360 
Proceedings of Royal Society of Edinburgh. [sess. 
g I i c i _ /(k 8 ) 
k 2 -a k 2 -b k 2 -c ' ' ' ' r(K 2 )’ 
/(A) _ /(A) 
a- F(A) (A - B)(A - C)(A - D) . . . 
, /q>)_ m 
F(B) (B-A)(B-C)(B-D) . . . 
Now the set of equations here solved is more general than that 
with which we started, the latter being the particular case of the 
former where 
/(K) /(K,) 
F(K) F(Ki)“ 
To effect this specialisation it is only necessary to make the 
arbitrary function /(a;) equal to 
(x - A)(x - B)(x - C) - (x - K)(x - KjXa - K 2 ) 
or equal to 
F(aj)-f(a?) say; 
(where, be it observed, the condition as to the degree of f{x) is 
fulfilled); for then, since f(ar) vanishes when £C = K, K x , K 2 , . . . . 
we have 
/(K) = F(K), /(Kj) = F(Kj) , /(K 2 ) = F(K 2 ) , . . 
The corresponding change in the values of the unknowns is easily 
made : for example, in the case of a , we have only to substitute 
F(A)-f(A), — or, what is the same thing, -f(A), — for /(A) in 
the numerator, the result being 
(A-K)(A-Ki)(A-K 2 ) 
(A-B)(A-C) .... 
We have thus as Binet’s theorem : — 
The solution of the set of equations 
+ 
+ 
/q £q /? 2 b 1 
+ + 
^2 Pi t>2 ft 2 ^2 A 
+ .••• + 
h ~Pn 
X l . ^2 , .^3 , , - *n ■ 
b n -f3,b n -f3 2 + b n -/3 s + •" ' • ‘ + b n -p n ~ L 
