332 Proceedings of the Royal Society of Edinburgh. [Sess. 
During the period now occupying us, the earliest suggestion on the 
subject occurs in 1844 in Grelles Journ ., xxviii. pp. 191-192, the determinant 
being of the 8th order, and the specialisation consisting in having a zero in 
the places 
16,17,18,27,28,31,38,41,42 
83,82,81,72,71, 68,61,58,57. 
The proposer of the problem, so far as it appears, received no satisfaction. 
The next instance occurred to Sylvester, who in 1855 having hit upon a 
peculiar determinant whose terms were all positive, ascertained the number 
of them by evaluating a special circulant. (See Quart. Journ. of Math., i. 
pp. 42-56, or our notice of it given above.) The third instance, like the 
first, arose as a problem and remained long unsolved. It appeared in 1858 
in the Nouv. Annates de Math., xvii. p. 262, under the heading “ Question 
445,” the requirement being to find the number of terms remaining in the 
case of a determinant of the n^ order when all those terms have been 
deleted which contain two or more diagonal elements. The fourth instance 
— which is more closely connected with the third than might at first appear 
— was the actual but incidental determination by Cayley in 1859 of the 
number of terms in a zero-axial determinant whose order is not greater 
than the 7 th . The numbers found were 9, 44, 265, for the 4 th , 5 th , 6 th orders 
respectively. (See Quart. Journ. of Math., iii. pp. 275-277, or our notice of 
it given above.) 
LIST OF AUTHORS 
whose writings are herein dealt with. 
(a) 1855. Borchardt, C. W. 
PAGE 
. 312 
1856. JOACHIMSTHAL, F. 
. 312 
1857. Cayley, A. 
. 313 
(£) 1854. Hermite, Ch. 
. 314 
1855. Hermite, Ch. 
. 314 
1857. Rubini, R. 
. 315 
1859. Clebsch, A. 
. 315 
( 7l ) 1854. Cayley, A. 
. 316 
1858. Cayley, A. 
. 317 
1864. Sardi, C. . 
. 319 
( 72 ) 1855. Sylvester, J. J. 
. 320 
1859. Borchardt, C. W. 
. 321 
(5) 1845. Cayley, A. . 
. 323 
PAGE 
(5) 1852. Tissot, A 324 
1854. Bazin, H 325 
1855. Brioschi, Fr. . . . 325 
1855. Hermite, Ch. . . 327 
1858. Zehfuss, Gr. 327 
1859. Cayley, A. ... 328 
1859. Hirst, T. A. . . 329 
1859. Cayley, A. ... 330 
(*) 1825. Scherk . . .331 
1844. (Anon.) .... 332 
1855. Sylvester, J. J. . . 332 
1858. (Anon.) . . .332 
1859. Cayley, A 332 
(Issued separately March 27, 1911.) 
