( 519 ) 



Out of (19) now follows that the j)laitpoiiit 

 is a node of the curve of contact and at the 

 same time that the line OP itself is one of 

 the tangents. To investigate this curve of 

 contact further we write (2) (after having put 

 there p = 0, r, =r 0, c^ = () and d^ = Oj in 

 the form : 

 Fig. 10. 



A.x' + n.vy + C.v' + I).v'i/ + E.vy' + Ff + . . . = 0. . (20j 

 To satisfy this by 



we must have Bk-\-F=^0. F'rom this ensues now, as J3 = '2 fl.,(/ 

 and F = 4: qe^,: 



X- y' 



2i 



(21) 



thus reproducing the equation (12) of the binodal line in the vicinity 

 of the })laitpoint. 



Therefore the curve of contact coincides in the vicinity of the 

 plaitpoint with the binodal line. 



This coincidence does not hold for the higher terms, as is natural 

 and as is shown still more clearly by the following. 



We put namel}^ : 



X := ky"^ -f- my^ 



and we substitute this value in ('20). As will immediately become 

 evident, we must include in (20) still the term y\ We write for it Gif. 

 So we find : 



{Bk + F)y' + {Ak-' 4- Bm + Ek + G) ?/^ -f- . . . = 0. 



From this ensues: 



Ak^ -f Ek + G 





B 



(22) 



Now follows out of (2): A = d,q — c,, B=2d,q, E=3t\q—'2(/^. 

 If we calculate the coefficient G of //^ in (2) we find -. 



^ = (/s P + Ó M — 3 e,), 

 SO here, as p =z : 



G = ^/\q — 3e,. 



So we find, if we put for the curve of contact /n — m,.-. 



{'^^-d.'I) -T7-r(- 2^^3 + ^^.'l) ^ + 3., - 5/;^ 



d. 



m,- := 



2cL 



. (23) 



35* 



