Fuly 30, 1885] 
NATURE 
393 
holm. The commissioners having presented a report of their 
work to his Majesty, he has graciously signified his approval of 
the following final propositions of theirs. 
Having taken into consideration the questions which from 
different points of view equally engage the attention of analysts, 
and the solution of which would be of the greatest interest for 
the progress of science, the commission: respectfully proposes to 
his Majesty to award the prize to the best memoir on one of the 
following subjects :— 
1. A system being given of a number whatever of particles 
attracting one another mutually according to Newton’s law, it is 
proposed, on the assumption that there never takes place an 
impact of two particles, to expand the coordinates of each 
particle in a series proceeding according to some known func- 
tions of time and converging uniformly for any space of time. 
It seems that this problem, the solution of which will con- 
siderably enlarge our knowledge with regard to the system of 
the universe, might be solved by means of the analytical re- 
sources at our present disposition; this may at least be fairly 
supposed, because shortly before his death Lejeune-Dirichlet 
communicated to a friend of his, a mathematician, that he had 
discovered a method of integrating the differential equations of 
mechanics, and that he had succeeded, by applying this method, 
to demonstrate the stability of our planetary system in an abso- 
lutely strict manner. Unfortunately we know nothing about 
this method except that the starting-point for its discovery seems 
to have been the theory of infinitely small oscillations.1 It may, 
however, be supposed almost with certainty that this method 
was not based on long and complicated calculations, but on the 
development of a simple fundamental idea, which one may 
reasonably hope to find again by means of earnest and persever- 
ing study. 
However, in case no one should succeed in solving the pro- 
posed problem within the period of the competition, the prize 
might be awarded to a work in which some other problem of 
mechanics is treated in the indicated manner and completely 
solved. 
(2) Mr. Fuchs has demonstrated in several of his memoirs ? 
that there exist uniform functions of two variables which, by 
their mode of generation, are connected with the ultra-elliptical 
functions, but are more general than these, and which would 
probably acquire great importance for analysis, if their theory 
were further developed. 
It is proposed to obtain in an explicit form those functions 
whose existence has been proved by Mr. Fuchs, in a sufficiently 
general case, so as to allow of an insight into and study of their 
most essential properties. 
(3) A study of the functions defined by a sufficiently general 
differential equation of the first order, the first member of which 
is a rational integral function with respect to the variable, the 
function, and its first differential coefficient. 
Mr. Briot and Mr. Bouquet have opened the way for such a 
study by their memoir on this subject (owrnal de Ecole poly- 
technique, cahier 36, pp. 133-198). But mathematicians 
acquainted with the results attained by these authors know also 
that their work has not by any means exhausted the difficult and 
important subject which they have first treated. It seems 
probable that, if fresh inquiries were to be undertaken in the 
same direction, they might lead to theorems of high interest for 
analysis. 
(4) It is well known how much light has been thrown on the 
general theory of algebraic equations by the study of the special 
functions to which the division of the circle into equal parts 
and the division of the argument of the elliptic functions by a 
whole number lead up. That remarkable transcendant which is 
obtained by expressing the module of an elliptic function by the 
quotient of the periods leads likewise to the modulary equations, 
that have been the origin of entirely new notions and highly im- 
portant results, as the solution of equations of the fifth degree. 
t See p. 35 of the Panegyric on Lejeune-Dirichlet by Kummer, ‘‘ Ab- 
handlungen der K. Akademie der Wissenschaften zu Berlin,” 1860. 
2 These memoirs are to be found in—(r) ‘‘ Nachrichten von der K. 
Gesellschaft der Wissenschaften zu Gottingen,” February, 1880, p. 170; (2) 
Borchardt’s ‘‘ Journal,” Bd. 89, p. 251 (a translation of this memoir is to be 
found in the ‘‘ Bulletin” of Mr. Darboux, 2me. série, t. iv.); (3) ‘‘ Nach- 
richten von der K. Gesellschaft der Wissenschaften zu Géttingen,” June, 
1880, p. 445 (translated into French in the “Bulletin” of Mr. Darboux. 
2me série, t.iv.): (4) Borchardt’s ‘‘ Journal,” Bd. 90, p. 71 (also in the ‘‘ Bull- 
etin” of Mr. Darboux, 2me série, t. iv.) ; (5) ‘‘ Abhandlungen der K. Gesell- 
schaft der Wissenschaften zu Gottingen,” 1881 (“ Bulletin” of Mr. Darboux, 
t. v.); (6) ‘‘ Sitzungsberichte der K. Akademie der Wissenschaften zu Ber- 
lin,” 1883, i. p. 507; (7) The memoir of M. Fuchs published in Borchardt’s 
* Journal,” Bd. 76, p. 177, has also some bearings on the memoirs quoted. 
But this transcendant is but the first term, a particular case and 
that the simplest one of an infinite series of new functions intro- 
duced into science by Mr, Poincaré under the name of 
“*fonctions fuchsiennes,” and szccessfully applied by him to the 
integration of lineary differential equations of any order. These 
functions, which accordingly have a vé/e of manifest importance 
in analysis, have not as yet been considered from an algebraical 
point of view as the transcendant of the theory of elliptic 
functions of which they are the generalisation. 
It is proposed to fill up this gap and to arrive at new equations 
analogous to the modulary equations by studying, though it were 
only in a particular case, the formation and properties of the 
algebraic relations that connect two ‘‘ fonctions fuchsiennes” 
when they have a group in common. 
Tn case none of the memoirs tendered for competition on any 
of the subjects proposed above should be deemed worthy of the 
prize, this may be adjudged to a memoir sent in for competition 
that contains a complete solution of an important question of the 
theory of functions other than those proposed by the Com- 
mission. 
The memoirs offered for competition should be furnished with 
an epigraph and, besides, with the author’s name and place of 
residence in a sealed cover, and directed to the chief editor of the 
Acta Mathematica before June 1, 1888. 
The memoir to which his Majesty shall be pleased to award 
the prize as well as that or those memoirs which may be con- 
sidered by the Commission worthy of an honorary mention, will 
be inserted in the Acta Mathematica, nor can any of them be 
previously published. 
The memoirs may be written in any language that the author 
chooses, but as the members of the Commission belong to three 
different nations the author ought to subjoin a French transla- 
tion to his original memoir, in case it is not written in French. 
If such a translation is not subjoined the author must allow the 
Commission to have one made for their own use. 
THE EpIToRS OF THE ‘‘ ACTA MATHEMATICA” 
DR. PERKIN ON THE COAL-TAR 
COLOURS * 
TAKING a precedent from some of those who have occupied 
this chair before me, I have selected for my few remarks 
to-day the subject in relation to Technical Chemistry, with 
which I have been personally connected—namely, the colouring 
matters produced from coal-tar products, with some of the 
lessons its development appears to me to teach is in connection 
with industrial chemistry. Sir Frederick Abel, in his address in 
1883, when speaking of the history of gunpowder, said that “It 
is one of the most remarkable features connected with the 
history of gunpowder, that until the last quarter of a century no 
radical changes should have been introduced into the manu- 
facture and modes of applying this, the first known practically 
useful explosive agent.” It appears to me that this is more or 
less true of all the older industries, which resulted simply from 
experiment and observation without any other basis to work from. 
They have had long histories in which little progress has been 
made, but of late years, owing to our advanced and rapidly in- 
creasing scientific knowledge, they are undergoing great, and in 
many cases radical, changes. : , ip 
The coal-tar colour industry stands in a very different position 
to our older ones. It has a sharply-defined origin, and a very 
short history dating back only to 1856, and it is not yet twenty- 
nine years since the date of the first patent. It is an industry 
which has been founded on scientific discovery, and has developed 
side by side with it, being in fact a most important handmaid to 
research, which in its turn has repaid it by new discoveries. At 
the date of its introduction yery little was known of the chemistry 
of colouring matters ; they were always found difficult bodies to 
investigate, and when produced in reaction were generally re- 
garded as secondary products, and every endeavour was made to 
get rid of them so that the other products associated with them 
might be examined ; but now, owing to the very extended study 
which has been made of these bodies, on account of this in- 
dustry, and the relationships which have been found to exist 
between the colour of the compounds and the chemical constitu- 
tion, it is possible with more or less certainty to predict the 
colour a compound will have before it is produced, and the 
means which can be used to modify it. 
1} The President’s address, Institute of Chemistry. 
