JRoj/al Astronomical Society. 69 



By means of these expressions, W may be calculated with great 

 facility when m is at all large. The author has given a table of the 

 roots of the equation W=0, from the second to the fiftieth inclusively, 

 calculated by a formula derived from the former of the above expres- 

 sions. This formula was not sufficiently convergent to give the first 

 root to more than three places of decimals ; but this root may be 

 obtained more accurately from Mr. Airy's table. 



The method by which the author has treated the integral W ap- 

 pears to be of very general application, and he has further exem- 

 plified it by applying it to the infinite series 



which occurs in a great many physical investigations, as well as to 

 the integral which occurs in investigating the diffraction produced 

 by a screen with a small circular aperture, placed in front of the 

 object-glass of a telescope through which a luminous point is viewed. 



ROYAL ASTRONOMICAL SOCIETY. 

 [Continued from vol.xxxvi. p. 477-] 



Jan. 11, 1850. — An account of some Improvements in a Speculum 

 Grinding and Polishing Machine. By John Hippisley, F.R.A.S. 



" The machine, as far as regards the general action of two excen- 

 tric pins, which transmit motion to the grinding and polishing tools, 

 is similar in principle to that adopted by Lord Rosse, and other ma- 

 chines previously in use. 



" The improvements consist in the arrangement of the parts so 

 as to effect cheapness and facility in the construction, with general 

 convenience in use, and especially in the manner in which the polish- 

 ing tool is connected with the apparatus which gives it motion. 



" This connexion is made by a ball and socket-joint of a novel 

 construction, which, while it transmits a perfectly equable motion, 

 without jerks or irregularities, to the polisher, and leaves it free also 

 to revolve about its own centre, as the friction between it and the spe- 

 culum may direct, facilitates the application of counterpoise weights, 

 so as to counterbalance in any required degree the weight of the 

 polisher, especially in the very last period of the polishing. 



" A polisher of considerable lightness has justly been deemed in- 

 dispensable, and for this purpose wood has been used instead of metal 

 in its construction. This material is, however, obviously liable to 

 unsymmetrical alteration of figure, from unequal expansion and con- 

 traction by moisture and heat. 



" Nor does it, it is submitted, adequately fulfill the condition re- 

 quired, namely, that of sufficiently removing pressure from the spe- 

 culum. It appears almost demonstrable that the last finishing action 

 of the polisher will be exerted with most advantage, to the perfection 

 both of figure and polish, when it moves without any vertical pressure 

 whatever. Those who are familiar with the adjustment of reflecting 

 telescopes are aware that the slightest confinement or pressure on 

 the speculum, when in its place in the telescope, is sufficient to im- 

 pair its power of definition, in other words, to alter temporarily, in 



