164 
MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
13. The constants k, k' may be easily found in various ways, perhaps the simplest 
means is to suppose a' = 0 to find k, and a" = 0 to find k'. 
If we put x 3 = 0 and x l = 2 x v we have «! — 0, and a" = 4 k x hence a = 2 x^ V k, 
(3 = — 2 x^ V k, and substituting in the formula for x 2 , we have (since a = — 3 xj 
2 x 1 = x l x { 0 2 V k — x l 0 2 *y 2 V k 
or 
therefore 
2 v / h — __i 1 
(- Sf ~ 3 -8) 
and 
4 ^ ~ 9 (0 — 2 + 0 2 )’ 
but 1 + 6 -j- 6 2 — 0 ; therefore 
k 
K — 2 2 .3 3- 
14. In like manner to find k' suppose x 2 — x 3 — 0, which makes a" = 0, and 
a' + 2 k ' . x j 3 , and u = (3 = a’, therefore 
Xj = y + 2 a:i 2 k' 
or 
4 = ^2^, 
whence 
1 
K ~ 2 . 3 a 
15. The preceding analysis furnishes the following formula, 
3 x l = (x x -f a? 2 + x 3 ) + J/ | -jjr (2 x l - x 2 - x 3 ) (2 x 2 — x 1 — x 3 ) (2 x 3 - x l - x 2 ) 
+ 4 V' — 3 - x 2 f {x 1 ~ x 3 y (x 2 - * 3 ) 2 j 
+ \/ { Y ( 2 “ ^2 ~ x 3 ) ( 2 ^2 — *1 “ ^ 3 ) ( 2 % “ *1 “ ^ 2 ) 
O 
— Y — 3 — x 2 ) 2 (x 1 - tf 3 ) 2 (x 2 — ^ 3 ) 2 j- 
the corresponding formulae for 3 a? 2 , 3 x 3 being obtained by writing & and (P before 
the cubic radical signs. 
In consequence of the negative multiplier — 3 under the sign of square root it is 
visible that this formula is not arithmetically applicable when the three roots are real 
and unequal, which is usually termed the irreducible case. 
] 6. The cubic surds in the formula above given are actually extractible, which 
verifies the solution. 
