332 Transactions of the Boyal Society of Soutli Africa. 
24 
234 
125 
14 
25 
235 
134 
34 
245 
12 
45 
123 
25 
134 
235 
35 
24 
135 
124 
25 
235 
25 
25 
245 
234 
24 
13 
235 
23 
345 
345 
12 
234 
15 
23 
134 
235 
14 
14 
145. 
Here the axisymmetry is first departed from in the case of the elenients 
in the places (9, 11), (11, 9). 
17. When the elements of fx are real we see from (X.) of § 7 that ju will 
simultaneously reach its row-by-row limit and its column-by-column limit 
when 
a^_h^ _c^ 
^^2 ^2 ^2 
JB„ Co 
a. 
B.~C. 
in other words, when the elements of jx are proportional to the correspond- 
ing elements of the adjugate determinant. This property, we know, is 
possessed by an orthogonant, the common ratio, in the case where the 
name orthogonant is strictly applicable, being the orthogonant itself. A 
set of interesting examples is to be found in the series of skew deter- 
minants" 
a h I 
h a\, 
a 
h 
c 
d 
-b 
a 
d 
- c 
- c 
-d 
a 
b 
-d 
c 
-b 
a 
a 
-b 
- c 
-d 
- e 
-f 
-9 
-h 
b 
a 
-d 
c 
-f 
e 
h 
-9 
c 
d 
a 
-b 
9 
■h 
e 
f 
d 
- c 
b 
a 
-h 
9 
-/ 
- e 
e 
f 
9 
h 
a 
b 
c 
d 
f 
■ e 
h 
9 
b 
a 
d 
c 
9 
-h 
- e 
f 
c 
-d 
a 
b 
h 
9 
-f 
e 
d 
c 
-b 
a 
* Another form of the second of the series, which looks essentially different, is not 
really so, the one 0', being obtainable from the other O2 by altering the signs of the 1st 
row and 1st column, and then changing columns into rows. As regards the third of the 
series, its first quarter is 0^, the quarter to the right is of the same type as O2 but with 
the signs of three rows changed, the last quarter is 0^, and the remaining quarter is of 
the same type as 0^ with the signs of one row changed : other forms, however, are 
obtainable, the first quarter being always 0^ or 0^, and all the other elements being 
determinable when the signs of the places (2, 5), (2, 7), (3, 5) are known. 
