( ^^21 ) 



where a (Icnotes an ;irl»ili"ai-v fiiiictioii and // and r the fuiietlonS 

 of j) and 7, derived t'roni I he e(|nati(»ns (4j. 



Ill ihe partk'idar ease that the timctioii 6* satisfies the two nieiiihers 

 of the equation (3) 8eparately, we have to distinguish two cases 

 acrordiug- to the double sign in 



dn _ &,., ± \/ 8,,^ - e,, d,, djj ^ ^^ 



The general integral can be written in both cases respectively 

 1 



q — v ], . (5) 



6^ = (2 ir ,1" _ 4 ^/ r/ + 4 t/ rp u' v) p — 2 Lu ;,'" ip v)h p d u 



Avhere g = (j [ii) has the same meaning as before, whilst /' and r 

 represent two new arbitrary functions of the arguments jilaced after 

 these symbols. 



Ill these cases the two intermediate integrals are 



^ =F i/ ('^ ii" — 9') — ■'■(" ''"' + ^'") = ^' (")' 



.'/ + 'V' [" {'■ =F ;/")] = 5 [" (t- =F ƒ/)], 

 the yalnes of u and v being expi'essed in [> and 7 with the aid of 

 the formulae (5). 



The condition (3) appears, although in a diiferent shape, already 

 in the excellent dissertation of J. Valyi (Klausenburg 1880). 



Mathematics. — "^Thr siiKjidaritu's of the. focal ciirrc of a phinc 

 l/eneral curve tOKchiiu/ the Ibic at mfinitji a t/'iucs ai/d pa.is/i/g 8 

 times through each of the iwaginary curie points at i)ifinitg.'' 

 By Dr. W. A. Versluys. (Communicated by Prof. P. H. Scholtk). 



(Communicated in the meeting of January 30, 1904). 



In "Verhandeling" 5 of the "Kon. Ak. v. W." at Amsterdam Vol. 

 VIII, I have deduced some formulae expressing the singularities of 

 the focal developable and of the focal curve in function of the sin- 

 gularities of a plane curve having no particular position. 



In a similar way it is possible to deduce the following formulae 

 exjU'essing the singularities of the focal de\'elopable and of the focal 

 curve of a i)laiie curve touching tiie line at infinity times and 



