( 323 ) 
NOTE ON PALMSTE^M'S GENERALISATION 
OF LAME'S EQUATION. 
By Sir Thomas Muir, LL.D. 
(1) In 1903 I communicated to the South African Philosophical Society* 
a paper on " Factorisable Continuants," in which a theorem of considerable 
generality was established, including as simple instances two isolated results 
due to Sylvester and Painvin. The fact that the theorem did not also 
include a similar result given by Heine in a paper on Lame's functions 
led to further investigation, the outcome of which was communicated to 
the Eoyal Society of Edinburgh! in 1904. Unfortunately I was not then 
aware that Heine's paper had been thrown a little out of date by later 
work, a very noteworthy generalisation of Lame's equation having been 
made by Palmstr^m in 1897 and been published in the year-book of the 
Bergen Museum. X This publication having recently come by accident into 
my hand I hasten to show how the method employed in 1904 in connection 
with Heine's result may be equally effectively used on the generalisation 
arrived at by Palmstr0m. 
(2) The determinant on which the generalisation turns is no longer 
a continuant, there being in it four contiguous non-zero diagonals instead 
of three. As Palmstr^m writes it the order is the (n + 1)*^, the main 
diagonal consists of B's, the upper minor diagonal is 
(271 -f ^1 - 2)/3, 2(2# + t,- 3)/3, 3(2/^ + -4)/3, 
the two lower minor diagonals 
7i(Sn + -3)/3, (n-l)(Sn + - 6)/3, (n-2)(Sn + - 9)f3, 
27i(n - l)/3, 2{7i — l){n-2)^, 2(n - 2)(7i - 3)/3, ..... 
and the factors of the result 
B + {2^^^+ {t,-2)7i}^, 
B + {2/^2 + _ 8)/, - 2t^ + 6}^, 
* Transactions, vol. xv, pp. 29-33. 
t Transactions, vol. xli, pp. 343-358. 
X " Sur nne generalisation de I'equation de Lame/' Bergens Museums Aarhog' 
for 1897, No. xi, 9 pp. 
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