457 ) 



'''P "^ ~S [(2/— '^><r-4(4y'-3x') (2y'-3x') + lüy'>^, . . (48) 



The formulae (12), (15) and (16) of tlic first dcscriplive pai-l of tiiis 

 j)aper may be derived from tliese formulae by means of tiie re\'erse 

 transformation into the original ifvsurfjice witli the aid of the foi'inulae 

 (26). Applying equation (31) we may also derive formula (17). In 

 the course of this we get first at formula (23), which is given at the 

 end of the descriptive part as serving also for the calculation for 

 coexisting phases. The last statement might be objected lo, l)ecause 

 for those phases not >/ but v'" is a quantity of the same order as 



a^andt'; but this objection loses its force when we observe that in — - 



dv' 



no term occurs with v'^ alone. 



18. From these formulae (46), (47) and (48) follows now imme- 

 diately the classification of the plaitpoints accoixling to the eight cases 

 and all the particularities of the corresponding graphical representation, 

 as described in § 2 — 9. It is only necessary to say a few words 

 about the constructioji of the cubic border curve. 



(2y'— 3x')=' — 4(4y'— 3x')(2y'-3x') + lGy'=iU. . . (49) 



A closer examination of this equation shows, namely, that the 

 curve possesses a double point, i.e. the point at infinity oftiie straight 

 line 2 y' — 3 x'^O. A simple parameter representation is therefore 

 possible and it is really obtained by putting 



2y'— 3x' — *• (50) 



from which follows : 



,.='_46-(.,-|.2y') -f lüy'i=0 (51) 



hence: 



y = X = .... (52) 



' 8(5—2) 12(s-2) ^ ^ 



