 §62 Mathematical Correfpondence. 3 {Sure 
And «6 + 26 + 461 364 6 == 11390625 — 25818750 -+- 4556250-4- 14260725 — 
3431760 — 936635 == 302076c0 --- 30187085 =20516. 
_. The fum required then is—= 20515, which may,be eafily fhown to be accurate in this cafe, as 
the roots of the equation are 1, 2, 3, 4, 5, and confequently ; 
a A646 +96 1. 6 — 15625 +4096 4 729-4 64-41 — 20615 as per theorem. 
WI. The ules to which this rule may be applied are many and various. It fuggefts, in the 
firft place, an eafy and expeditious method for finding the limits between which the roots of an 
equation are contained, when none of them is impoffible. For in this cafe the {quayes, the biqua- 
drates. the cubo-cubes, &c. of all ciie roots will be affirmative, aud therefore greater than the fame _ 
power of the greateft root. Hence, in order to determine a number greater than any of the roots 
of an equation, find by the theorem the fam of the fquares, biquadrates, &c. of the roots, and 
extract the fame root of this ium. The refult will be the number required. Thus, in the fark 
I 
example, «?-|- 2? = 4o, a4 | B4 == 1312, &c. and confequently 4/(a? + 6) = 4/4o= ue 
nearly, 44/ (a4 Bt) = 4/1312 = pons nearly, Sc. which fhows that the greateft root muft 
100 : 
be lefs than a 6 &c. Alfo, inthe third example, as a+ £61464 26+ 620518, 
i : 
6,/20515, Or § A nearly, is greater than any of the roots of the equation, 
In this way, it is evident, we may often find a near value of the greateft root, and afterwards 
by the common methods of approximation determine it to any degree of exa€tnefs. If one of 
the roots be much greater than the others, this method may.be employed with much fuccefs: as 
for example in the equation x?—~ ro1 x1 ICO == 0, where /p = 101, g == 100, and a*-+ f7== 
i? — 29 = C201 — 200 = 10001, the {quare root of the fum of the fquares is nearly equal to 
: th part of the whole. 

T . : - 
#00— , which differs from the greateft root of the equation only by 
200 2.0000 
Vii. A fecond ufe to which this rule may be applied, is to inveftigate general properties of 
curve lines. Harriet, by pointing out che genefis of equations from the combination of inferior 
ones, and thence the formation of the coefficients, fuggefted a great number of fuch properties 5 
from the preceding theorem, which is founded spon this genefis, it is nanifeft, that many more 
may be deduced. We might exemplify this by demonftrating fome of thofe curious properties of 
the circie given by that excellent geometer Dr. Matthew Stewart, in his book of General Theo- 
zems, but this we fhali leave to fome other occafion. 
VIII. The lait application which we fhall make of this theorem is to the analyfis of a certain 
clafs of problems belonging to the higher geometry. When it is requirea to determine the equa- 
tion of a curve, from having given a certain relation between the fegments of a variable line, 
which-meets the curve in two or more points, the inveftigation will be much fhortened by a 
knowledge of fuch theorems as the above. The cafes in which it will be ufeful are thofe where 
the fum of any powers of the fegments are given. ‘The following problem may be given as an 
example: . 
“‘ ‘et the fixed point A be the pole of an indefinite 
svumber of right-lines, as ABB’, it is required to deter~ 
mine the curve line BRB’ which all thefe lines cut in ; » 
the points B,B’, é&c. fo that the fum of the mh powers ; Bea 
of PB, PB’, &. may be given.” Pade ee 
Let AC be taken for the axis, and from B,B’, &c. 
draw the perpendiculars BC, B/C’, &c.: thenif be 
the number of points in which AB cuts the curve, the 
number cf lines AB, AB’, &c. will be likewife — », 
and confequently by a well-known property of curve- A: fe 
tines the relation between AB and the angle BAC ¢ ne 
will be’ expreffed by an equation of the ath degree. Let AB therefore — x, and let a Pee 
&c. be certain functions of the fine, cofine, tangent, &c. of the angle BAC 3 then may the 
relation between x and this angle be exprefied by the equation x" —fxn—I fe gx8 2 ee x3 
+} &c. =0, by which affumption the firft condition of the problem is anfwered. 
TX. The roots of this equation, it is evident, are equal to the fegments AB, AB’, &c. and the 
fecond condition requires that the fum’of the m‘> powers of thefe roots fhall be conftant : let this 
furn== A, and by means of the theorem Seét. IV, the relation between g, 9, %, &C. will be given. 
‘We have then «® — part 1. gyn? . rxn-3 — &c. = 0, and ph — mgs"? A mrp™—3 
wom Wisp 4 : 
ey a &e. F=A 
” pStex 
which two equations anfwer all the conditions of the problem; 

