6l8 PHILOSOPHICAL TRANSACTIONS. [aNNO 1722. 



(where a, b, c, &c. represent the coefficients of the preceding terms) may have 

 its root determined in the following manner : viz. 



Putting a + ^art + ^ = v m the 1st case, and a -{• ^ aa — 1 = v m the 

 2d. Then will y = ^"^v — 4-;/-^ in the 1st case ; and y = ^^^v + ^l/-L in 

 the 2d. Which solutions were inserted in N° 309, of that year. 



Now by what artifice these formulas were discovered, will clearly appear from 

 the demonstration of the following theorem. 



Let X denote the versed sine of any arc, and t that of another, the radius 

 being 1 ; and let the former arc be to the latter, as 1 to w : then, assuming the 

 two equations, which may be called know, viz. •*■■> 



1 — 2z" + z^" = — 2z''/, and 1 — 2z + z^ = — 2zji7 ; 

 by expunging z, there will arise the equation by which the relation between x 

 and t will be determined. 



CoroL 1. — If the latter arc be a semicircle, the equations will be 

 1 -f z" = O, and I — 2z -f z^ = — 2z^: 

 from which if z be expunged, there will arise an equation, by which will be 

 determined the versed sines of the arcs, which are to the semicircle, taken 

 once, or thrice, or 5 times, &c. as 1 to n. 



CoroL 2. — If the latter arc be the whole circle, the equations will be 

 1 — z" = O, and 1 — 2z + z^ = — 2za: : 

 from whence expunging z, there will arise an equation, by which are deter- 

 mined the versed sines of the arcs, which are to the circumference, taken 

 once, twice, thrice, 4 times, &c. as I to n. 



CoroL 3. — If the latter arc be of 6o°, the equations will be 

 1 — z" + z^" = 0, and 1 — 2z + z^ = — zj:: 

 from which, by expunging z, there will arise an equation, by which are deter- 

 mined the versed sines of the arcs, which are to the arc of 6o°, multiplied by 

 1,7, 13, 19, 25, &c. or by 5, 11, 17, 23, 29, &c. as 1 to n. 



If the latter arc be 120*^, the equations will be 



1 — z" + z^" = O, and 1 — 2z -I- z^ = — 2z : 

 from which if z be expunged, there will arise the equation, by which are 

 determined the versed sines of the arcs, which are to the arc of 120°, multi- 

 plied by 1, 4, 7, 10, 13, &c. or by 2, 5, 8, 11, 14, &c. as 1 to n. 



An Account of a new sort of Molosses, made of Apples ; aiid of the degenerating 

 of Smetts. By the Hon. Paul Dudley , F. R. S, N° 374, p. 231. 



The apple, that produces the molosses, is a summer-sweeting, of a middling 

 size, pleasant to the taste, and full of juice, so that 7 bushels will make a 

 barrel of cyder. The manner of making it is thus ; you grind and press the 



