302 



May 6, 1841. 



Sir JOHN W. LUBBOCK, Bart., V.P. and Treas., in the Chair. 



The Right Honourable George Stevens Byng, M.P., was balloted 

 for and duly elected into the Society. 



The following papers were read, viz. — 



1. "Investigation of a New and Simple Series, by which the 

 Ratio of the Diameter of a Circle to its Circumference may easily 

 be computed to any required degree of accuracy." By William 

 Rutherford, Esq., of the Royal Military Academy, Woolwich. 

 Communicated by Samuel Hunter Christie, Esq., M.A., Sec. R.S. 



Among various formulfee for the rectification of the circle discovered 

 by the author, he has found the one given in this paper to be that 

 best fitted for computation : and he has been enabled by means of 

 it, with comparatively little labour, to extend the number, express- 

 ing the ratio of the diameter to the circumference, to 208 places of 

 decimals, a degree of accuracy hitherto unattainable, without a great 

 amount of labour, by means of any of the series which have yet been 

 employed. 



The celebrated series of Mr. John Machin, for the rectification of 

 the circle, is derived from the formula 



^ —4 tan — —tan , 



4 5 239 



which converges with considerable rapidity, but gives rise to tedious 

 computations, in consequence of the divisor 239 being a prime num- 

 ber. But by converting the above formula into the following, 



_ =4 tan — ^— tan — -^-tan — , 

 4 5 70 99 



a series is obtained by which the extended computation above men- 

 tioned was readily effected. 



The methods of computation are then stated in detail, and the 

 resulting value of vr is given to 208 places of decimals, which is pre- 

 sumed to be accurate to the last figure, the computations having 

 been actually carried as far as 210 figures. 



2. "On the Phenomena of thin plates of Solid and Fluid Sub- 

 stances exposed to polarised light." By Sir David Brewster, K.H., 

 D.C.L., F.R.S., & V.P.R.S. Ed. 



From a theoretical investigation of the phenomena described in 

 this paper, the author deduces the important law, that when two 

 polarized pencils, reflected from the surface of a thin plate, lying on 

 a reflecting surface of a different refractive power, interfere, half an 

 undulation is not lost, and white- centred rings are produced. When 

 the inclination is exactly 90°, the pencils do not interfere, and no 

 rings are produced. 



