1896-97.] Relation between Permanents and Determinants. 135 
For the case of the 4th order it is 
+ + + + 4 + 
\a x bfodt\ - o^gc^l - t>o\a^c ? dt - c 3 - rf 4 |a d & 2 Co| + ja 1 6 2 | \ c z c h\ + \ a \ c ^\ p 2 ^ 4 | + |%d 4 j |& 2 c 3 | 
+ + ++ + + + + ++ 4 - 4 - 4-4- 4-4-1 
+ jVo^ 4 ! - - ^K 6 ^! - c z\ a A d i\ - + \ a A\ \ C ?M + | a i c a| |M 4 | + M 4 | | Vs! * 
where again the verification is easily made hy observing that the 
coefficients of a v a 2 , a 3 , are 
| 6.,c 3 cZ 4 I - | b 2 c z d A j — b ( },, c^d A | — c 3 ! b 2 d 4 1 - d 4 j & 2 c 3 1 + & 2 | c s d 4 I + c 3 | b 2 d 4 1 F d A | & 2 r 3 | 
+ +■+' + + + + + + + + + + + + + / ' 
F ! 6 2 C 3 d 4 i - I VA i - \\ L h d 4 I “ C sl h d 4 I — C h\ h 2 c 3 i + ! + c sl M 4 ! + ^4 | & 2 C S I ) ’ 
“ b i c s d 4 1 F ^3! 1 F d^ Vs 1 + &jj c 3 d" 4 j 1 
+ 1 V A j - c 3 | \d± | - t? 4 1 \c % | -- b x j c z d A j j ? 
I VA i F 6 2 | c A I - d,\ b x c 2 | F c 4 | b 2 d 4 1 ^ 
4 + + + 4-4- + + r 
+ 1 VA | - b 2 \ c :i d A 1 - d A \ 6 4 c 2 . j - Cjl&g^l J 
or 
- | VA j + b 2 \ c 2 d A | + Cj| b 2 d 4 j + d±\ b 2 e x | | 
+ + + + + - •+ + + / 
F 1 Vi d 4 I “ h 2 l C \ d 4 1 “ C ll b 2 d 4 t “ d 4\ h C l I j , 
- I b x c 2 d 2 1 - b 2 1 e x d ?) I - c 3 | b x d 2 ! + d x | V3 1 'j 
+ + + + + + + + r 
F ! VA I “ h\ C l d Z I ~ 0 8 | \ d 2 I - ^ll Vs I j 
or 
— ! b 2 c 2 d 2 j F b 2 1 c%d l | + c 3 j ! F d 2 1 & 2 c 3 1 ^ 
F | & 2 cA j - & 2 | cA j - e 3 l &A 1 - riy Vs 1 j 
and that hy the previous case each of these vanishes. 
3. For convenience in making the verifications the identity in 
these special cases has been written in two lines. For other pur- 
poses, however, it is better not to make this bisection, but to write 
as follows : — 
a^btyCod^ | 
| « x V ~ 2V A + \a^b 9 1 = 0, 
+ + + + 
| a-j&gCg i - 2«il ^2 C 3 ! + 2 a x b 9 \ c 3 - j ajfegCg | = 0, 
+ + + + + + 
2a x j b 2 c z d x | + 2 a Y h 2 1 . | c 8 c? 4 1 - 2 a i b 2 c ?> ' d 4 + ! | = 0. 
