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XIII.—Continuants resolvable into Linear Factors. By Thomas Muir, LL.D. 
(MS. received August 22, 1904. Read November 7, 1904. Issued separately January 13, 1905,) 
(1) It is known that a continuant whose three diagonals are formed of certain 
equidifferent progressions is resolvable into linear factors, the earliest specimens 
placed on record being those of SytvesreR and Parnvin.* The object of the present 
paper is to show that there are continuants of quite a different type which are also 
so resolvable, and to expound a general mode of investigating the subject. 
(2) The continuant of the n” order whose main diagonal is 
G@, G@+2-12-¢, a+2:22-¢, “a+2-d%c¢, .... . 
and whose minor diagonals care 
2(n-1)b, (m-—2)(b+c), (m—3)(b+ 2c), 
m(b—c), (m+1)(b—2c), (n+2)(b-3c),..... 
is equal to the product of the n factors 
{a+2(n—1)d} - {a+ 2(n — 3)b + 2(2n — 3)c} 
- {a+2(nm—5)b + 4(2n - 5)e} 
- {a+ 2(n—7)b+ 6(2n — 7)c} 
. CEG Li aah = 2)1c} : ' mee ey) 
Taking for the purposes of illustrative proof the case where n= 5, viz. 
a 2.46 ; : : | 
B(b-c) at+2c 3(b+c) ; : 
6(b-2c) at+8c 2(b+2c) ; | 
7(b-3c) a+18e 1(b+3¢c) | 
8(b-—4c) a+32c © 
and performing the operation 
col, + col, + col, + col, + col, 
we find we can remove the factor a+ 8b and write the cofactor in the form 
a-—8b+2c 3(b+c) 
—2b-12¢ a+8ce 2(b+2c) 
— 8b 7(b-3c) a+18e 64+38¢ 
| — 8b : 8(b-— 4c) a+32c 
Performing now on this cofactor the operation 
col, + 4 col, + 9 col, + 16 col, 
* (SyivestEr, J.J.] “Théoreme sur les déterminants de M. Sylvester,’ Nouv. Annales de Math., xiii. p. 305. 
PAaINvVIN, . “Sur un certain systeme d’équations linéaires,” Journ. de Liouville, 2° séx., iii. pp. 41-46. 
Moir, THomas. “ Factorizable Continuants,” Trans. S. Afr. Philos. Soc., xiv. pp. 29-33. 
TRANS. ROY, SOC. EDIN., VOL. XLI. PART II. (NO. 138). 51 
