1905-6.] Dr Muir on the Theory of Alternants. 
389 
simplifiable into a multiple of D : so that there is obtained the 
second result 
A 2 = 2* n(M “ 1) • sin a Q sin oq . . . sin a n • D . 
The two results are Prouhet’s, who set them for proof by others. 
Salmon (1859). 
[Lessons introductory to the modern higher algebra, xii + 147 pp., 
Dublin.] 
The determinant form of the difference-product and the 
determinants in s 0 , Sj , . . . are given, but merely as illustrative 
examples of the general subject. 
Liouville (1846).* 
[Sur une classe d’equations du premier degre. Journ. (de 
Liouville ) de Math ., xi. pp. 466-467 ; or Nouv. Annales 
de Math., vi. pp. 129-131; or Archiv d. Math. u. Phys., 
xxii. pp. 226-228.] 
The set of equations referred to is that dealt with by Binet in 
1837. Chelini and Liouville arrived at a new solution, much 
simpler than Binet’s, and related to that used by Murphy in 1832 
in solving other sets of linear equations. 
LIST OF AUTHORS 
whose writings are herein dealt with. 
1832. 
Murphy 
page 
. 357 
1854. 
Brioschi 
PAGE 
. 370 
1837. 
Binet . 
. 358 
1 1854. 
JOACHIMSTHAL 
. 371 
1841. 
Haedenkamp 
. 361 
1854. 
Brioschi 
. 373 
1845. 
Borchardt 
. 362 
1855. 
Borchardt . 
. 374 
1845. 
Rosenhain 
. 362 
1856. 
Prouhet 
. 376 
1845. 
Sturm . 
. 363 
1856. 
ScHEIBNER . 
. 377 
1846. 
Terquem 
. 363- 
1856. 
JOACHIMSTHAL 
. 377 
1846. 
Cayley 
. 364 
1857. 
Bellavitis . 
. 381 
1846. 
Chelini 
. 366 
1857. 
Betti . 
. 383 
1846. 
Liouville 
. 389 
1857. 
Baltzer 
. 384 
1847. 
Grunert 
. 366 
1857. 
Brioschi 
. 387 
1849. 
Rosenhain 
. 367 
1857. 
Prouhet 
. 388 
1850. 
Mainardi 
. 368 
1859. 
Salmon 
. 389 
1853. 
Cayley 
, 369 
* This should have been inserted under but not separate from ‘ 1 Chelini 
(1846),” the proper joint heading being “ Chelini and Liouville (1846).” 
{Issued separately November 16, 1906.) 
