30 REPORT— 1876. 



On a 'fheorem in the Mensuration of certain Solids 

 By Professor James Thomson. 



On Division-remainders in Arithmetic. By W. H. Walenn. 



The author referred to a series of papers of his on unitation recently published 

 in the 'Philosophical Magazine/ and to some remarks published in the I3rit. Assoc. 

 volume for 1870. If x divided by 8 leave remainder y, then the author calls y the 

 unitate of x to the base S, and writes V^x^y. The results of " unitation " may be 

 conveniently applied to the verification of many numerical operations. The method 

 of unitation is practically equivalent to the theory of congruencies, viz. the equa- 

 tion Uja;= y would be written x^y (mod S) ; and many of the results are iden- 

 tical with those given by Gauss. 



On Many-valued Functions. By M. M. U. Wilkinson, M.A. 



General Physics, &c. 

 On the Transformation of Gravity. By James Croll, F.R.8.* 



On the Influence of the Residual Oas on the Movement of the Radiometer. 

 By William Crookes, F.R.S. 



The author's recent experiments show that the movement of this instrument is 

 not due to a direct repulsion exerted by light on the vanes, but to a mutual action 

 called out between these vanes and the" very attenuated gas remaining in the instru- 

 ment. It is well known that, with a moderately good vacuum, the motion becomes 

 more rapid as the exhaustion proceeds ; but he has recently succeeded in producing 

 such a complete exhaustion that he not only reaches the p'oint of maximum effect, 

 but goes so far beyond it that the effect nearly ceases. The vacuum is measured by 

 means of a special apparatus, in which a moving plate, instead of continuously rota- 

 ting in one direction, as in the ordinary radiometer, is suspended by a glass fibre, 

 which it twists in opposite directions alternately. The movement is started by 

 rotating the whole apparatus through a small angle, and the observation consists in 

 noting the successive amplitudes of vibration when the instrument is left to itself, 

 a mirror and spot of light being employed for this purpose. The amplitudes form 

 a decreasing series, with a regular logarithmic decrement. The logarithmic decre- 

 ment is nearly constant up to the point at which the vacuum is apparently equal to 

 a Torricellian vacuum, the mercury in the gauge standing at the same height as a 

 barometric colunm beside it ; but as the exhaustion proceeds beyond this point, the 

 logarithmic decrement becomes smaller— in other words, the amplitude diminishes 

 less rapidly. By plotting the observations and supposing the ciu've continued, it is 

 indicated that, if a perfect vacuum were attained, and the glass fibre had no visco- 

 sity, the logai'ithmic decrement would be zero, we should have perpetual motion 

 with constant amplitude, whilst, at the same time, the radiometer would cease to 

 act. Other gases as well as air have been tried. Aqueous vapour is very unfavour- 

 able to the action of the radiometer ; hydrogen, on the contrary, gives the best 

 result of all. Several experiments have been already described, which seem to 

 point to the true explanation of the action of the radiometer ; but the author thinks 

 !Mr. Stoney's explanation the clearest. According to this, the repulsion is due to 

 the intei'ual movements of the molecules of the residual gas. When tlie mean 

 length of path between successive collisions of the molecules is small compared 

 with the dimensions of the vessel, the molecules, rebounding from the heated sur- 

 face, and therefore moving with an extra velocity, help to keep back the mora 



* Printed in extmso in the Phil. Mag. 1876, ii. p. 241. 



