348 PHILOSOPHICAL TRANSACTIONS. [aNNO 1 772. 



both these works, it is considered only as a variety, or different in sex. Mr. 

 Graham has the same opinion. It lives on fish, frequenting the lakes near the 

 sea coast. It lays its eggs in water, and cannot rise off dry land. It is seen 

 about the beginning of June, but migrates southward in autumn. It is called 

 sekeep, by the natives. Its eyes are small, the irides red; it weighs 1 pound, and 

 measures 1 foot in length, and one-third more in breadth. 



23. Larus, Gull. 57. Parasiticus. 226. JO. Arctic Gull. Br. Zool. Faun. 

 Am. Sept. l6. Edw. 148. 149. Churchill river, N° 15. 



This species is called a man of war, at Hudson's Bay. It seems to be a fe- 

 male, by the dirty white colour of its plumage below; it agrees very well with 

 Edwards's drawing, and that in the Br. Zool. illustr. 



r 24. Sterna, tern. 58. Hirundo (variety), 227. 2. The greater tern. Br. 

 Zool. Faun. Am. Sept. 



The number belonging to this bird is lost, perhaps it is N° 17, from Churchill 

 river, called ' a sort of gull, called egg-breakers, by the natives.' The feet are 

 black; the tail is shorter and much less forked than that described and drawn in 

 the Br. Zool. The outermost tail-feather also wants the black, which that in 

 the British Zoology has. In other respects it is the same. 



XXX. Geometrical Solutions of Three Celebrated Astronomical Problems. By 

 the late Dr. Henry Pemberton, F.R.S. Communicated by Matthew Raper, 

 Esq., F.R.S. p. 434. 



Lemma. — To form a triangle with two given sides, that the rectangle under 

 the sine of the angle contained by the two given sides, and the tangent of the 

 angle opposite to the lesser of the given sides, shall be the greatest that can be. 



Let the two given sides be equal to ab, and ac fig. 4, pi. 7 : round the centre 

 A, with the interval ac, describe the circle cde, and produce ba to b; take bf a 

 mean proportional between be and bc, and erect the perpendicular fg, and com- 

 plete the triangle AGB. 



Here the sine of bag is to the radius, as fg to ag; and the tangent of abg to 

 the radius, as fg to fb: therefore the rectangle under the sine of bag and the 

 tangent of abg, is to the square of the radius, as the square of fg, or the rect- 

 angle EFC, to the rectangle under ag or ac and fb. But, eb being to bf as bf 

 to bc, by conversion, eb is to ef as bf to pc, and also, by taking the difference 

 of the antecedents and of the consequents, ef is to twice af as bf to fc ; and 

 twice afb is equal to efc. 



Now, let the triangle bah be formed, where the angle bah is greater than 

 bag. Here, the perpendicular hi being drawn, the rectangle under the sine 

 of bah and the tangent of abh, will be to the square of the radius, as the rect- 

 angle Eic, to the rectangle under ac, ib. But if is to fb as 2APi to 2afb, or 



