394 Proceedings of Royal Society of Edinburgh. [sess. 
the following is given (p. 375), being the fourth of the said 
instances : — 
“ Let 1 , q , i 2 , . . . , i n be the n+ \ roots of the equation 
x n+1 -l = 0, 
then, whatever be the values of A , A 1 , A 2 , . . . , A n 
= ( A + A 1 + . . . + A n )( A + i x A 1 + . . . + i\ A n ) 
• (A + f n A T + . . . + i n A n ). } 
A 
A, • ■ 
, . A„ 
A, 
A 2 . . 
.. A t 
A n 
A,.. 
■ • A n _x 
No word of proof is added: probably the result was reached by 
Sylvester’s “ dialytic ” method of elimination. But however this 
may be, it should be noted that resolvability into linear factors 
soon came to be looked on as the fundamental property of the 
circulant. 
It has to be noted that Spottiswoode makes a slip in omitting 
the sign-factor (- l)^ (n_1) from the right-hand member; and that 
he writes his determinant in such a way as to have it per- 
symmetrie with respect to the principal diagonal, whereas Catalan 
wrote his so as to have it persyinmetric with respect to the 
secondary diagonal. Putting C' for the functional symbol in 
the former case we have 
C (a 1} a 2 , . .., O = (-lp-^-^.C \a li a 2i ..., a n ) . 
If therefore Spottiswoode had followed Catalan’s mode of writing, 
his result would have been strictly accurate. 
Cremona (1856). 
[Intorno ad un teorema di Abel. Annali di sci. mat. e fis. y 
vii. pp. 99-105.] 
To prove the theorem of Abel referred to in the title, Cremona 
starts by establishing three lemmas, the first of which is Spottis- 
woode’s theorem regarding circulants. Taking any n quantities 
«] , « 2 » * ' • ’ a n-\ 
and denoting 
$q -f- a^cf -}- a^a 2 r -f- . . . + a n _i (P ^ by 0 r 
