420 Mr. Lubbock on the Perspective 

 Let a = volume of gas at a standard temperature as 32° ; 

 V = the volume at some other temperature : — = increment 

 by elevating the standard 1°; 7« = number of degrees above 

 or below the standard. Then a + = V . or a = V x 



— n 



"" Yours, &C. T T3 17 



„ + ,„ • J. B. Lmmett. 



LXIL On the Perspective Bepresentation of a Circle. By 

 John William Lubbock, Esq. F.B.S. S,- L.S.* 



LET $ {x,z,i/), <^' {x,y, z) be the equations to any curve line ; 

 let the eye of the spectator be situated at the origin of the 

 co-ordinate axes, and let x',y', z' be the co-ordinates of any 

 point in the conical surface whose vertex coincides with the 

 origin, and whose base is the curve in question. The equa- 

 tion to this cone is found by eliminating x,y, z from the equa- 

 tions ^xyz), <^'{x,y,z), and x' = —-, x' = —^. 

 Let $ and (p' be the equations to a straight line 



ay =z bx -h a. /,% 



az z= ex + ^ ^ ^ 



The equation to the plane passing through the origin and 

 this straight line will have for its equation 



/3 {ay' — bx') = a{az' — c x') 

 The equation to any straight line parallel to line ( 1 ) will be 

 ay = bx + a! ,2) 



az =z ex + ^' ^ ' 



a, 6, c which depend on the direction of the line being the 

 same, and the equation of the plane passing through this line 

 will be ^1 {ay>-bx') = «' {az'-cx') (3) 



The equations of the line which is the intersection of these 

 planes are ay' — bx' = 



az' — ex' = 

 which are the equations of a straight line parallel to lines ( 1 ) 

 and (2) and passing through the origin. 



The representation of any line is the intersection of the sur- 

 face of the picture, and the plane (3 {ay'— bx') = a.{az'— ex'). 

 But it is evident, in consequence of the preceding theorem, that 

 the representations of all lines which are parallel, will meet in 



the picture in the point where the line -! "^ ~ > cuts the 



surface, whatever this surface may be. If the picture is a 

 plane of which the equation is x = A, the equation to the re- 



• Communicated by the Author. 



pre- 



