1905-6.] Dr Muir on the Theory of Alternants. 
357 
The Theory of Alternants in the Historical Order of 
Development up to 1860. By Thomas Muir, LL.D. 
(MS. received July 2, 1906. Read July 18, 1906.) 
My last communication in reference to the history of alternants 
dealt with the period 1795-1841 ( Proc . Roy. Soc. Edin., xxiii. 
pp. 93-132). The present paper continues the history up to the 
year 1860, but in addition contains an account of three writings 
belonging to the previous period, namely, by Murphy (1832), 
Binet (1837), and Haedenkamp (1841). 
Murphy (1832, Nov.). 
[On elimination between an indefinite number of unknown 
quantities. Transactions Cambridge Philos. Soc., v. pp. 
65-76.] 
Murphy’s third example in illustration of his method is the set 
of equations 
1 “I" X-^ "I - + 
1 + 2x 1 + 2% 2 + 
1 + 3aq + 3 2 x 2 + 
1 + nx 1 + n 2 x 2 4- . 
+ x n — 0 
+ 2 n x n = 0 
+ 3 n x n — 0 
+ n n x n = 0 
which he neatly and easily solves, giving the value of x m , and 
thus in effect evaluating 
1 
1 
1 . . . 
. 1 
1 . . 
. . 1 
1 
1 . . . 
. 1 
1 
2 
2 2 . . . 
9m-l 
2^+i 
e yn 
2 
2 2 . . . 
. 2 n 
<-r 
1 
3 
3 2 . . . 
. 3 m_1 
3 m+1 . . 
. . 3 n 
4- 
3 
3 2 . . . 
. 3 n 
1 
n 
n 2 . . . 
. n m ~ x 
% m+1 . . 
. . n n 
n 
n 2 . . . 
. n n 
His connection with our subject is thus seen to be similar to 
Prony’s. 
It should be carefully noted, however, in passing, that Prony’s 
