1900-1901.] Dr Muir on a Peculiar Set of Linear Equations. 257 
in the expansion is easily determined. In the case of the 4th 
order it is 
(2 3 - 3) + (3 + 6 + 3) i.e. 17; 
in the case of the 5 th order it is 
(2 4 - 4) + (4 + 12 + 12 + 4), i.e. 44; 
and for the n th order it is 
- 1 -jT=T) + { (n - 1).+ (n - l)(n - 2) + (»-l)C„_ 2 , 2 + («-l)C„_ 2 , + . . 
which = 2 n_1 - (n - 1) + (n - l)2 n-2 , 
= (n+l)2 n ~ 2 -(n-l). 
(14) The D and any two of the N’s associated with such a set 
of equations are connected by a simple relation, the only other 
magnitudes involved being the elements in the place 1,1 of the 
two H’s. For, taking any two of the equations, say the second 
and third of a set of four, and subtracting, we have 
(i-s' 2 )* 2 -(i-9' 3 K + 
and therefore by substituting for x 2 and x 3 
(! * ~ (l-9 , 3) N fc?4>0i>£/ 2 ) = (92-93) D (9vMs>ffi)- 
(15) Returning now to § 1 we see that the four expressions 
obtained for any one of the six quantities, x 1 , x 2 , x 3 , g x , g 2 , g 3 , 
give rise to six equations, four of which involve only four of the 
said quantities. Thus from the expressions for x 1 we have 
<£(01 j 02 j 9z) ~ 4 > (9 i 5 x 2 > 9z) j 
<£(0i> 02>03) = <£(0i > 9 2 > x z ) > 
= <l>(9l> X 2> X s)> 
$(9 1 5 92 5 ^3) = ^(^1 5 X 2 > X i) i 
each of which involves only four quantities, while the others 
<K0 i>02 >0 s) = <K9i, x 2> x s)’ 
<£(0 i , ^2 > 0s) = <i>(9 1 , x s) > 
involve five each. Taking the first of these six, which involves 
9n 92 j 03 > x 2 ’ an d wr 14ing it in the form 
-^(01 J 02 » 03) -^(01) X 2 > 03) — ^”(01 > X 2 > 03^ -^(01 ’ 02 j 03) = 
VOL. XXIII. 
E 
