Mr. H. Hilton on van der Waals* Equation. 585 



(•333; -27) ('4; -12-5) ('5; -4) (-667; 0) (-8; -781) 

 (1 ; 1) (1-333 ; '844) (1*5 ; -741) (2 ; -5) (2*5 ; -352) 

 (3 ; 259) (4 ; 456) (5 ; '104), and is shown in fig. 1. 

 Differentiating (a) again twice we have 



<Py_ 1440 18. cPy = -12960 72 



dx 2 ~ (3a?— I) 3 ** ; dx % > (3<^-l) 4 a?' 



.-. At a point of inflexion (3.27 — 1) 8 = 80# 4 . Eliminating 

 between this equation and (a) we have 



yx t =Qx 2 — Qx + 1 ...... (7) 



as the curve through the inflexions of the family. For a 

 point where the tangent has 4-point contact we must have 



<Py _ d 3 y 



dx 2 ~ da? 



,=0, 



and hence we have #=t; = 2ol8 ; hence we see that the 

 curve (7) touches the member of the family for which 

 0=7^ at the point (3 ; 2^); it also touches the branch 

 ya? 2 = — 3 of the curve of the family for which = 0, at the 

 point (5 ; —27) (but elsewhere lies wholly above the branch), 

 but touches no other member of the family at a finite point. 

 Differentiating (y) we have 



it has therefore tangents parallel to y = where <ff=*271 and 

 1*229 (approximately), and inflexions at the points (3 ; —27) 



(J-; '9936). 



It passes through the points ( — 5; *290) (—4; *519) 

 (-3; -901) (-2; 2-312) (-1; 13) (-1; 4600) ('2; 25) 

 (•211; 0) (-25; -68)(*3; -41*975) (-333; -27) (-4; -17*188) 

 (-5; -8) (-789; 0) (1 ; 1) (1-2; 1*178) (1*333; 1-160) 

 (1-667 ; -994) (2 ; -812) (3 y -457) (4; -285) (5 ; -194), and 

 is shown (as far as possible) in fig. 1. 



It is of degree 5, class 5, and deficiency zero ; it has 3 

 double points and 3 cusps (coinciding in the multiple point at 

 infinity) ; 3 bitangents (two of which are imaginary) ; and 3 

 inflexions (all real, one at infinity). It has a quadruple point 

 at infinity at which the tangents coincide with ^ = 0, and an 

 inflexion at infinity at which the tangent is y = 0. 



It meets the curves of the family (a) for which = 

 - (± >/ 3—1) ['824 and —3 074 approx.] in the axis of x. 



Phil. Mag. S. 6. Vol. 1. No. 5. May 1901. 2 Q 



