399 
1909-10.] Dr Muir on the Theory of Bigradients, 
and forming therefrom the series of functions 
y 
V' 
xV 
V 
xY' 
V' 
a 2 V 
xY 
y 
x 2 Y 
xV' 
a 
a 
, 
a 
a 
a 
a 
b 
a 
V 
a 
b 
a 
b' 
a 
c 
b 
G 
1) 
G 
b 
a 
c 
b' 
d 
G 
b 
d' 
G 
e 
d 
G 
e 
d! 
etc., which he denotes by — F p F 2 , — F s , . . . . , lie affirms that there is a 
homogeneous linear relation connecting every consecutive three of the 
latter functions,* namely, 
P^Fs + (aP^ + P^ + P^F, + P 2 2 Fj = 0, 
P 2 2 F 4 + (*P 2 P 3 + P 2 P>PVP 3 )F 3 + P 3 2 F 2 - 0, 
where by P l5 P 2 , P 3 , . . . . are meant the determinants 
a a 
b b' 
a 
a b' a' 
b c 
c d! 
V 
a 
b a 
c b 
d c 
e d 
f e 
a 
b’ a 
a c 
b d' 
c e 
d f 
and by P], P' 2 , P' 3 , . 
altering the last rows into 
the determinants got from P p P 2 , P 3 , . . . . by 
c c' ; e d e d ! ; y f e g' f e ; 
No proof is given of the relations ; indeed, after pointing out that they 
involve the proposition that the first and last of three consecutive functions 
are of opposite sign for every value of x that makes the intermediate 
function vanish, Cayley adds : “ Je n’ai pas encore reussi a demontrer dans 
toute la generality l’equation identique d’ou depend cette propriete.” 
The case which brings him into closer contact with Sturm, namely, 
where V' is the differential-quotient of V, is dealt with in some detail. 
Heilermann, [H.] (1852, December). 
[Independente Berechnung der Sturm ’schen Reste. Crelles Journ xlviii. 
pp. 190-206.] 
The subject of this paper is of course closely connected with that of the 
author’s previous work (1845). Like Cayley, he begins with two functions 
that are unrelated, and subsequently passes to the special case where the 
* The signs require verification. 
