310 Scientific Intelligence. {Oct, 
seems to have made a mistake in one of the signs of the root con- 
nected with ¢: when corrected, &c.” I should have said, ‘* Mr. 
Lockhart seems to have made a mistake in one of the signs of the 
yoot connected with ¢, when the equation is reducible by Cardan’s 
rule : when corrected, &c.” So that the roots of 
sfc 3 63 a z fF 
eS “ — =, if the given equation 2° — l x = c, be 
' rr z o7 9 W the g q 
reducible, will be but if irreducible, the roots 
will be 
x ce b 
$4 /5-F 2 z 2 
g t as EAS 
trish b 
Tay lige CBee wae t {2 b 
-$4j/4-4 
v a? 6 s 
-t<\/7-+ sist e b 
BRE OP a dca” Sabet) 
- You will perceive that, when the above omission is supplied, the 
observations in Mr. L.’s last letter lose all their weight, and the 
conclusions I have come to in my former letter remain in full force. 
Iam afraid, however, that I have not expressed myself with that 
perspicuity which I ought to have done, when pointing out the part 
of Mr. Lockhart’s demonstration, where the error appears to have 
originated; for if I had expressed myself properly, Mr. L. must 
have seen that there was to be a distinction between equations which 
are reducible by Cardan’s rule, and those belonging to the irre- 
ducible case: but that he did not perceive it, is manifest from the 
observations in his last letter. ; 
. In endeavouring to supply the above defect, I shall begin by 
premising, that in equations belonging to the irreducible case, ¢* is 
always greater than 5, but less when the equation can be reduced 
by Cardan’s rule. Hence in irreducible equations the quantity 
ae me ‘ ; : : 
(e = 5) is always postive ; but in reducible equations, negative. 
. Be bes 
_-Mr. L., in No. 30 of your Annals, has shown that a as sts 
iaies 23 2 B83 ae g a 2bt 
+ Ale Behe or, which is the same thing, y Tham 
e b ce 38 
: OPES 8 Be, : . : 
+ tH) x (5 — =) 7 Now in extracting the square 
root of these equal quantities, it is plain that the roots on both sides 
of the equation must be of the same kind; that is, if one be a 
positive quantity, the other must be positive also; or if one be 
negative, the other must be negative likewise. . Now the roots are 
b i? b (nd b3 é 
es = (ea) BE a + a ee oes 
ck (é 3) * / 4 gad Vv (5 x): But it 
é , 8 RAs ir 
has been noticed before, that the quantity (¢* — 5) is in itself a 
