55 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
and subtracting b 0 times the first from a 0 times the second, and dividing 
the result by x, we obtain 
a 0 
a i I 
+ 
a 0 ^2 
X + 
I a o a s 
\ 
b Q ^2 
1 b 0 b 3 
or / 2 say ; 
and by subtracting | ajb x | times the second from b 0 times this third series 
and dividing by x there results 
a o a Y 
■ \ 
« 2 
\ 
+ 
tt 0 CLy 
• \ \ 
x + 
CIiq CL-^ CL 4 
• ^1 ^3 
%? + • • • , or / 8 say ; 
\ ^1 ^3 
\ \ h 4, 
and so on. The outcome is 
a 0 4- a x x + a 2 x 2 + • • • • 6 1 q ^ 
b 0 + bjX + b 2 x 2 + • • • 0 o - -j- _ O 3 P q^q^ x 
1 °2 ~ -q— - 
C7 3 
where 0 O , 6 V 0 2 , • • • • are the first terms of f 0 , f v f 2 . . . respectively. 
Yent^jols, . (1877). 
[Sur un probleme comprenant la theorie de l’elimination. Gomptes- 
Rendus . . . Acad, des Sci. (Paris), lxxxiv. pp. 546-549.] 
Ventdjols’ subject would have been much better described by Lemonnier’s 
title of 1875. In substance nothing fresh is brought forward. 
Dickson, J. D. H. (1877). 
[A class of determinants. Trans. Roy. 80c. Edin ., xxviii. pp. 625-631.] 
[The numerical calculation of a class of determinants, and a continued 
fraction. Proc. London Math. Soc., x. pp. 226-228.] 
The determinants here considered are the bigradients dealt with by 
Heilermann (1845) and Muir (1876). They also arise in the same connection. 
Mansion, P. (1878). 
[Sur l’elimination. Bulletin . . . Acad . . . . de Belgique, xlvi. pp. 899— 
903.] 
What is interesting here is Mansion’s mode of obtaining the evanescent 
bigradient array that results from the existence of common roots. The 
equations being 
A(aj) = a 0 cc 5 + ... +a 5 = 0, B(x) =* + . . . + & 4 = 0 
