366 Proceedings of Royal Society of Edinburgh. [sess. 
= ar? r v M + + 
It only remains now to multiply by f(x) in the form 
x n +p 2 x n ~ 2 - 
obtaining 
f m (x) = x n ~ m -Y m _ 1 + 
+ a;»-j*-2(V m+1 -jo i y m +p 2 Y m _!) 
+ 
and then to condense the coefficients, — an easy operation, since 
all the V’s are identical save in their last columns : for example 
^m + 1 "b ]?2^ m — 1 
5 0 
S 1 • • 
• • 2 
5 1 
\ *• 
• • S m- 1 
S m - 1 
S m • • 
• • S ‘2m-i 
S wi+1 — P\ s m +i^2 s m-l 
S m+ 2 —]?l s m+ 1 + P^m, 
^2 m _Pl^2m-l "b-^2®2m— 2 
Chelini (1846). 
[Determinazione geometrica in coordinate ellittiche . . . . 
Raccolta sci. di Polomba, ii. pp. 109-113, 126-131; 
see also v. pp. 227-263, 333-374.] 
Grunert (1847). 
[Vollstandige independente Auflosung der n Gleichungen der 
ersten Grades . . . Archiv d. Math. u. rhys., x. pp. 
284-302.] 
The equations are 
A 1 + A 2 a r + A 3 a r + . . . + A n a r — CL r (v — 1 , 2 ,..., ?i) 
that is to say, are of the type to which Murphy’s belong, and with 
which a problem in interpolation is connected ; and the solution, 
rather tardily reached (p. 301), is 
m 
K(a s 
> , a 3 , a 4 
> * * 
• > a n) 
( a l- 
- a 2 )( a l - 
a 3 )( a l - 
a 4 ) • 
• « ‘ ( a l 
-a«) 
a 3 , a 4 , 
> a n) 
a 
( a 2“ 
" a l)( a 2 ~ 
a s)( a 2 “ 
a 4 ) . 
. . (a 2 - 
„ \2 
a n ) 
m 
K(a,, 
<M 
S 
. . . 
J a n) 
a, 
( a 3“ 
- a l)( a 3 “ 
a 2 )(a 3 - 
a 4 ) . 
. . (a 3 - 
\“3 
a«) 
