390 TRANSACTIONS OF SECTION A. 
5. On Ideal Numbers. By J. H. Grace, M.A., F.R.S. 
6. On a Correspondence in the Theory of the Partition of Numbers. 
By Major P. A. MacManon, F.R.S. 
7. A Continuant of Order N+I which is expressible as the Product 
of N+I Facters. By Professor W. H. Merzier, Ph.D. 
The theorem of this paper may be expressed as follows. The continuant :— 
Bi aa. 0 0 0 
ma(a—B) r—ap 2aa 0 0 
0 (n—1)a(a—B) 7 —2aBp 3aa 0 
0 0 (n—2)a(a—B) r—3aa 0 
: : : 
: . . *—(2u—1)aB naa 
0 ; a(a—B) r—nap 
= {r+na(a—B)} {7 +na(a—B) —2aa+aB} {r+na(a—B) —4aa + 2ap} 
nf Arennay ae Ss. o P8 . (1) 
It will be observed here that each factor differs from the preceding by the 
quantity «8 —2aa. 
If in this theorem we put a=8=1 it reduces to 
7 a 0 0 0 
n(a-—1) r—1 2a 0 0 
0 (w—1)(a-1) r—2 3a 0 
0 e . 
= {r+n(a—1)} {r+n(a—1-2a41}...fr—na}. . © 2 « (2) 
a theorem due to Painyin.! 
If in (1) B=2a all the factors become equal to (*—zaa), and the theorem 
becomes 
r aa 0 . 0 
—naa —2aa 2aa ° 0 
0 —(n—l)aa + —4aa 3a 0 
0 j i — ae r—2naa 
= (r —naa)r*! . . r ° . (3) 
Tf in (3) we put aa=1 it reduces to a special case of (2) noted by Painvin 
(loco citato). 
Tf in (1) we put 8 =0 it reduces to 
r aa 0 0 0 
naa r 2aa 0 0 
0 (n—1)aa r 3aa 0 
0 , f r naa 
0 e * aa r 
= (7? —n?a%a*) (1? — (m —1)?a70").. ee eeee eee (7? — aa") or 
= (7? —n?a7a*) (1? — (1 —1)?a70")..0.. ese eee (7? —2?a’?a?)r 
according as x is even or odd, the factors coming together in pairs . . (4) 
If in (4) we put aa=1 it reduces to a theorem given by Sylvester.? 
1 Journ. de Liowville, iii. 2 Now. Annales de Math., xiii. 
