348 DR THOMAS MUIR ON 
and learn (1) that one of the equations is not independent of the others, (2) that the 
B’s are connected by the equation = 
2, — 2B, + Bs = T(Bo- 283+ 4), 
and (3) that o is expressible in terms of any three of the 6’s, for example, 
o = —26,+9B,—5f,. 
The conclusion thus is that in the continuant with which we started we can retain © 
any three of the 6’s, and express in terms of these the fourth 8, all the ys,andp,q, 
r, $,—thus obtaining a function of fowr variables which is resolvable into linear 
factors. 
(8) Had the determinant operated upon been of the sixth order, we should still 
have found s = —28,+98,—58, and the first four 6’s connected by the same equation 
as in the preceding case, but there would have been a fresh equation of condition E 
connecting the second set of four consecutive (Sis. wal, 
By — 2B, + By = 3(Bs— 28, + Bs) - = 
Similarly the case of the seventh order would be found to differ from that of the sixth 3 
merely in having the additional equation 
5(B, fa 26,+ Bs) = 11(B, a 265 ats Bs) 3 
and SO on. { 
As the result of all this we therefore affirm that — Jf the continuant 
a 2(n—-1)P, ins! Game) 2 
ny a+p (n — 2)B, Ae Spee ead 
(n+ 1)y5 a+q (SB). Blt ovens 
: (1 + 2)y5 Gate Ne i cre 
be resolvable into linear factors by means of the set of multipliers 
tok ] AF ee IESE page rs 
a) (Aes. Ol RG aed Uae 
te Ge QR iowean 
lh: See eee 
re Eee 
then (1) every four consecutive 8's are connected by a linear relation, viz. 
1-(B, — 2B, + Bs) = 7-(By— 283+ B4), 
3-(By — 2B, + B,) = 9-(B, — 28,4 Bs), 
5-(Bs - 28, + Bs) = 11-(B,- 285+ Bo), 
thus making all the B’s expressible in terms of any three; (2) all the y’s are expressible 
im terms of the same three 8's because of the fact that Bmy+%m= —28,+98,—58; for 
all values of m; and (3) p,q,zr,... are also so expressible because the sum of the 
elements of any row of the continuant is-constant. . Oi 
