Vou. IV.] Matiematical Correfpondence. 563 
Had the fum of more powers of the roots of this equation becn given, the values of p 9 ry &C, 
would have been more reftriéted. 
From this equation, the relation between AC and BC may be eafily deduced; for if AC—, 
and BC ==y, x? will bex= v? pe y?, ~ =4/(v? + y?), and == the tangent of the angle BAC, 
4 
which values being fubftituted in the foregoing equations, the relation required will be found.. We 
fhall now confider fome particular cafes of this problem. 
X. Let the number of points B, PB’, &c. be two, and let the (um of the fquares of AB, AB’ be 
given ; then will n=2, m2, and the two equations a? — fw ++ ¢ == 0, and f?—~27== A, 
Flence 2g = p?m A, I=; L —=A; and Pp = P= A=>0o. 
Let n — 2 and ws = $3} then will x? — pxtg= 0, and £3 = 39/.—A. Hence q =f? 
— = and x?— fx. = po tao. 
Let = 2, or let the line AB cut the curve in two points; then will Xe? amen ft ol 9 == 0, and 
pr — mg tm, — PL 4m, st gene q3h™—6 1 &e. = A, In this cafe 
the latter equation may be exprefled differently ; for the two values of x being equal to 
I I T I screen ny of T eae NNT ! 
ah Sa f? — 9) en aii hs — 7), AB™ will bea={ = sma ari q))",AB™ 
I ; I ane x . : : 
werk i tie DP ah Gree YA Ged te) i ok Ow eet VAG eat? atc 
But a much fimpler folution of this cafe may be given by afluming an equation of the forne 
“ies a Of x?m— px-}-¢ == 0; for then AB™ - ABI". fy and confe- 
quently 47" — Ax™ + 9 = 0, which equation, expteffing the nature of the curve, is infinitely 
more general than thote of Bernoulli, Leibnitz, and de !'Hofpital. 
Aberdeen, Aug. 1796. ; fi. CYGNI. 
AP PEN DAX, 
XI. In.Se&. VI, a method is pointed out by which the greateft root-of any equation may be 
, found by repeated approximations, when none of the roots are impoflible. This is done by finding 
the fum of any power of the roots of the equation by the general rute, and extracting the fame 
root of the fum; that is, if «, B, y, &, &c. be equal to the roots of the given equation, a being 
the greatef, and m = any number fufficiently large, « will be nearly equal to ™4/(a™ —- B™ 
fy" +5", &e.). Now it is manifet that, if m be fuppofed infinitely great, « will be exaétly 
equal to the preceding exprefiion, and confequently if its value can be determined in this cafe, we 
will have a general rule tor finding a. 
XA. But by Set. IV, the value of ¢™ + 6" + y™ + 3, &. is equal to.~™ — mgh™—> 4. 
mp3 — (iS — as 9°) f° —4 4. &c. 3 whence, by means of De Moivre’s theorem, the 
' hae 2 t r 
m'® root of this expreffion is found — f—= ; Zr et i os 
2 p ro) 
2 aT ie 21» 73 
Therefore a Se — muro : as 2 rns &c. which will give 
7 £ i 2 : : 
the true value of «, if the number of terms of this feries be infinite, which is the cafe when m 
is infinitely great. 

r s+? t-Lagr 
By dividing this feries into fa€tors, we have « —= (f —2 (x +7) (I— + ) a 
a vhsebertsy 
aca ) &e. 
Now it is evident, that if the values of 93 Ty 5, ty &ec. be fmall in comparifon of fy the feries 
expreffing the value of « will converge very quickly: in fuch cafes, therefore, it may be ufed 
with advantage. . : ‘ 
In this value, if g, r,s, 7, be taken — 0, & is equal to / exactly, which, it is manifefl, muft be 
true, as the given equation is then a fimple one. Alo, by making 7, or r, or s, or ¢, &c. to va~ 
nith with all the following coefficients, we fhall obtain the following particular theorems : 
‘ 
2 3 
x. In quadratic equations “ — a. as =m OCC, 
v4 
24593 
2. In cubic equations woe Ie Deel hep lear a + &c, 
f Vi f f* 5 a ; 
2 27742 
3. In biquadratic equations = pe Pcie os a + rime a &e. 
