10 REPORT 1842. 



temperature of winter and 96° Fahr., it was proved that the rate of going of chrono- 

 meters, furnished with the gold-covered steel spring, was not injured by that covering. 



On the Rate of a Patent Compensating Pendulum. By E. J. Dent. 



The invention consisted in giving impulse to the pendulum at the centre of percus- 

 sion instead of the usual place, which is near to the centre of suspension. Mr. Dent 

 stated the performance of clocks thus made to be very satisfactory. He mentioned, in 

 connection with this subject, the invention of a new escapement by the Astronomer 

 Royal, which had been found of advantage in preventing the stopping of clocks at low 

 temperatures. 



On a New Chronometer Compensating Balance. By E. J. Dent. 



The ordinary compensation balance is constructed of two segments of a circle, of 

 brass and steel, and from the circular form of these pieces, the compensation weights 

 attached to them are moved out too far from the centre of motion in cold, and not 

 sufficiently in toward the centre in warm weather. Hence chronometers thus con- 

 structed gain on their rates at the mean temperature, and lose at both extremes. 



Mr. Dent stated, from experiment, that the compensation weights were carried over 

 nearly equal spaces for equal increments and decrements of heat, whereas the mathe- 

 matical rule requires that they should be carried over spaces proportioned to the 

 square of the distance from the centre of motion. 



Mr. Dent's invention of a new compensation balance consisted in the compensation 

 pieces being made into such curves as would practically meet the case required. The 

 experimental investigations on which this construction was founded, were described at 

 length. 



On a Mode of expressing Fluctuating or Arbitrary Functions by Mathema- 

 tical Formulae. By Sir W. Hamilton. 



A simple Method of arriving at the decimal part of the Sine or Tangent below 

 a second of a degree, to the xo£ TO th or j]jo^oo"o tn P art °f^' By Moses 



HOLDEN. 



On Decimal Fractions. By Anthony Peacock. 



If the successive remainders in finding the reciprocal of a prime number be placed 

 over each quotient figure, the whole will form a table of the decimals equivalent to such 

 number as a denominator, and the respective remainders as numerators. By this ar- 

 rangement the several properties of pure repetends may be studied with great advantage. 

 J I. 10. 13. 14. 24. 8. 22. 17. 25. 18. 6. 2. 20. 26. 



(.9 6 5 5172413 7 931 

 The figures in the upper line are the several remainders, or numerators, and may 

 be called the indices of the series ; the under figures are the repetend : thus — 



=c -1379310344, &c., — — -93103448, &c, — = -03448275, &c. 

 ' ' 29 29 



From the two following original properties of a reciprocal repetend various practical 

 rules have been devised. 



1st. The product of any two numerators or indices is equal to the numerator stand- 

 ing in the sum of the places, reckoning from right to left, unless the product exceed the 

 denominator, in which case the numerator is the remainder from the product divided 

 by the denominator : thus, in 29ths, 4 and 7 arc the indices in the 6th and 8th places, 



