# TransBots

Jon Awbrey jawbrey at oakland.edu
Fri Sep 21 20:16:56 CEST 2001

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here are two very useful translator programs:

http://tr.voila.fr/

http://babel.altavista.com/raging/translate.dyn

automatic regards,

jon awbrey

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Antreas P. Hatzipolakis wrote:
>
> Dear Neil,
>
> I am copying the following formuale from the paper:
>
> Luigi Berzolari: Allgemeine Theorie der Ho"heren Ebenen Algebraischen Kurven.
> Encyclopa"die der Mathematischen Wissenschaften mit Einschluss ihrer
> Anwendungen. Band III_2. Heft 3.
> Leipzig, Druck und Verlag von B.G. Teubner, 1906. SS. 313 - 455.
>
> In p. 341:
>
>      Hesse'sche Kurve | Steiner'sche Kurve  | Cayley'sche Kurve
> __|__________________________________________________________________________
>   |
> N |3(n-2)       | 3(n-2)^2     |    3(n-2)(5n-11)
>   |
> N'|3(n-2)(3n-7) | 3(n-1)(n-2)  |    3(n-1)(n-2)
>   |
> D |0  | 3(n-2)(n-3)(3n^2-9)(3n^2-9n-5)/2 | 9(n-2)(5n-13)(5n^2-19n+16)/2
>   |
> D'|27(n-1)(n-2)(n-3)(3n-8)/2 | 3(n-2)(n-3)(3n^2-3n-8)/2 | 9(n-2)^2(n^2-2n-1)/2
>   |
> R |0            |  12(n-2)(n-3) | 18(n-2)(2n-5)
>   |
> R'|9(n-2)(3n-8) | 3(n-2)(4n-9)  |    0
>
> -------------
>
> In p. 352:
>
> n(n+1)(n+2)/3
>
> n(n+1)(n+2)(2n^3 + 6n^2 + 7n -3)/36
>
> n(n^2 - 1)(n+2)(2n^5 + 14n^4 + 49n^3 + 91n^2 + 90n + 18)/324
>
> -------------
>
> In p. 353:
>
> n(n+1)(n+2)/6
>
> n(n^2 - 1)(n + 2)(n^2 + 4n + 6)/72
>
> 280C(n+4,9) + 280C(n+4,8) + 105C(n+3,7) + 77C(n+3,6) + 43C(n+2,5) -
> - 16(n+2,4) + 20C(n+1,3) - [n(n^2 - 1)(n^2 - 4)(n-3)/360]
>
> Sorry, I have no time to translate (with the help of dictionary, since
> I don't speak German) the German text explaining the meaning
> of the formulae.
>
> Greetings from Athens
>
> Antreas
>
> PS: There are no typos in my typing of the formulae.
> (I corrected them before sending the message!)

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