344 DR THOMAS MUIR ON 
we obtain 
(a+4b4+14¢) | 1 364 8¢ 
4 a+8e 2b+4¢ 
9 7Tb-2le at+18e 4c 
16 . 8b-32¢ a+32c |, 
of which the determinant factor is reducible to the three-line form 
a— l26—4¢c 2b+4¢ 
—206-48c a+18e 6+3c 
—48b—48c 8b—32c¢ a+32c 
The next operation 
col, + 6 col, + 20 col, 
gives in like manner 
(a+20c) | 1 2b+4e 
6 a+18e 6+3¢ 
20 8b-32¢ a+32c 
or 
(a+20c) | a—12b-6¢e b+ 3c 
—32b-112c¢ a+32c 
and finally the operation 
col, + 8 col, 
enables us to change this two-line determinant into 
(a—4b4+18c) 1 b+386e 
8 a+t32¢ 
or 
(a— 4b + 18c)(a - 8b+ 8c). 
The desired result thus is 
(a+ 8b)(a+ 4b + 14c)(a + 20c)(a— 4b + 18c)(a - 8b + Be). 
(3) The continuant of the n® order whose main diagonal rs 
a, @+2(1-3)c, a+2(2-4)e, a+2(3-5)c,... 
and whose minor diagonals are 
(n—1)b, (n-2)(b+e), (m-3)(b+2c),.... 
(n+2)(b-3c), (n+3)(b-4c), (n+4)(6-5c), . 
is equal to the product of the n factors 
{a+2(n—1)b}, 
{a+2(n — 3)b+ 2(2n—-1)e}, 
{a+2(n—-5)b+4(2n—- 3)c} , 
Plot b+ ae Ne} ’ . (i 
This is established by proceeding in the same way as in § 2, the sets of column-— 
multipliers now being 
is Qi” con Ones Tees AL; +, Misael ene 
LAO, 20 eee | I Ale Oe eG 
lela eo taf instead of 19.5 20 
esha Res 
