283 



R = — — sin c. 



When = 90°, the general equations become 



tan /3i ^/(tan gi) . 



'*" ^ ~ V(tan gi) - V {(tan j3i + tan gn)) ' ^ ^ 



_ «<7/i2 sin j3i tan j3i / 1 \2 



~ 2 cos gi \ V (tan /3i + tan gi) - V (tan gi)J ' ^' -^ 



If the force in this case be supposed to act horizontally 

 (gi + /3i = 90°), these equations may be reduced to 



tan = cot (c - §) ; (22) 



R=^-co4.-f). (23) 



If the face be vertical, then /3 = c, and the equations may be 

 further reduced to 



tan ^ = cot ^c ; (24) 



R=^cot2ic- (25) 



The Rev. Charles Graves communicated the following 

 note respecting geodetic lines on surfaces of the second order. 



" At a meeting of the Academy which took place in last 

 June, I stated a general theorem, from which I am able to de- 

 duce Joachimsthal's theorem respecting the geodetic lines 

 traced on a central surface of the second order; and at the 

 same time to show geometrically the reason why the property 

 enunciated in it is common to geodetic lines and to lines 

 of curvature. From the general theorem to which I refer, 

 the following proposition is a corollary: 



" If a central surface of the second order (A) he circum- 

 scribed by a cone (a), the quantity PD is the same for L, L', 

 L", L'", four sides of the cone which make equal angles with 

 its internal axis : P denoting the perpendicular from the centre 



VOL. IV. z 



