224 Proceedings of the Boyal Society of Edinburgh. [Sess. 
There is a similar set beginning with 
Ei ~ c n~ ~ a n + j 
E 3 + E 4 
+ d 9 o + d 
23 ~ 24 
16. Taking the equalities occupying the diagonal of the square of 
15 we obtain 
El E 2 d" E 3 ~ *01 ~~ ^22 C 33 = ~ 2(^22 d - E 3 d - El) 
E 3 El E 2 El E 2 E 3 ’ 
which on substituting for 3u n the elements of the last three rows of | d u 
becomes 
Ei d - E 2 d - ^ 
33 
C 11 C 22 C 33 ~~ (E 2 d* E 3 d" E 4 ) d" El d" El El > 
= GLyy — {dyy + ^ 2 ^ ^33 ^ El) > 
or, say, 
a 
n 
li 
+ 2% 2Bi~o 
a tlieorem established by Pascal for the purpose of applying it to a special 
form of skew determinant.* 
17. In the same way it is shown that 
E 2 C 12 l 
d" El ~ C 21 ) 
^13 ~ C 13 l 
+ El “ C 31 J 
E 3 _ f> 23 l 
+ E 2 ~ 6 *32 i 
= — Ei + E2 d~ E3 ~ E4 5 
= — Ei d - E2 — E33 + E4 ’ 
= — El — E 2 d~ E 3 d~ E 4 J 
and hence by addition 
c / , Ei ) ( JL c 2 L c n ) — °Ei d~ E 2 + E 3 + E 4 
Further, as we know from § 14 that 
it follows that 
2Ei- 4 «ii ^L/Ei > 
(2>-2>n)-(2>-2E+(2^ 
2 rf n ) _ En -/i.) . 
* Pascal, E., Sui determinant! gobbi a matrici. Rendic. . . . Accad. . . . di Napoli (3), 
xxiii, pp. 40-47. 
Notwithstanding this paper it would still seem a most natural and simple way of treating 
Voigt’s determinant (1) to make the odd-numbered rows occupy the 1st, 2nd, 3rd, . . . places 
and to reverse the order of the remaining rows, (2) to do the same in the case of the columns, 
(3) to multiply the displaced rows by J - 1 and the disjilaced columns by - J — 1, and (4) 
to use Zehfuss’ theorem on the centrosymmetric determinant thus resulting. 
