a 

 singular coincidence, Professor Pierce, of Cambridge University, U.S., had studied 

 this question and obtained a complete solution of it, which he had communicated to 

 the author during the present meeting of the British Association. A slight con- 

 sideration served to show that precisely the same solution as Professor Pierce had 

 found for the problem of points in a plane was applicable with a merely nominal 

 change to the sphere also ; and thus the solution of a question set almost in sport 

 was found to supply an essential link for the complete development of a method of 

 considerable importance in practical mechanics. The author stated that it would 

 be easy to draw up tables of the values of the constants appearing in the linear 

 function, representing the resultant of three forces at right angles to one another, 

 for the principal cases likely to occur in practice, the values of these constants 

 depending solely upon the condition of relative magnitude to which the component 

 forces are supposed to be subjected. 



Light, Heat. 



On the Influence of very small Apertures on Telescopic Vision. 

 By Sir David Brewster, K.H., F.R.S. 



[The manuscript of this paper has been lost.] 



On some Optical Illusions connected with the Inversion of Perspective. 

 By Sir David Brewster, K.H., F.R.S. 



The term "Inversion of Perspective" has been applied to a class of optical 

 illusions, well known and easily explained, in which depressions are turned into 

 elevations, and elevations into depressions. One of the most remarkable cases of 

 this kind, which has not yet been explained, presented itself to the late Lady 

 Gcorgiana Wolf, and has been recorded by her husband Dr. Wolf. When she was 

 riding on a sand-beach in Egypt, all the footprints of horses appeared as elevations, 

 in place of depressions, in the sand. No particulars are mentioned, in reference to 

 the place of the sun, or the nature of the siu'rounding objects, to enable us to form 

 any conjecture respecting the cause of this phenomenon. Having often tried to see 

 this illusion, I was some time ago so fortunate as not only to observe it myself, but 

 to show it to others. In walking along the west sands of St. Andrews, the foot- 

 prints, both of men and of horses, appeared as elevations. In a short time they 

 sank into depressions, and subsequently rose into elevations. The sun was at this 

 time not very far from the horizon, on the right hand ; and on the left there were 

 large waves of the sea breaking into very bright foam. The only explanation which 

 occurred to me was, that the illusion appeared when the observer supposed that 

 the footprints were illuminated with the light of the breakers, and not by the sun. 

 Having, however, more recently observed the phenomenon, when the sun was very 

 high on the right, and the breakers on the left very distant, and consequently very 



