94 
Proceedings of Royal Society of Edinburgh. [sess. 
simplest form of alternant. The full solution is given for the 
first four cases, but without any indication of the method employed. 
Thus for four variables the results appear in the form 
_ ~ P2P3P4Z0 d~ (P2P3 ~b P2P4 ~h PsP^l ■ ~ (P 2 T Pg ~b p 4 )z 2 + Z g 
^ (Pi — P2XP1 “ P3XP1 “ P4) 
= ~ PlP 3 pA Z 0 + (P1P3 + P1P4 + P3P4K ~ (pl+ p 3 + Pi) z 2 + 
1X2 (P 2 -Pl)(P 2 “P 3 )(P 2 - P4) 
P '3 = 
P 4 = 
and the writer then adds : — 
“En general, quelque soit le nombre w, pour avoir le 
numerateur de la fraction qui donne la constante g K} il faut 
prendre toutes les racines, excepte la racine p K , et des n - 1 
racines restantes, en trouver le produit total, la somme des 
produits n- 2 a n — 2 , n- 3 a n — 3, n- 4 a n — 4, . . . ., 
2 a 2, 1 a 1, multiplier, respectivement, le produit total et 
chacune des sommes par z 0 , z v z 2 , . . . ., z n _ 2 , ajouter z n - 1 , 
et donner a tous les termes des signes alternatifs, en com- 
mengant par - ou + , selon que n est pair ou impair. 
“ Pour avoir le denominateur, on soustraira, successivement, 
de p K chacune des autres racines, et on fera un produit de 
toutes les differences donnees par ces soustractions.” 
It is, of course, quite possible that Prony was not acquainted 
with Vandermonde’s memoir of 1771, or Laplace’s of 1772, or 
Bezout’s of 1779 ; and, further, that in seeking for the solution of 
his equations he was lucky enough to hit upon the set of multipliers 
which, being used, would, on the performance of addition, eliminate 
all the unknowns except one - e.g., in the case of four variables 
the multipliers 
“ P 2 P 3 P 4 > 
+ (P 2 P 3 d" P2P4 d~ P3P4) ) 
~ (P2 d- P3 + P 4 ) , 
1. 
If, however, he was familiar with the method of any one of 
these memoirs, and applied it to the set of equations under discus- 
