358 Proceedings of Royal Society of Edinburgh. [sess. 
set of equations is not the same as Murphy’s, the determinant of 
the one being conjugate to that of the other.* When the use of 
determinants is debarred or avoided, this difference is far from 
unimportant, — a fact which might readily be surmised from the 
present instance, since Murphy’s mode of procedure, though 
strikingly effective upon his own set, is quite inapplicable to 
Prony’s. It should also he observed that the solution of 
Murphy’s set is not essentially different from the solution of 
the familiar interpolation-problem to determine , a 2 , . . . , a n , 
so that a Y + a 2 x + a s x 2 + . . . + a n x n ~ l or y may have the values 
y 1 , y 2 , . . . , y n when x has the values x x , x 2 , . . . , x n 
respectively , — a problem which had been solved in one way by 
Newton (1687), in another way by Lagrange (1795), and in a 
third way to a certain extent by Cauchy (1 8 1 2). f 
Binet (1837). 
[Observations sur des theoremes de Geometrie, enoncees page 
160 de ce volume et page 222 du volume precedent. 
Journ. (de Liouville) de Math., ii. pp. 248-252 : or, in 
abstract, Nouv. Annates de Math., v. pp. 164, 165.] 
The main object of this short paper of Binet’s was to draw 
attention to the fact that a. theorem regarding homofocal surfaces 
* The two sets of equations are 
Ci r s% + 
. +.«”*„ = u r 
| r=n 
\ r= 1 
. . (I) 
a r 1 x l + 
+ . ■ 
£ r=n 
\ r=l . ; 
. . (J) 
The former is substantially the interpolation-problem which goes back to 
Newton, and which may therefore for distinction’s sake be associated with 
his name : the latter being first found solved by Lagrange (Recherch.es sur les 
suites recurrentes . . . Mem. de Vacad. de Berlin , 1775, pp. 183-272 ; 1792, 
pp. 247-299 : or (Euvres completes, iv. pp. 149-251 ; v. pp. 625-641) may be 
called Lagrange’s set,, provided we remember that he also gave a solution of 
the other. The first to deal with both of them in more or less general form 
by means of determinants was Cauchy (1812):° but in saying so, a mental 
reservation must be made in view of Cramer’s mode (1750) of Continuing 
Newton’s work. 
t Newton, Principia, lib. iii. lemma v. : also Aritlimetica Universalis , 
probl. lxi. Lagrange, Journ. de Vec. polyt., ii. cah. 8, 9, pp. 276, 277 : or 
(Euvres completes, ,vii. pp, 285, 286. Cauchy, Journ. de Vec. pclyt., x. 
eah. 17, pp. 73, 74 : or Euvres completes, 2 e ser. i. 
