1887 .] 
Dr T. Muir on Simple Alternants. 
433 
14. On the Quotient of a Simple Alternant by the Difference- 
Product of the Variables. By Thomas Muir, M.A., LL.D. 
On 17th March 1879 I gave to the Royal Society of Edinburgh 
an account of some researches on the subject of Alternants. A 
short descriptive abstract of the communication was published in 
the Proceedings , vol. x. pp. 102, 103, the concluding paragraph of 
which is as follows : — 
“ Vow it is well known that the alternant whose indices are 
in order 0, 1, 2, 3, . . . . is equal to the difference-product 
of its variables. In regard to every other alternant it is 
evident that it must contain the said difference-product as a 
factor, but what the co-factor should be is not so readily seen. 
In particular cases, doubtless, it can be found without much 
difficulty, but a general method of obtaining it has hitherto 
been a desideratum. Such a general method the author has 
discovered along with a number of less important results bear- 
ing on the same special form of determinant.” 
These results were never published, as, very shortly after the date 
referred to, the volume of the Giornale di Matematica for 1878 
arrived, and I found that what I had looked upon as my most 
important theorem, was given and proved by Garbieri at the begin- 
ning of the volume. 
Vow, however, that fresh interest in the subject has been 
awakened by the very able papers of Professor Woolsey Johnson, 
which have recently appeared in the American Journal of Mathe- 
matics (vii. pp. 345-352, 380-388) and Quarterly Journal of 
Mathematics (xxi. pp. 217-224), I desire to resuscitate my method. 
Happily there is no question of priority involved. The methods are 
diverse, and the superiority lies entirely with Professor Johnson’s. 
His reduction-theorem 
<*(0, P c l) = H 3 - 2 ,*>-i + <Ac . a( 0, p - 1, q - 2) , 
his discovery of the existence of the symmetric function Q which 
is the difference of a{ 0, p, q, r) and a( 1, p, q , r— 1), and his mode 
of determining the said function, are things of which I had not 
dreamed, and which I consider the most notable results added to 
the theory of alternants for a great many years. 
