( 184 ) 



osculating plane of C^ in R. Through R and five points of 6', con- 

 secutive to R a twisted cubic C'^ can be brought, on the condition 

 that R, I and V are an ordinarv point, an ordinary tangent and 

 an ordinary osculating plane of C-^. The developable IJ^ formed by 

 the tangents to C^ and the developable Z), have in common the 

 line / and four consecutive generating lines.' 



If / must count for 2, 3 or 4 common tangents of 6" and S, 

 this is also the case for C\ and >S'. The theorems proved in ^ 6 

 and 7 for C^ hold good for any twisted curve. This gives rise to 

 the following theorems : 



//' the developable D^ correspondinu to curve C^ touches any 

 surface S in point P whilst the <jenerating line I of D^ through P 

 is no injlexional tangent of S, the line I counts for two or for three 

 coininon tangents to C\ and S according to the surfaces having in P 

 an ordinary or a stationary contact. 



If the point of contact P of D^ and IS he a parabolic point on S, 

 then I counts for four or for two common tangents of C\ and S 

 according as the inflexional tangent of S in P coinciding luith I or not. 



J f the point of contact P of D^ and S be a hyperbolic point on 

 S and if the tangent I of C\ coincides with an inflexional tangent 

 in the point P of S, then I counts for four or for two common 

 tangents of C^ and according to R coinciding ivith P or not. 



If C\ touches S in P, whilst the osculating plane of C^ in P 

 coincides with the tangent plane of S in P, then the tangent I in P 

 to C\ counts for four or for three common tangents of C\ and 0, 

 according to I being an inflexional tangent of in P or not. 



The theorems proved here for curves in space hold with a slight 

 modification (see § 1) still for plane curves. They can be easily 

 proved by taking for C\ tirst a parabola // after \vhicli they can 

 be directly extended to an arbiti-ary conic section and after this to 

 an arbitrary plane curve. 



Belft, June 1905. 



Physics. — The shape of the sections of the surface of saturation 

 normal to the x-axis , in case of a tJiree phase pressure betioeeii 

 two temperatures.'' By Prof. J. D. van der Waals. 



In these Pi-oceedings of March 1905 I have (fig. 4, 5 and 6) 

 represented in a diagram some sections of the {p, T, ,i')-surface normal 

 to the 7 -axis for three temperatures, at which three |)hases can 

 exist simultaneously. The three temperatures chosen were: 1**^ the 



