364 Proceedings of 'Royal Society of Edinburgh. [sess. 
we conclude therefrom that the equation 
+ Xb Y + jiuq 4- vd^)x l + (a 2 + Xb 2 4- pc 2 + vd 2 )x 2 + 
= a 5 + Xb 5 + gc 5 + vd 5 
holds for all values of X, p, v ; and in order to obtain the value of 
Xj we have to solve the set 
cl 2 + b 2 X 4- c 2 /x + dp/ — 0 
CIq 4“ &gA. + Cg/A 4" dp/ = 0 i ) 
cl ^ 4* b^X + c^/x 4- dp/ = 0 J 
where the determinant of the coefficients of the unknowns is the 
conjugate of the complementary minor of a 1 in | a^b 2 c 3 d^ |. With 
this fact in view, and along with it the nature of the relation of 
Murphy’s set to Prony’s, it will he readily seen that both sets 
appear in Sturm’s procedure. 
Terquem follows Sturm, and extends his method to the set of 
n equations 
x 1 + 0.x 2 4- x 3 4- . . . + x n = b 0 y 
a 1 x 1 + l.x 2 +a 3 x 3 + . . . + a n x n = b Y 
a\x x 4- 2 a 1 x 2 + a \ x 3 4- . . . 4 -a 2 n x n = b 2 > 
a\x x 4-3 apc 2 4- a\x 3 4- . . . 4 - a 3 n x n = b 3 
where the coefficients of x 2 are the differential-quotients of the 
corresponding coefficients of x{. The possibility of this solution 
rests on selecting x 2 as the first unknown to be determined, and on 
the set being thus reducible to one of the previous type. 
Cayley (1846, Aug.). 
[Note sur les fonctions de M. Sturm. Journ. ( de Liouville) de 
Math., xi. pp. 297-299 : Collected Math. Papers , i. pp. 
306-308.] 
The functions referred to, which are really Sylvester’s substi- 
tutes * for Sturm’s functions, are introduced in the form — 
* Sylvester. On rational derivation from equations of existence, 
Philos. Mag., xv. (1839), pp. 428-435: Collected Math. Papers, i. pp. 40-46. 
Sturm. Demonstration d’un theoreme d’algebre de M. Sylvester. Journ. 
( 1 de Liouville ) de Math., vii. (1842), pp. 356-368. 
