( 27 ) 
NOTE ON EECUREENTS EESOLYABLE INTO A SEQUENCE 
OE ODD INTEGERS. 
By Sir Thomas Muir, LL.D. 
(1) In an examination of the properties of the series 
1, 3, 10, 35, 126, , 
the r*'^ term of which is Cor—i, r—i, a determinant made its appearance which 
for the 4th order is 
1 
1 
2-3i 
1 
1 
4-2-52 
4-3^ 
1 
1 
6-4-2-73 
6-4-5, 
6-3^ 
1 
and has the value — 9 11-13. The fact that the value was the product of 
a sequence of odd integers led to the discovery that under another form it 
had in effect been already studied. The alternative form in the case of the 
4th order is 
1 
2 
31 
1 
4 
5o 
^1 
1 
6 
7;, 
5o 
3i 
1 
and is clearly the more convenient of the two. It was brought forward in 
1892 in a paper* connected with the theory of integers, where it had for a 
companion the determinant 
1 
-2 
3i 
1 
-4 
5o 
3i 
1 
-6 
5o 
3i 
1 
These for the n^^' order, which we may denote by R„ and R',,, their dis- 
coverer had quite skilfully evaluated, his results being 
R„ = (-1)^-^27^ + 1) (2n + 3) . . . {4>n - 3), 
R',, - (271 + 3) (2n + 5) . . . (U - 1). 
(2) The main object of the present paper is to establish a pair of much 
more general equalities, and to do so not by a mere extension of Reich's 
* Eeich, K., " Zur Theorie der quadratischen Ueste," Archiv d. Math. u. Physik. 
(2), xi, pp. 176-193. 
