16 
ME. W. SPOTTISWOODE ON AN EXTENDED POEM OF 
And so also generally, 
V, 
i-1 
0 
. 0 
0 
, or = 
Vi 
i-1 
0 . 
. 0 
0 
V 
V. 
i-2 
. 0 
0 
V 
Vi 
i-2 . 
. 0 
0 
V. 
V 
V: 
. 0 
0 
Vi 
V 
Vi . 
. 0 
0 
V. 
V 
Vi 
. Vi 
1 
V 
Vi 
V . 
Vi 
1 
V 
V, 
V 
. V 
Vi 
Vi 
V 
Vi . 
V 
Vi 
according as ^ is even or odd. And the expression for H-'Si is to be fonried from this 
by adding the following ^ rows (each consisting of places), "viz. 
V *+i— 1 0 . . 0 0 
V V 2 .. 0 0 
V V V . • 0 0 
and repeating the first column of Hi j times, so as to complete the square. 
By means of these formulse the expression 
7i 
=(«, h ..IS, H,)” 
may be exhibited as a function of V? Vi- 
There is, however, another way in which this transformation may be effected. Let 
a, j3 ; Ki, (3,; .. be the roots of the equation 
(«, ^,..IS, 
solved with respect to S : S,. Then 
{a, h, CIS, S,)^=(«,S+/3,H,)(ccH+(3S0 
=(VV,I«./30(«H+/3H0-a,(aH+/3H,)-/3,(«H.+i3S) 
=(VViI«ii30(VV.I«^)-(VV,Vl«,i8,I«i8). 
{ahcdX^B.f 
= (“2S+/32H,)(a,H+/3iHi)(aS+|3H) 
= («2V+/32Vj(«,H+/3,H0(«S+j3H,) 
-2«2 («,H +i3 S+/3H,) 
— ( 3 ^ («iH +/3iHj)(a H,+^S) 
Similarly, 
