TOL. LXVJ] PHILOSOPHICAL TRANSACTIONS. 647 



The following are my observations. 



. _j r Emersion of « almost certain ; the immersion was not 



• . • • 1^ observed on account of clouds. 



52 Immersion, 1 ^^^ ^^^^^-^ 



44 Jimersion, J 



34 Immersion of <e, very certain. 



'20 Emersion of the same. 



Immersion of Flamstead's 1 15 in . 



31 Immersion of a star of the 6th magnitude in n . 



20 Immersion of m Virginis, very certain. 



I have lately discovered two curious theorems, which I shall here communicate 

 to the R. s. 



Theorem. — Let a, b, c, d, e, f, be a polygon whose 

 sides are named a, b, c, d, e,f\ and the exterior angles 

 «, (3, y, i, t, ^, so that the side a be placed between 

 the angles « and (3, b between (3, y, &c. 



1 . a X sin. a + ^ X sin. (a -f (3) + c X sin. (« + (3 + y) 

 + dx sin.(« + p + y + J)+e X (sin.a + (3 + y + ,J+ 

 +/X sin. (a + |3 + y + <r+ « + = O. 

 2. a X COS. x-\-bX cos. (« + (3) + c X cos. {a. -{■ ^ + y) -\- d X cos. (« -f j3 + ^ _f. <j) 

 + e X cos. (a + |3 + y + <r + +/X cos. (* + p + y + <r+£ + ^) = 0. 



In fact it is sin. (a + (3 + y + <^+£ + C) =^sin. 36o° = 0, and cos. (a + p 

 + 7 + <^+« + C) = + '» ^^^ ^^ order to give the same form to the two ex- 

 pressions, I rather chose to represent them as I have done. By means of these 

 two theorems the solution of polygons will be as easy as that of triangles by 

 common trigonometry. 



XXVL Investigation of a General Theorem for finding the Length of any Arc 

 of any Conic Hyperbola, by Means of two Elliptic Arcs. With some other 

 New and Useful Theorems deduced from it. By J. Landen, F.R.S. p. 283. 



1 . From the theorem noticed in art. 1 of the author's paper in the Philos. 

 Trans., 1771, (p. 150, of this abridged vol.) it follows, that in the hyperbola 

 AD (pi. 12, fig. ll), if the semi-transverse axis Ac be ^ wi — n; the semi-con- 

 jugate = l^mn; and the perpendiclar cp, from the centre c on the tangent dp, 

 =: v'(n» — ny — t"^); the difference dp — ad, between the said tangent dp and 

 the arc ad, will be equal to the fluent of >/ ™ ~ "{^ ~ ,„ X '• 



2. It is well known, that in any ellipsis whose semi-transverse axis is m, and 

 semi-conjugate n\ if a;- be the abscissa, measured from the centre on the trans- 

 verse axis, and z the arc between the conjugate axis and the ordinate correspond- 

 ing to X, '/ "'^r_% X i' will be = i, g being = '■^-^. 



Hence, / ) ^ ' — ^ X t being = v' ^^ — _, X — ^— f, it ap- 



