8 THOMAS MUIR ON SOME TRANSFORMATIONS CONNECTING 
The corresponding relation for two rows of jive elements is 
b, |b,e,{ 0 0 0 
a, |ay¢,| |c¢d,| 0 0 
[bs | |,65| [eds | [dyes] =] O |b; | |B,d5| | ares | 
0. 0 (de,| lee] @ 
0 0 0 leds | 
oO 
(3’) 
i) 
= 
and from this and (3) the general theorem is apparent. 
Again, taking the complementary of (2) with respect to |m,a,.b,¢,d,|, and, 
for the sake of comparison with (3), changing the suffixes 4, 5 into 1, 4 respec- 
tively, we have 
| myb,| |O,e,| 9 0 
|mya4| |Ay¢4| |qa,| 0 
O fads] 0,4.) 
0 On [Oyegh & 
m, ab, |be,| |qd,|= 
(4) 
where the right-hand member differs from that of (3) only in the first and last 
columns. 
§ 5. By taking the complementary of (3) with respect to | @,b,c,d,| we should 
of course return to (1); by taking the complementary, however, with respect 
to | a,b,¢,d,e, | (or | a,b.¢,0,e, fg... |) we obtain a new result, viz., 
Ayloses| Agdse,| 0 0 
| Dycodlse, | | bo@s¢5| | aab5¢, | 0 
0 | Cb3@5 | | ACs, | | dadaeyey | 
0 0 = |a,dae,| | aab,a,e, | 
| @b.¢,d,e, |= = 15) 
| obg¢5| | adses| | e03¢, | 
which it is interesting to compare with both (1) and (1’). 
§ 6. If in (2) we write 4, ¢, d, e for a, 6, ¢, d respectively, a for m, and 2, 3, 4, 5 
for 1, 2, 3, 4 respectively, the left-hand member of (2) will be a principal minor 
of the left-hand member of (1’) and the first determinant on the right-hand of 
