1902-3.] 
Dr Muir on a Special Circulant. 
547 
A Special Circulant considered by Catalan. 
By Thomas Muir, LL.D. 
(MS. received June 29, 1903. Read July 6, 1903.) 
(1) In his “Recherches sur les determinants,” published in the 
Bulletin of the Royal Academy of Belgium in 1846, Catalan gives 
a proposition in regard to circulants of the type 
C(-l, i, i, i, .... ) 
C( - 1 , - 1 , 1 , 1 , . . . . ) 
C( — 1 , — 1 , — 1 , 1, .... ) 
which deserves some special attention, as there can be little doubt 
that it is seriously misleading. Unfortunately, too, it has actually 
already misled to a considerable extent, the proposition having 
been reproduced by quite a number of writers without any care 
being taken to verify it. Even the very latest text-book, that 
written by Professor Pascal of Pavia and translated into German 
by H. Leitzmann, gives importance to the proposition, and by 
omitting the so-called proof diminishes the chance of detection of 
error. 
(2) Having proved satisfactorily in § 17 that 
C(-l,l, 1,1, . . . )„ = (n - 2)( - 2) w_1 
Catalan proceeds in § 1 8 as follows : — 
“Soient, plus generalement, les equations 
- x 1 - x. 2 - • • 
■ * — X P + X P + + • • 
1 • + X n = U j 
00-^ tXjC) " " 
+ 
+ 
5s 
I 
sf 1 
1 
" +x n = u 2 
- aq — x 2 - • ■ 
• +X p + X p+x + • • 
• -x th m u n 
dans lesquelles nous supposerons p<\n\ si p etait egal a 
ces equations seraient evidemment indeterminees. Pour 
obtenir le determinant A, formons la somme de ces equations 
et retranchons-les deux et deux : nous aurons 
