264 Transactions of the South African Philosophical Society. 



2. The essence of the solution consists in noting that to assert 

 the validity of the given set of equations is the same as to say that 

 the expression 



x z x 2 x 3 x n _ -. 



z 7j r ~z ~5 — r ~z 75" ~r •• • i~ - j- — J- 



S — Pi h ~~ P2 h ~ P3 h ~ Pn . 



vanishes for n values of £, namely, the values b z , b 2 , ..., b n ; and that 

 therefore 



*,(£-&) (£-/3 3 ) ...(if- A,) + ^(S-ft) (£-£,) •••(*-&) + 



(|-/3,)(4-/3 2 )...(4-A,) 



must be identically equal to 



A(J- 6.) (J- 6.) ...($-&„). 



Since the only term containing l n in the left-hand member is - l n , it 

 follows that A is - 1. We have only then to put %=}3 I} j3 2 , ... in 

 succession in this identity and we obtain 



(/3,-MQ3,-6 i ,)-G3,-ft„) 

 x '~~ (/3 I -A,)...03 1 -/3„) ' ■•- 



3. As a second example, which readily suggests others, let us 

 take the set 



a?! + 2a I jr 2 + Sa 2 1 x 3 + . . . + na%~ J x n = 



Lt'2 t ^ / T ~~ T~ ^2 2 *T" 2 ^-'Q "" P • • * "T~ Lvn^ll -^~ 



<x n _jii? I + a~ n _ 1 x 2 + <x, l _ I 3?3 + ... + a n _ 1 x n — — 1 / 



Here the equivalent assertion is that the expression 



1 + £«, + £ 2 z 2 + ^ic 3 + . . . + £X 



vanishes for w ^- 1 values of £, namely, the values a It a 2 , ..., a n _ n and 

 that its differential-quotient with respect to £ vanishes for t, = a x . 

 We thus learn that it is of the form 



A(£ - a,) 2 (t, - a 2 ) (| - a 3 ) ... (4 - a n _ x ) : 



and observing that it becomes 1 when £ is put -—0 we learn further 

 that 



A= . ^ - 



1 2 ^ • • * *~™H : i 



