( 201 ) 



it is evideiil lliat llie lianiioiiic* curves of P willi ivs[»e('( lo llic 

 cnrves of (he cubic [)eiicil also foriu a sysleiii witli index two. 



For /;" i)assiiig tlirongl» P the curve Ji^ breal^s \\\) into the system 

 of the i)ohai' conic and the polar line of P with respect to that 

 cur\e which (ouch each other in P. 



As Jv' and Jt^ have in common the tangents out of P, being thus 

 of the same class, tlie harmonic cnrve has only then a node when 

 this is the case with the original cnrve. 



5. If with respect to a given /:'' we detei-mine on eacii riglit 

 line thi'ough P the poijits B^, B.^, B^ in such a way that />/ is 

 harmonically separated by .1/ from .1^ and /I/, we get as locus ot 

 tlie points JJ a curve of order .sac, A", with a threefold point in /^ 

 For, if Bj^ coincides with P, then A^ is one of the points of inter- 

 section of k'^ with the polar line of P and the reverse (see § i). 



As the points B correspond one by one to the points A, the cnrve 

 h'' is of the same genus as ^■^ so it has still 6 donble points or 

 cusps. This last is exclnded because in that case not a single tangent 

 conld be drawn from /-* to h\ whilst it is clear that the tangents 

 out of F to /c' also touch h'\ 



From the definition of A" follows immediately that this cur\e can 

 meet the curve k' only in the points of contact R of the above 

 mentioned six tangents: so in each point R they have three poijits 

 in common. The right line PR having in R two points in common 

 with /•', bnt thi-ee points with h\ R must be one of the six nodes 

 of h'' and BR one of the tangents in that nide. 



Chemistry. — " PtYp'iratlon of ci/clohcjuino!.'' By Prof. A. F. 



HOLLEMAN. 



The preparation of ketohexamethylene in somewhat large quantities 

 is one of the most lengthy operations, whatever known process 

 ma^' be used. 



Since, by means of the addition of hydrogen to benzene, by the 

 process of Sabatiek and Senderens, Iiexa-hy(h-ol)enzene has become 

 a readily accessible substance, it was thought adxisable to use this 

 as a starting point for the preparation of tlie said ketone by first 

 converting it into moiiochlorohexamethylene, converting this in the 

 usual manner into the corresponding alcohol and then oxidising this 

 to ketone by the process indicated by Baever. Mr. van üeh ï^aan 

 has tried, in my laboratory to realise this. 



