: 12 THOMAS MUIR ON SOME TRANSFORMATIONS CONNECTING 
Taking the complementary of (9) we have 
| E 2 0 0 
& 23 E. 3 
0 & Ns E 4 
0 0 im 15 
a =. —_ .. : : (12) 
LASS) leafs) Leds fs| lestaSs| | oecsta Ss! | rests Ss | 
and thus again, by division, there comes 
Lesbstst fs! teed vst , gr Bey (13) 
ay 9202 
3 nae: EG 
1s 
—the complementary of (10). 
If the values of the €’s and @s be compared, it will be seen that there is 
something abnormal in the second line. This is not due to an error; the 
factor |a,b,c,d,f,| must appear in one of the two elements &, ¢, and may 
appear in either, but not in both. 
§ 10. In (10) we have a continued fraction found as an expression for the 
quotient of a determinant by a differential coefficient of it with respect to one 
of the elements. This was obtained from the two distinct theorems, (8) and 
(9), by division, &c. Owing; however, to a peculiarity of (8), we do not need 
the assistance of (9) to obtain such a result. Taking, instead, the identity 
corresponding to (8) for the case of the determinant |@,b,c,d,|, we have from 
it and (8), by division, 
my, % O 0 
Gq, iy aa 
% Y. % O 
ee | Dyeods | =|0 x 3 wt 0 |: ie Yara, ae 
| oC, | 0. 2 a) ea 
eo Ya, Me 
O % Ye 
0” 0 OF 2.7 
and, continuants being unaltered in substance by having the order of the 
elements in their diagonals reversed, there thus results 
| aybateds fs | | debsl4 | | dicots | =y,—- Uses 
| a,b.cod, | 5 an HERES 
Ya 
(14) 
Ley 
Ys - 5% 
_ 
Ze 1 
There is evidently no theorem corresponding to (6) or (7), as this corre- 
sponds to (10), the continuants employed in finding the former having a 
symmetry with respect to both diagonals. 
