110 Proceedings of Royal Society of Edinburgh. [sess. 
i.e., product of the differences of a , b, c , Sylvester denoted the other, 
viz., the determinant 
l a a 2 
1 b b 2 
1 c c 2 , 
by £P D(abc), £ being his sign for multiplication according to the 
law a r . a s = a r + s . Using this notation he rediscovered, as has also 
already been seen, Schweins’ theorem regarding the multiplication 
of the alternant 
| aWd 4 . . . .| 
by such symmetric functions as 
(a + b + c + . . . ), (ab + ac + . . . + be + . . , ), 
his form of statement being 
£(S r (abc ...l). £PD(0 abc ...l) = £_ r PD(0 abo . . . Z), 
where £_ r implies that after ‘zeta-ic’ multiplication the subscripts 
.are all to be diminished by r. 
His attempted generalisation of this theorem has likewise been 
spoken of, its validity, however, being left undecided upon. Instead 
of the multiplier S r (abc . . .1) he proposed to take any symmetric 
function whatever of a, b, c, . . ., Z, — or, rather, any function 
ivliatever followed by any symmetric function. This would have 
been a most noteworthy extension which Schweins had not fore- 
seen, but unfortunately there are grave doubts as to the truth of 
it, — indeed, one may go so far as to say that there would be no 
doubt whatever about the author’s inaccuracy, were it not that 
there are doubts also as to • his meaning. By way of test let us 
take the case where the multiplier of | a 1 b 2 c d d 4: \ is the symmetric 
function %a 2 bc. From later work* it is known that 
| flWdl . = | aW| - 3|aWd 5 | , 
whereas, according to Sylvester, there ought to be on the left only 
one alternant. Now although we know that Sylvester was in the 
habit of making guesses, and that these guesses though often 
brilliant were not always so,f it would be next to impossible to 
* See Muir, “Theory of Determinants,” p. 176 (1882). 
t See Crelle's Journal, lxxxix. pp. 82-85. 
