1907-8.] 
Dr Muir on General Determinants. 
697 
chapters of a text-book planned on a fairly large scale. The theorems, 
twenty-three in number, are carefully enunciated and are printed in italics; 
but, although the proofs receive every attention, it is very doubtful whether 
the object aimed at was to any extent accomplished. There is at any rate 
nothing sufficiently fresh in the treatment to warrant attention here. 
Zehfuss, G. (1858). 
[Ueber die Auflbsung der linearen endlichen Differenzengleichungen 
mit variabeln Coefficienten. Zeitschrift f. Math. u. Phys., iii. 
pp. 175-177.] 
His solution suggests to Zehfuss the remark (p. 177) that every 
determinant can be expressed as a multiple integral. It will suffice to 
give the result in the case of a determinant of the 4th order. Denoting 
cos 27 t6 + J — 1 sin 27rd by l 9 , and putting P for 
F F F I s (I s - F) (I s - F) (I s - l a ) (F - F) (F- l a ) (F - F) 
and Q for 
(i a jl- a +&F-0 + Cl l-y + d F- 5 ) 
x (a 2 l _2a + b 2 \~W + c 2 1-2v + cU- 2fi ) 
x (a 3 F 3a + b ? l~W + c 3 1 _3 v + c7 3 1-35) 
x (a 4 l" 4a + b 4 V^ + c 4 l~*v + d 4 l~ 4S ) , 
Zehfuss says that 
■ ■■■ i 
2±«iVs rf 4 = Iff) PQ JadfldydS. 
0 
He does not, however, note in passing that 
p - 
F 
F 
D 
F 
! 2a 
12/3 
l 2 v 
125 
13a piS 
l 3 v 
14a 
F/3 
1 4 V 
145 
Zehfuss, G. : Mainardi, G. (1858). 
[Ueber die Zeichen der einzelnen Glieder einer Determinante. Zeit- 
schrift /. Math. u. Phys., iii. pp. 249-250.] 
[Una regola per attribuire il segno proprio ad ogni parte di un 
determinante numerico. Atti . . . Istituto Lombardo (Milano), 
i. pp. 105-106.] 
Neither of these communications is of importance. Zehfuss, using the 
recurrent law of formation and giving “ derangement the very opposite of 
