662 



etc. and from (7) the ratios between the increments. In (10) and (11) 

 the sign d indicates now that we have to differentiate with respect 

 to all variables, except to P and T. We now have ii^ -\- J equations 

 between ?2* — 1 ratios of the if increments; consequently relations 

 must exist between the coefficients. It follows again from (10) and 

 (11) that .I'l?/! . . . have to satisfy (2), which is also the case here. (7), 

 however, must also be satisfied. When we compare (6) with (10), 

 then we see in connection with (Jl) that we may put 

 fij zzz a(l{x)^ = ad{x)^ = . . . 

 H, = ad(y), = ad{y), = . . . 



When we substitute those values in (7) then we find: - 



A.d'Z, -f- X,d"-Z^ -f = . . . = :^^ ?.d'Z= ... (1 2) 



[Prof. W. VAN DER WouDE lias drawn ray attention to the fact 

 that we are able to express generally in a determinant the conditi- 

 ons in order that (7), (10), and (11) may be satisfied. We then obtain 

 the same determinant as that one to which attention is drawn in 

 the previous communication. We have, however, to add to this 

 determinant still a series, which follows from (7). The conditions 

 sought for are then, that all determinants which may be formed 

 from this, are zero]. 



The turning-line Er has, therefore, a singular point when (12) 

 is satisfied, then it follows from (8) : 



— :s'(aF). AP-f :e (xH) . i\T -^ ^ :s {).d*Z) -\- ... . = o 



dP 

 Consequently for — the same values as in (9). AP and A7Mtself 



are values of the second order ; when we express them in one of 

 the others, e. g. in A^\ then we may write 



AP = a Aa.\' -^ b A.i\' -{- ... 



LT=a,Lx,-' + ^,A.i',='-f . .. 



It is apparent from (9) that a: a^ must be = ^ (^//) : ^ (AF). 

 We now give to A.i'i the two opposite values -\- IS and — S; in 

 the one case we go along curve Er starting from the singular point 

 towards the one side, in the other case towards the other side of 

 the curve. It follows for Lx^ =z -\- S that : 



LP=aS-' -\-bS' -^ ... and LT = a,S' ^ b,S' -^ . . . (13) 

 for Lx, = — S that : 



hP=aS' —bS' ^ ... and A7 = a,S' — b,S' + . . . (14) 



Consequently AP and A 7' have the same sign in (13) and (14), 

 curve Eji consists, therefore, in the vicinity of the singular point 

 S of two branches Sri and Sv with the common tangent Siv; the 



