304 Proceedings of the Royal Society of Edinburgh. [Sess. 
XVII. — The Theory of Recurrent Determinants in the Historical 
Order of Development up to 1860. By Thomas Muir, LL.D. 
(MS. received June 13, 1910. Read July 4, 1910.) 
Like Wronskians, and for the same reason, recurrents were at first dealt 
with among “Miscellaneous Special Forms” : their previous history is thus 
to be found under Wronski 1812, Scherk 1825, and Schweins 1825 in the 
chapter so entitled. ( History , i. pp. 472-474, 478-481.) 
The name is quite recent, having been first proposed by E. Pascal in 
1907 in a paper published in the Rendiconti .... 1st. Lombardo, (2) xl. 
pp. 293-305. 
Spottiswoode, W. (1853, August). 
[Elementary theorems relating to determinants. Second edition, . . . 
Crelles Journ., li. pp. 209-271, 328-381.] 
In the last chapter or section (§ xi.), which is headed “ Miscellaneous 
Instances of Determinants,” Spottiswoode gives (pp. 373-374) an expression 
for the n th differential-quotient of u/v in terms of the n ih and lower differ- 
ential-quotients of u and of v. The first four cases are 
( 
uY 
— 
V V 
V 2 , 
(~y = - 
V 
2 V 
V ~ 3 , 
V 
V J 
u u 
\vj 
V 
V 
v" 
u 
u 
u" 
V 
3v 
v-\ 
(y r - - 
V 
4v' 
V 
2v 3v" 
V 
3v 
6r” 
V 
' V 
V- 
v'" 
V 
2F 
3v” 
±v" 
u 
u 
u 
u" 
V 
V 
u 
V 
v r ' 
v iv 
u 
u 
u" 
u"' 
u iv 
where the arithmetical coefficients appearing in the elements of the 
determinants are those of the binomial theorem. 
He also notes that any binary quantic may be expressed as a determi- 
nant : thus a 0 x n + a x x n ~ l y + x 2 a n ~h/ 2 + .... is written by him in the form 
a 0 
a 2 a s . . 
a n _ j a n 
y 
X 
y 
X 
y x . . 
?/ X 
