VOL. LV.J PHILOSOPHICAL TRANSACTIONS. 223 



above method; inscribe also another polygon ahklm &c, of n sides, in any other 

 manner, but having one angle at the point a. Then the sum ah -\- be -^ cd ■{- 

 de + &c, will be greater than the sum ah -{• hh -\- kl -\- Im -\- &c. 



Theor. 2. — In fig. 7, describe about the given ellipse two polygons arcde 

 8cc, and pgrst &c, of n sides, of which the points of contact are respectively 

 a, h, c, d, e, &c, and p, q, r, s, t, &c. And let the 



(tang. + sec.) comp. Z_ aiab : (tan. + sec.) comp. ^ ccb :: be : bs, and 

 (tang. + sec.) comp. Z. ccb : (tan. + sec.) comp. ^ cod :: cv : cc, and 

 (tang. + sec.) comp. ^ cod : (tan. + sec.) comp. ^ eEd :: de : do, &c. also 

 (tang. + sec.) comp. Z. />«? : (tan. + sec.) comp. ^ qRr :: ^r : ga, and 

 (tang. + sec.) comp. Z. qRf • (tan, + sec.) comp. ^ ssr :: S5 : rR, and 

 (tang. + sec.) comp. Z. ssr : (tan. + sec.) comp. Z. 't* - Tr : js, and so on. 

 Then will the sum of the sides ab + bc + cd + de + &C' = Pa + QR + 

 RS + ST + &c. 



Carol. Let there be described about an ellipse, as before, a polygon of n sides 

 ABCDE &c, fig. 9 ; describe also about the ellipse another polygon ghklm &c, 

 of n sides, in any other manner, but having one point of contact a the same as 

 the former. Then will the sum ab + bc + CD + de -{- &c. Ije less than the 



sum GH + HK + KL + LM + &C. 



Similar properties may also be affirmed of the polygons described in the hy- 

 perbolas. 



XXIII. A Dissertation on the Nature of Evaporation and several Phenomena of 

 Air, Water, and boiling Liquors. By the Rev. Hugh Hamilton, D. Z>., 

 F.R.S. p. 146. 



This dissertation may be consulted with advantage in a much improved edition 

 of it, in the author's volume of Philosophical Essays. 



XXIF. Physical and Meteorological Observations, Conjectures, and Suppositions. 

 By Benjamin Franklin, LL.D., and F.R.S.* p. 182. 



The particles of air are kept at a distance from each other by their mutual re- 

 pulsion. Every 3 particles mutually and equally repelling each other, must form 

 an equilateral triangle. All the particles of air gravitate towards the earth; 

 which gravitation compresses them, and shortens the sides of the triangles, other- 

 wise their mutual repellency would force them to greater distances from each 

 other. Whatever particles of other matter, not endued with that repellency, are 

 supported in air, must adhere to the particles of air, and be supported by them; 



» On reading the preceding paper in the Society, it was recollected that this paper, similar in 

 some particulars, had been communicated to the Society about nine yeais before, though not till now 

 printed.— Orig. 



