1915-16.] The Theory of Circulants from 1880 to 1900. 151 
VI. — The Theory of Circulants from 1880 to 1900. 
By Sir Thomas Muir, LL.D., F.R.S. 
(MS. received November 8, 1915. Read December 20, 1915.) 
My last paper on the “Theory of Circulants” appeared four years ago 
(. Proc . Roy . Soc. Edin., xxxii, pp. 136-149), and brought up the history 
of the subject to 1880. I have now completed the research up to the 
close of the nineteenth century — the goal originally aimed at. It is con- 
sidered that from the beginning of the present century the indexing of 
mathematical writings is sufficiently full and detailed to make it com- 
paratively easy for a worker to follow up for himself the recent lines of 
development. 
Studnicka, F. J. (1880). 
[Ueber eine neue Determinanteneigenschaft. Sitzungsb. . . . Ges. d. 
Wiss. (Prag), pp. 50-54.] 
The circulant dealt with, namely, C(— 1, 1, 1, 1, . . . , 1) is a case of 
Sylvester’s of 1855 (Hist., ii, p. 407). 
Scott, R. F. (1880). 
[Note on a determinant theorem of Mr Glaisher’s. Quart. Journ. of 
Math., xvii, pp. 129-132.] 
Taking C(a, b, c, d, e, f) as an example of an even-ordered circulant, 
and performing the operations 
colj + col 4 , col 2 + col 5 , col 3 + col 6 , 
row 3 - rowj , row 4 - row 2 , row 5 - row 3 , 
Scott finds it equal to 
a + d 
b + e 
c+f 
a - d 
b-e 
e~f 
e+f 
ci + d 
b + e 
f~ c 
a — d 
b-e 
b + e 
c+f 
ci + d 
e-b 
f-° 
a — d 
The form of the second factor leads him to draw attention to what he calls 
a cyclically skew determinant, namely, 
a i 
CL<2 
« 3 • • 
• a n 
~a n 
a i 
• • 
• «n- 1 
~ a»-i 
- a n 
a Y . . 
• a „-2 
- a 2 
~a 3 
- ° 4 ■ • 
■ «i 
