HISTORICAL SKETCH. 
11 
In the same year Rehorovsky 9 published the eleventhic and the 
twelfthic, with the aid of an additional formula giving the sum of the 
constituents in a row or column; for example, it gives the sum of the 
constituents in 3 2 2 2 1 as 
( 1N2+2+l ( 2 + 2+l) ! 
^ ’ 2! 2! 1! 
-30 
(14) 
In the same year Durfee showed that the scheme of arrangement in a 
triangle is always possible in two ways, and in the American Journal of 
Mathematics he published the twelfthic arranged in one of them, checking 
his results by the Cayley-Betti law of symmetry (10) as well as by the law 
for the sum of the constituents in a row or column (14). 
Major MacMahon, in his article in The Proceedings of the London 
Mathematical Society for 1883-84 (vol. 15), gives several symmetric func¬ 
tion formulas, four of which will be here noted. The first formula enables 
one to write down all those terms of the highest degree in the n’s of a sym¬ 
metric function from a function in a table of lower weight. For example: 
[3 2 2 2 ] = a\a 2 — 2 a b a 3 a 2 — a 5 a 4 a 1 + 5a| + 2 a 6 a| -(- 3a 6 a 3 a L — 9cc 6 a; 4 
—7a 7 a 2 a l + 6a 7 a 3 + 7a 8 a\ + a 8 a 2 — lSagftj + 15a 10 
Therefore 
[3 3 2 2 ] = ( — 1) 13-10 (a5« 3 — 2a 6 a 4 a 3 — a 6 a 5 a 2 + + 2a 7 a| + 3a 7 a 4 a 2 
— §a 7 a b ax — 7a 8 a 3 a 2 + §a 8 a i a l + 7cc 9 a 2 + — \ba w a 2 a l + 15a n a 2 ) 
+ terms of lower degrees. 
(15) 
Another of his formulas enables us to compute any function from a 
single function of lower weight by means of a differential operator V _ r . 
In particular 
F _ 1 = cr 1 
d 
da 0 
+ 2 a 2 
d 
da 7 
+ 3c 3 
d 
dan 
+ 
and by means of his equation 
V- r ty] = ( — l) K x[_x l r] (16) 
we may calculate, for example, [41] from [4]. We first write in the 
homogeneous form 
[4] = — 4a 4 «o + 4a 3 a; 1 ao + 2a|«o — 4a 2 a 2 a 0 + cc\ 
Then 
[41] = —\V _ l[4 ] 
= — 7 [ — 12a 4 a 1 ao + 8a 3 ala 0 + 4 ala 7 a 0 — 4 a 2 a\ + 8a 3 a 2 al — 1 
+ 8a 2 a\ 4- I2a 3 a 2 al — 12 a 3 a\a 0 + 16a 4 a,rto — 20a, 5 a ( ] ] 
or 
[41] = 5 a 5 — a 4 a! — 5a 3 a 2 + a 3 a\ + 3ala 7 — a 2 a^ 
•Rehorovsky: Wien Denkschriften der Kaiserliche Akademie der Wissenschaften in Wien Mathe- 
matisch-Naturwissenschaft-liche Classe, vol. 46, Vienna, 1882. 
