TRANSACTIONS OF THE SECTIONS. 9 



The problem may be transformed by supposing two of the points of the group to 

 range on the contour itself, according to a law which may be expressed by saying 

 that the probability of their being found on any arc shall vary as the product of 

 the segment included between the arc and its chord, multiplied by the time of 

 describing the arc about any centre of force arbitrarily chosen within or upon the 

 contour, — a theorem which, accepting the idea of negative probability, admits also 

 of extension to the case of a centre of force exterior to the contour. 



Among other problems which the author readily resolves by aid of his principle 

 of transformation, may be mentioned that of determining the mean value of a 

 triangle whose angles are taken at random anywhere within a given triangle, 

 parallelogram, ellipse, or ellipsoid. In this description of questions a peculiar 

 difficulty arises, from the fact that the figure which is to be integrated in order to 

 determine the numerator of the fraction which gives its mean value must always be 

 taken positive, whereas its algebraical expression will repeatedly change its sign, 

 •according to a more or less complicated law. This quality of the analytical expo- 

 nent of the arithmetical value of the figure constitutes, in fact, a sort of polarization 

 which has to be got rid of; and the depolarizing process is effected with great ease 

 by virtue of the simplified form impressed upon the data by the method set forth 

 in the paper. 



The author further took occasion briefly to allude to the form in which his own 

 problem of four and Mr. Woolhouse's problem of three points were originally pro- 

 posed, viz. in each case without a specified boundary, and to express his opinion 

 that the principle which had been applied to them, and in which he had formerly 

 .acquiesced, was erroneous, as it could be made to lead to contradictory conclusions, 

 and must be abandoned. He was strongly inclined to believe that, under their 

 original form, these questions do not admit of a determinate solution. 



On Professor Trice's Modification of Arbogast's Method. 

 By Professor Sylvester, F.li.S. 



Light. 



On Moving PJiotographic Figures, illustrating some Phenomena of Vision con- 

 nected with the Combination of the Stereoscope and the Phcnalistoscope by 

 means of Photography. By A. Clafdet, F.lt.S. 



From the beginning of photography it must have struck those who were ac- 

 quainted with the phenakistoscope, invented by Plateau, that photography could 

 produce with advantage series of pictures for that instrument with greater accu- 

 racy than any made by hand. M. Duboscq made some ingenious attempts in this 

 direction, but not entirely satisfactory. The author had also turned his attention 

 towards the subject, and had practically experienced the difficulty of obtaining 

 together the phenakistoscopic and stereoscopic effects. He desired to lay before the 

 Association an attempt he had made some years ago, illustrated by an instrument 

 showing how easy it was to obtain the illusion of moving figures, but without 

 stereoscopic effect, — two pictures of different positions being sufficient to elicit the 

 phenomenon, although the illusion suffers from the deficiency of intermediate posi- 

 tions. Nothing, however, is easier than to employ eight different pictures in as ninny 

 different stages of the action, and the effect will be sufficiently complete. Fortius, 

 Mr. Claudet, having placed in his stereoscope two separate cubic frames, has only to 

 fix on their four sides, at right angles, two sets of four pictures, which are made to 

 pass in consecutive order before the lenses, and the figure assumes consecutively 

 eight different stages of the whole action. The instrument with only two pictures 

 will suffice to illustrate the principle, and, at the same time, elicit some curious 

 phenomena of the perception of vision. The author has constructed his instrument 

 in such a manner that by means of a slide with one hole he can, by moving it 

 rapidly in a reciprocating horizontal direction, shut one lens while the other remains 

 open ; and in continuing that motion, while one eye sees one of the two pictures, 



