220 Mr. J. J. Sylvester on Poncelet's approximate 



drantal triangle ZYR, 



cosZR=sinRYcosRYZ, 



sinRY 

 ■. RY=tati 





tan p = tan RO = tan RY - tan Y = 



When c=\/2, this vanishes ; and when c > y/2, the conditions 

 become incompatible. 



/ 2 g Q 



The equations tan <£== a/ -^ — - , or cos 2</>= § — ^, and 



p = £- tan- 1 ^2 = 0-54° 44' 



are well adapted for logarithmic computation. Suppose 



c=$, cos2<£=-±}=-, 44 2<£ = 180°— 63° 54'= 116° 6', 

 <£ = 58°3', / d = 3°19', 



giving a maximum error tan (1° 39' 30") 2 = -0008375. The 

 linear form corresponding to this is 



i + rc* ^ r + y + ^} = ' 5778 ^ + ,57 ' 7 % + ,5 778^ 



If c< 1, the formula changes; the limiting area from a tri- 

 angle becoming a hexagon through all the angles of which a 

 circle will admit of being drawn, which circle will give the limit- 

 ing segment, p becomes the third side of a spherical triangle 

 of which the other two sides are tan -1 \/2 and tan -1 c respect- 

 ively, and the included angle 45° ; so that 



""'- \/ 3 irW\/3(TT?r( 1+ ^mm> 



and the maximum error, i. e. I tan 2 ^ V becomes 

 •3(T+?j + l + V'c 



