PROCEEDINGS 



OF 



THE ROYAL SOCIETY. 



1832-1833. No. 11. 



November 15, 1832. 



JOHN WILLIAM LUBBOCK, Esq. M.A., V.P. and Treasurer, 

 in the Chair. 



A paper was read, entitled "On some Properties of Numbers in 

 Geometrical Progression." By Charles Blacklewar, Esq. B.A. Com- 

 municated by J. G. Children, Esq. Sec. R.S. 



This paper contains the demonstrations of the three following 

 theorems ; namely, 



1°. If the terms of a geometrical series be raised to 2 , then any 

 odd number of them is divisible by the corresponding terms of the 

 original series. 



2°. If each term of a geometrical series be raised to any odd power, 

 the sum of the terms so raised is divisible by the original series, if 

 the number of terms taken be any power of 2. 



3°. If the number of terms of a geometrical series be any power of 

 2, the sum of the terms raised to the power m is divisible by the sum 

 of the same terms raised to the power n, provided m divided by n be 

 a whole number. 



November 22, 1832. 

 JOHN BOSTOCK, M.D. Vice-President, in the Chair. 



A paper was read, entitled "Account of an Improvement in the 

 Machine for producing Engravings of Medals, Busts, &c. directly 

 from the Objects themselves, in which the Distortions hitherto at- 

 tending such Representations are entirely obviated." By Mr. Bate. 

 Communicated by J. G. Children, Esq. Sec. R.S. 



Some printed representations of medals having been received from 

 America, about fifteen months ago, evidently effected by some pro- 

 cess of ruling, Mr. Bate, jun. constructed an instrument for accom- 

 plishing the same object j but the results, both of the American method 

 and of the one invented by Mr. Bate, were attended with a degree 

 of distortion. This the author has ingeniously obviated, by giving an 

 inclination of 45 degrees to the plane in which the tracing-line is 

 moved over the surface of the object of which a representation is to 

 be given. 



