174 
MR. MURPHY ON THE ANALYSIS OF THE ROOTS OF EQUATIONS. 
these differences may be taken either way, we have also from the same equations 
five other factors equal to the former and with contrary signs. 
In this manner ten factors of a lv may be found by equating any pair of the four 
quantities a, (3, y, c>, and the number of pairs being six, the whole number of factors 
of « lT is sixty, these factors are very easily formed, and here we present the first thirty 
factors, the remaining thirty being formed merely by changing the signs of these, or, 
which is the same, inverting the order of the differences of the roots in each factor. 
37. For greater clearness we shall subdivide these thirty factors into five groups, 
from the first of which x x is excluded, from the second x 2 , and so on ; each subdivi- 
sion contains six factors, four of which are of one form, and two of a different form ; 
they are as follow : 
O 2 — X 4 ) + (x 3 — x 4 ) -f co 2 (x 3 — x 3 ) 
Os - X 2 ) + " Os — X i) + » 2 Os - x i) 
04 — X o) + » O2 — •%) + O2 — *3) 
0 5 - X 3 ) + a 0 4 — •%) + 0 4 — X 2) 
(x 2 ~ <%) + (» 2 + co 3 ) Os — * 4 ) 
Os — x 4 ) + 0 2 + *> 3 ) Os — x z) 
Oi — ^ 4 ) + ® Os - x d + Os - x s) 
Os — x & ) + a (x 4 — x b ) + co 2 (x 4 — x x ) 
(x 4 — x 3 ) + a Ol — X 3 ) + or Ol — x 5 ) 
Os - a?j) + a 0 3 — * 1 ) + " 2 O 3 — * 4 ) 
Oi — ^ 3) + O 2 + ^ 3 ) Os ~ *4) 
Os - X i) + O 2 + " 3 ) O3 - x \) 
01 — x i) + ” 0 4 — x 2) + ^ O4 - %») 
0 2 — ®s) + » Oi — *5) + Oi — x d 
(x 4 — x 4 )-{-A> (x 5 — x 4 ) + A) 2 Os — x i) 
0 5 ~ * 4 ) + M O 2 ~ * 4 ) + " 2 O 2 — * 1 ) 
Oi — •%) 4- 0 2 4- *» 3 ) 0 4 — x z) 
(x 4 — X 2 ) 4- O 2 + CO 3 ) Os — x x ) 
01 - X b) + * Os - X b) + ^ ( X 3 - X 2 ) 
02 — X 3 ) -f M O5 — ^3) + *> 2 O5 ~ ^1) 
03 - x x ) + u (x 2 — x{) 4- v 2 O2 “ *5) 
0 5 — X 2 ) + co Oi — #2) + 131/2 Ol — x i) 
Oi — x 2 ) 4- O 2 4- co 3 ) 0 3 — <%) • 
Os ~ X b) + (" 2 + " 3 ) O2 ~ *l) • 
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