VOL. LXVII.] PHILOSOPHICAL TRANSACTIONS, 157 



±yr' 



s only as variable, — - X the whole fluent of * -/ (y' — r) X (<" + :r'-) will de- 

 note a force which, acting at the distance g from c, would be equivalent to the 

 motive force of all the particles in the section hi whose radius is hi and thickness 



x'. Which fluent is = Af X (x' + ^') = ^' X (A' - x') X {x- + ^^ X {b' 



— x"")). Consequently ^^ X the whole fluent of .r X (b' — x-) X (x- + 



i_. X {b"^ — x')) will denote a motive force which, acting at the distance o from 

 c at right angles to a ray from it, would be equivalent to the whole motive force 

 urging the spheroid to turn as above mentioned. Such equivalent force will 

 therefore be = -^^'-^ X {r't + r"-b') = r^ X {r' + b-) X s; and this being put 



log og 



fmn 



A V' ■ 



'"' — b'^ 1 • 1 -11 1 si>ine- . . ;■- — b- 



7~-\ 

 computed in art. 1 



loted hv — V — 



= -^ X ('■" — y^) X s (the value of the same force found in art. 5,) we find 

 F = fgmn X -.-T-Ai ; which will be = 2—- X , ,-7, if/be= - its value 



cd r^ — b- 



Or p will be denoted by - X -rT~p 5 ^^ ^ ^^ to e as m to d, and as ?j to c; 

 and a and g'be each := r. 



7. Fig. 25, the body being a cylinder whose centre of gravity is in c, and 

 whose proper axis pn is lb, and diameter 1r ; the accekrative force (p) at the 

 distance g from c, will in like manner be found = ~f^ X '., ~ ' ; the cylin- 

 der being considered as urged to turn about a diameter passing through c. 



\i I : e :: m : d :: n : c, and a and g be each ^ r, p will be =r 

 c£ . 3)-" - 4 fc' 

 ■ r ^ 3r^ + ib"-' 



XVI. Directions for making the Best Composition for the Metals of Reflectin'r 

 Tel^copes ; with a Description of the Process for Grinding, Polishing, and 

 giving the Great Speculum the True Parabolic Curve.* By Mr. John Mudge. 

 p. 296. 



As the method of casting, grinding, and polishing the specula of reflecting 

 telescopes, by Mess. Molyneux and Hadley, published in Dr. Smith's Optics, is 

 generally followed by, and well known to the workmen, I shall avoid a repetition 

 of the general directions there given, and only remark on such parts of that 

 process as I think, are essentially defective, and supply them by a method of my 

 own, which, from repeated trials, I have found completely to answer the pur- 

 pose. After therefore referring to the above account for the manner of making 

 the gages, patterns, the method of casting, as well as a great many other par- 

 ticularSj I will begin with 



* For this paper Mr. Mudge was hcaioured with the Society's prize gold medal. 



