OF THE DIFFEEENCES OF THE BOOTS OF A GIVEN EQUATION. 
57 
= 25a/3(a — — a,S(a+/3)(7+^) + (a'+/3%§ + a/3(7 + ^)' — y^(« + /3j(7-f ^) + /o'} 
= 2scc^l3%a-f3y, ^iz. 
2 , 2 a ®/ 3 " 
— 2ea^f3'(a — |3)^(«+f3)(y+§), viz. — 
+ apyhlsi«—(if{c(^-{-f^% viz. 
+ - 32 ,«^ 
' a ^ 
+ 2 |V^i 3 ^ 
+ 2a«^/T(«-/3)^(y + §f, viz. 
: 2„,2 
,2 
4- 2 2i2a^/3^y 
- 22 , 2 «^f 3 V 
+ 2 | 2 . 2 a ^ i 3 
— aj3y^26(a— i3)"(a+^)(y+^)’ viz. — 2-2,2a'/3 
+ 2 | 2 . 2 «^ i 3 y 
4-a/3y526(a— /3)V^, viz. 
+^ 2 , 2 a "| 3 y 
- 12 ^ 
a "' 
2 , 2 a ^, 3 ® 
- 226 ^^ 3 ^ 
- 22 ,«^ 3 V 
+ 22405 * 3 V 
+ 22 . 2 «W 
- 22 . 2 «W 
+^ 324 «^ 
2 ^ 2 , 2«^/3 
-2~2ecc^(3^ 
+ S%a^f3y 
- 12 ^ 
(53) 
-2(4^) 
-1(521) 
+ 1(431) 
+2(42^) 
-2(3^2) 
+ 3e(4) 
-2<31) 
- 2 e ( 2 ^) 
+ 3e(2P) 
- 12 e ^ 
where for a moment a is put equal to unity. 
The value of the last-mentioned expression is then calculated as follows :- — 
— 4 

12 
- 8 
— 8 
-8 
— 8 
-12 
_8 —4 —12 
— 12 = 
—96 
bde 
+ 1 
+ 16 
+ 10 
+ 10 
+ 2 
+ 12 
+2 +4 +3 
_ 
+ 60 
<?e 
+ 8 
+ 
8 
+ 4 
+ 8 
+ 4 
+ 6 
+ 4 -2 
= 
+ 40 
b'ce 
-9 
— 
8 
— 11 
— 1 
-4 
-12 
— 2 
= 
-47 
Ve 
+ 3 
+ 3 
+ 3 
= 
+ 9 
cd!^ 
—7 
— 
8 
- 5 
— 1 
-4 
-2 
= 
—27 
bH^ 
+ 3 
— 
4 
+ 1 
— 2 
+ 2 
=Z 
0 
bed 
+ 6 
+ 
8 
+ 3 
+ 1 
= 
+ 18 
b^cd 
-3 
— 1 
= 
— 4 
e 
-2 
— 
2 
— 4 
h^e 
+ 1 
= 
+ 1 
And restoring the powers of a by the principle of homogeneity, and putting 
