704 PHILOSOPHICAL TRANSACTIONS. [aNNO 1780. 



X - ; whence the tangent of mtn, the angle sought, is Jj^^ML-zJfl g^j jj^jg 

 t ttt — axx ' 



ttt ttt 



is a maximum when ;t^7^, "''r+~- ~ '^■^' "'' ^^'^'-'" T+~ —y^J' °'' ^'^en cp and 

 PM have such a proportion that cp- : pm" :: ca^ : cb^. 



Let AtA^ab (fig. 5) be an ellipse whose centre is c ; draw the circumscribed 

 and inscribed circles as before ; the former cutting the '2d axis produced in d the 

 latter cutting the first axis in d, and the 2d axis in b. On nb as a diameter des- 

 cribe a circle cutting the first axis xn in q, draw do. and Z'q. Set off cr = dq 

 join DR, and draw dp at right angles cutting the first axis in p, draw pm an or- 

 dinate to that axis ; then m will be the point in the oval line where the angle of 

 deviation is greatest. Otherwise, on cb produced set off cr = ba, join dr, and 

 draw dp at right angles cutting the 2d axis in p : draw pu an ordinate to that 

 axis, and m will be the point where the angle of deviation is greatest. 



At the maximum (when xx = ;) pn^ = — —, pm'^ = — — , and tp^ = 



—— : whence tp'^ = pm X pn. Also, pn, pm, tp, are to each other as ca 

 CB, and V {c\ X cb), respectively. Therefore, -77^— — -r-i is the tangent of 



-y (Alt /\ no J *-^ 



MTN, radius being unity. Also \/ — is the tangent of ntp ; and mtp is the 

 complement of ntp : therefore mtn is twice the excess of ntp above 45°. 



In fig. 5, draw a circle through the points n, m, t ; and at the maximum 

 where tp" := pm X np, this circle will touch ca produced in t. Fi-om e the 

 centre of this circle draw ef perpendicular to nm, also the radii en and em • 

 then FN is the sine of nef, or half nem, or of its equal mtn, to the radius en. 



^ PN + PN J PV — PM rr, . 



But EN = et = pp = — , and fn = . Therefore pn + pm is 



to PN — PM, or CD -j- CB is to CD — CB, Or ca -|- cb is to CA — cB, as radius 

 is to the sine of the greatest angle of deviation, which is therefore equal to 



CA 



CA + CB 



, radius being unity. 



XXF. Of Cubic Equations and Infinite Series. By Charles Hulton, LL. D. 



F.R.S. p. 387. 



As this tract will be found at large inDr. Hutton's works collected, the abridg- 

 ment of it is omitted in this place. 



XXyi. Of a most Extraordinary Degree of Cold at Glasgotv in January last 

 (178O); tilth some New Eaperimenls and Observations on the Comparative 

 Temperature of the Hoar-frost and the ^ir near it, made at the Macfarlane 

 Observatory btlongi)ig to the College. By Patrick Wilson, M.A. p. 45 J. 

 On Tuesday, Jan. IJ, 1780, there was a slight frost, and, on the evening of 



