1903 - 4 .] Dr Muir on the Theory of Continuants. 
147 
Painvin (1858, February). 
[Sur un certain systeme d’equations lineaires. Journ. de 
Liouville (2), iii. pp. 41-46.] 
The system of equations referred to in the title of Painvin’s 
paper had presented themselves to Liouville in the course of the 
research which led to his “ Memoire sur les transcendantes ellip- 
tiques ...” {Journ. de Liouville (1), v. pp. 441-464). Painvin’s 
reason for taking up the subject was his belief that one of 
Liouville’s results could be more simply arrived at by the use of 
determinants ; and in a few lines of introduction he succeeds in 
showing that the result in question can be viewed as merely the 
resolution of the determinant 
r a ...... 
n{a - 1) r - 1 2 a .... 
(rc--l)(a-l) r- 2 3 a ... 
{n - 2 ){a — 1) r - 3 , . . 
... r - n + 1 na 
... a - 1 r - n 
into factors. 
In explanation of the process followed the case of the fourth 
order 
r a 
3 {a - 1) r - 1 
2 a 
2(a-l) 
y 2 
3 a 
a - 1 
r — 3 
will suffice. Increasing each element of the first row by the corre- 
sponding elements of the other rows, — an operation which we may 
for the nonce symbolise by 
rowj + row 2 + row 3 + , 
— he removes the factor r + 3a - 3 and finds left the cofactor 
1111 
3(a - 1) r- 1 2 a 
2 (a - 1) r — 2 3 a 
a- 1 r - 3 
