GENERAL DETERMINANTS WITH CONTINUANTS. 11 
The theorem related to (8), as (2), or rather that form of (2) used in § 6, is 
related to (1’), is : 
Yo Ly 0 0 
22 Y3 X 
&3 Y4 Uy 
0 7 Ys 
| bess f; |=— . : : (9) 
Gy | Mgbg| | Mqbgeg| By | Byco| | Dycodls | 
where 2, Y, %2,... have the same signification as before. This may be obtained 
after the manner of (8), but (8) having been proved, (9) at once follows as the 
result of differentiation with respect to a,. 
From (8) and (9), by division, there comes 
| aybyColy fs | __ hey 
Meni ok) ng ten 3 @4) to avnioy edt 10) 
Yo Uaey 
ae eee 
Ys Ys 
an identity more notable than those of like kind previously given. 
§ 9. The result of taking the complementary of (8) is peculiar, the left-hand 
member remaining unchanged. We thus obtain still another expression for 
| @,6,¢,4,f;|, which would not readily have been lit upon otherwise, viz., 
Uhl E, 0 0 0 
a 72 E, 0 0 
os "3 &, 0 
g; Up E, 
0 im 1s : 
| a,b,¢.4, 7; | = ; : (11) 
lai fs| lasfs! leds fs | asl, f, | | desta 75 | | Ores 7s | 
where 
&,=|d,¢,d,f51, &=|4,¢,0,f5 |, 
a =| eds; | | abcd f; le | az, f, ie 
E,=| a,¢3¢, fs | | dfs | ) G—| bye, fs | | a, fel» 
&,=| ad, fs| fi» €,=|¢dsfs| as, 
and 
=| bests | 
=| %40,4,f5| 5 
I3=| Dyess fg | | boda f5|—| Oicada fs | | tsds Js» 
m=|Cdsf5| |e,f5l—lerds fs! lest5l, 
Ns=| Ay fs |ta—| rs [ts « 
