1888.] 



On Hamilton's Numbers. 



99 



that the circuit which is best according to the rules given by these 

 equations is seven times as good as the best previously found. 



I have then shown that the mirror must be of such a size "as to 

 have a moment of inertia one-third of that of the active bar. In 

 the particular case considered, where the active bar consists of two 

 pieces, one antimony and one bismuth, 5 x 1 X \ mm., at a mean 

 distance of 1 mm. apart, the diameter of the mirror should be 2 J mm. 

 This size both theoretically should, and practically does, enable one 

 with certainty to observe a deflection of £ mm. on a scale 1 metre 

 distant. 



General considerations show that the antimony-bismuth bars cannot 

 have too small a sectional area, but that the length when already short 

 is only involved in a secondary manner. 



It is shown that the* heat in the circuit is equalised mainly by con- 

 duction, which is thirty times as effective as the Peltier action. 



It is found necessary to screen the antimony and bismuth from the 

 magnetic field by letting them swing in a hole in a piece of soft iron 

 buried in the brass work. 



I have shown that the instrument imagined in the preliminary note 

 would be so much more than dead beat that it would not be possible 

 to use it advantageously, but on making a corresponding calculation 

 for the best circuit, now found, using conditions which have been 

 proved by practice to work well together, a difference of temperature 

 of one ten -millionth of a degree centigrade is by no means beyond 

 the power of observation. 



The figures given by an actual comparison between the newest 

 instrument and one of the original pattern is very favourable to the 

 former. 



In conclusion, I have explained the peculiar action of the rotating 

 pile, and have shown that it is different from that figured in Noad's 

 ' Electricity and Magnetism.' 



II. " On Hamilton's Numbers. Part II." By J. J. Sylvester, 

 D.C.L., F.R.S., Savilian Professor of Geometry in the 

 University of Oxford, and JAMES Hammond, M.A., Cantab. 

 Received March 9, 1888. 



(Abstract.) 



§ 4. Continuation, to an infinite number of terms, of the Asymptotic 

 Development for Hypothenusal Numbers. 



In the third section of this paper ('Phil. Trans,,' A., vol. 178, 

 p. 311) it was stated, on what is now seen to be insufficient evidence, 

 that the asymptotic development of p — q, the half of any hypothenusal 



