410 PRIZE QUESTIONS. 
PRIZE QUESTIONS OF THE PHYSICAL-MATHEMATICAL CLASS OF 
THE ROYAL PRUSSIAN ACADEMY OF SCIENCES FOR 
THE YEARS 1866-7: 
PUBLISHED AT THE PUBLIC SESSION UPON THE LEIBNITZ FESTIVAL, JULY 7, 1864. 
I. 
(From the Steiner Legacy. ) 
In one of the monthly reports of the Academy, for January, 1856, as well as 
in an essay published in vol. LIIT of Crell’s Journal, Steiner communicated 
a series of the fundamental properties of the surfaces of the third order, arriving 
at the means for a purely geometrical theory of the same. 
The Academy desires that this remarkable labor of the distinguished geometer 
be carried out further, according to its, synthetic methods, and be perfected in 
several essential points. 
For this purpose it would be necessary, first, to give proofs, which are for 
the most part merely indicated or even omitted for the principal propositions ; 
but the investigation must be extended beyond the cases treated by Steiner to 
those surfaces in which the elements serving for the geometrical construction 
are in part imaginary. Beside this, the Academy would regard it as an import- 
ant perfection of Steiner’s theory, although not absolutely indispensable, if a 
complete characterizing of the different kinds of cones in space formed by the 
intersection of two of the surfaces here alluded to were given. 
Essays contending for the prize may be written in the German, Latin, or 
French languages, and the time during which they can be received will expire 
upon March 1, 1866. Every essay must be accompanied by a motto, which 
must also be placed upon the outside of a sealed envelope containing the author’s 
name. 
The prize of six hundred thalers will be conferred at the public session of the 
Leibnitz festival in the month of July, 1866. 
1 
The theory of the elliptic and Abel’s functions, which has already enabled 
the solution of problems in nearly every branch of mathematics, and for which 
former means for the analysis were insuflicient, is without doubt susceptible of 
numerous additional applications, and, therefore, the Academy offers the follow- 
ing prize questions : 
“ For any important problem in algebra, arithmetic, integral calculus, geom- 
etry, mechanics, and mathematical physics, which can be perfectly solved by 
aid of the elliptic or Abel’s transcendentals.” 
The essays may, at the will of the author, be written in the German, Latin, 
or French languages, and the time of their reception expires upon the lst of 
March, 1867. Every essay must be accompanied with a motto, which must 
also be written upon the outside of a sealed envelope containing the author’s 
name. 2 
The prize of one hundred ducats will be conferred at the public session on 
the Leibnitz festival in July, 1867. 
