MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
177 
[3, I, 1] = — 20 (x 2 x 3 -f x 2 x 4 + x 3 x b + x 4 x b — 4 x 2 x b — 4 x 3 x 4 ) 
— 20 x 2 3 (x x x 4 + x x x b + x 3 x 4 -\- x 3 x b — 4 x x x 3 — 4 x 4 x b ) 
— 20 x 3 3 (x x x 2 x x x 4 -\- x 2 x b + x 4 x b — 4 x x x b — 4 x 2 x 4 ) 
— 20 x 4 3 (x x x 3 -f- x x x b + x 2 x 3 + x 2 x b — 4 x x x 2 — 4 x 3 x b ) 
— 20 x b 3 x 2 + x x x 3 -f- x 2 x 4 -{- x 3 x 4 — 4 x x x 4 — 4 x 2 <z 3 ) 
[2, 2, 1] = — 30 {x x 2 x 2 (x 3 -f x b — 4 x 4 ) + x x 2 x 3 2 (x 4 + x 5 — 4 x 2 ) 
+ x x 2 x 4 2 (x 2 + x 3 — 4 x 5 ) -f x 2 x 2 (x 2 + x 4 — 4 x 3 ) 
+ x 2 2 x 3 2 ( x x + X 4 — 4 x b ) + x 2 x 2 (x x -\-x b — 4 x 3 ) 
+ x 2 x 2 (x 3 -f x 4 — 4 x x ) + x 2 x 2 (x 2 -f x 5 — 4 x x ) 
+ x 3 2 x b 2 (^! + x 2 — 4 x 4 ) + x 4 2 X 2 (^! 4- * 3 — 4 x 2 ) } 
[2, 1, 1, l] = — 60 2 x x 2 x 2 x 3 x 4 
[1, 1, 1, 1, 1] = 480 x x x 2 x 3 x 4 x b . 
The quantity a' is the sum of all these multiplied by a constant k'". 
42. It remains to give the value of the constants k, k', k", k"', which may be easily 
found by comparison of the above values with the constituent parts of the roots of a 
biquadratic ; they are as follows : 
and the term uninfluenced by surds is 
X x + Xcj + + x 4 + x b 
5 
43. All the constituent parts except the last-mentioned vanish when all the roots 
are equal, but for their separate evanescence the condition for the equality of roots 
is insufficient, the requisite conditions are easily seen from the factors of such parts 
already given ; they are of two classes, arising from the different relations of the 
couples {(a, a 2 ), (at, ax’), (to 3 , co 4 ), (co 3 , a 4 )}, and of the two couples {( co 2 , CO 3 ), (co, CO 4 )}, each 
member of which contains the same imaginary part with the other, which is not the 
case with the former couples. 
44. I cannot at present, from the pressure of other engagements, pursue these in- 
vestigations further ; if any additional light to the analyst is furnished by the pre- 
ceding imperfect reflections on a subject so often treated, the author’s object will in 
a great measure be attained; they at least tend to show the imperfection of our 
MDCCCXXXVII. 2 A 
