( 60 ) 



it can only he identified with Koktkwkc/s e(iuati()n (2), wlien loy d' 

 can he expanded in a series. This can only hai)i)en when the diflfe- 

 i-ence lietween .r and .vj^i is so small that ,r — .rjiji is vanishingly 

 small with regard to .ry^,/. We i-emain thns in the immediate neigh- 

 hourhood of the j)laitj)oint '). In this ease we (ind that the equation 

 of the tf>surface can be brought into the form 



V/' = c, .."^ + </, .,-- -f- d, .v"-^ v" + ^3 .; ' r"^ + ., ,■ '^ -f . . . 



where, to the first a|»|)roximation 



^ ^\ RTjc I ^ l^m\ ^^ ^ _ J^ m^ 



1 1 RTk 1 m„^ 



1 m\, 1 ] 



2 nTi.vT,,i 2R^-T'j, 



1 



./■« — — r '"^" ^^^'- ')• 



5 



J lie expression 4 rj^^_ — t/., =z — -— — is always positne hence 



2 .VT,d 



ni„„ is always negatiye; it follows that the j)laitpoint on the if'sui-- 



face is always of the first kind '). 



Since ^Z» = when )n\^ -\- iri\)ii^^ = 0. the second special case 



of border cnrye and connodal Wuq treated by me ■•) agrees with the 



1) The following expansion can tlius be used to determine the coordinates of the 

 critical point of contact, (cf. Keesom I.e. p. 34;2). 



~) The expressions for c/3 and e-^ agree witli tho.se foiuid by Keesom (1. c. p. 'Ml). 



'^) See Kortewect, Wien. Ber., p. 1158. 



■*) Proc. Amsterdam 27 Sept. 11)02 p. 329. The reference to this special case 

 allows me to correct some mistakes in the formulae which are connected with 

 this and the preceding special cases. In Proc. of 28 June 1902 p, 207 line 2, 

 read : 



/ 1 m, ,m,. 1 'm.'K.m.,,^ _ 



P, - PTk = m„, - -- -ii-^ + - -ii- - s;\ 



and in Proc. 27 Sept. p. 328, Hne 12: 



P — Plk = —^ m,.^ — — \- -^ \{v — vTkY- 



7/1 7/1 



Further in the last Proceeding p. 329 line 9 for the coefficient of -^^— 



5 

 read 4 instead of — 



