73Ö 



10. Any coincidence of llie Q^ is completed into a f|iiadrnple 

 by two complementanj points. The locus rf of those points which 

 we shall call the complementary curve has apparently quadruple 

 voints in Sk ; foi' /«' has four coincidences. Of the four coincidences 

 of jP^ four of the complementary points lie on rf ; with this conic 

 the curve ö has therefore 4 -]- 5 X 4 = 24 points in common. 

 Consequently the complemeniary curve is of order 12. 



The curves q\ which touch the plane ). in the points of the 

 curve of coincidences y\ intersect X moreover on the complementary 

 curve d"; so they form a surface of order 24, which passes eight 

 times through the curve (/^ 



This surface is intersected by a plane A' along a curve of order 

 24 with 5 octuple points Sk- As the curve of coincidences y'" lying 

 in A' has double points in Sk the two curves outside Sk have 

 24x6 — 5x8X2 = 64 points in common. Consequently there are 

 64 curves 9% touching two given planes. 



The surface A^ belonging to the straight line / intei-sects an 

 arbitrary plane (p along a curve (f^, which has 5 triple points on 

 q\ As the curve of coincidences 7:* lying in (p has 5 nodes on q^, 

 it intersects (f^ moreover in 9x6 — 5X3x2 = 24 points. From 

 this appears once more that the curves ()\ which touch a given 

 plane, form a surface of order 24. At the same time, the fact that 

 the complementary curve is of order 12, is confirmed. 



Chemistry. — "Equilibria in ternary systems". XII. By Prof. 



SCHREINEMAKERS. 



We have seen in the previous communication that the saturation- 

 curve under its own \apour-pressure of the temperature 7^n (the 

 point of maximumtemperature of the binary system F -{- L -^ G) 

 is either a point [fig. 5 (XI)J or a curve [fig. 6 (XI)]. We shall now 

 examine this case more in detail. 



dii 

 If we calculate — for this curve in the point II from (6) and (7) 

 d.v 



(XI), then we tind an infinitely great value. The curve going through 



H in fig. 6 (XI) and the curve disappearing in H of tigure 5 (XI) 



come in contact, therefore, in H with the side BC. Now we take 



a temperature somewhat lower than T//. The saturationcurve under 



its own vapour-pressure terminates then in two points n and h 



situated on different sides of and very close to H. [n and h in fig. 



4 — 6 (XI) may be imagined very close to ^.j As the saturationcurve 



