398 
Proceedings of Royal Society of Edinburgh. 
[sess. 
a l 
a 2 
• 
• 
a 3 
« 4 
a l 
a 2 
a 3 
a 4 
or 
a 3 
«1 
a 2 
«4 
^2 
h 
b l 
C 1 
C 2 
C 3 
C 4 
C 3 
C 1 
C 2 
C 4 
the fourth is a case of the fifth : and the fifth is itself a case of a 
theorem ( C' ) of Desnanot’s. 
CATALAN (1839). 
[Sur la transformation des variables dans les integrates multiples. 
Memoir es couronnes par VAcademie royale . . . de Bruxelles , 
xiv. 2 me partie, 49 pp.] 
The first of the four parts into which Catalan’s memoir is divided 
hears the title “ Valeurs generates des inconnues dans les equations 
du premier degre , et pr opr ietes des denominateurs communs,” and in 
the introduction it is said to contain several remarkable new pro- 
perties of the functions called resultants by Laplace “ et connues 
aujourd’hui sous le nom de determinants” 
His method of dealing with the opening problem is to derive the 
solution of n equations with n unknowns from the solution of 
n- 1 equations with n— 1 unknowns ; or more definitely, to show 
that if the multipliers X v X 2 , X 3 necessary for the solution of the set 
of equations, 
a Y x Y + bpc 2 + cpc 3 — a x \ 
apC-^ + bpC 2 d" CpC 3 — a 2 V 
i + b 3 x 2 + c 3 x 3 = a 3 / , 
be the determinants of the systems 
a 2 b 2 a 3 b 3 a l b x 
a 3 b 3i a l a 2 b 2i 
then the multipliers X 15 A 2 , X 3 , A 4 necessary for the solution of the 
set 
oqaq + \x 2 + cpe 3 + dpc± = a 4 ' 
a 2 x x + b 2 x 2 + c 2 x 3 + d£x± = a 2 
d“ b 3 x 2 + c 3 x 3 + d 3 x^ = a 3 
+ b^x 2 + cpc 3 -1- dpc^ = a 4 _ 
