564 — Anfwers - Quefiions propofed. | (Sur. 
. e4 : 
4. In equations of 5 dimenfions « == / —* a EE | eer vragen? t293 Pe 
ft . 5 A 
XIII. Example 1. Let the equation x*—~ 124%. 19 =o, be given, its ff == 124 and 
g==19. Here g being fmall in comparifon of /, the greateft root will be eafily determined by 
the preceding feries: 
. i tg? ays F9ts 
Therefore « = 124 Pele pt A 19 
124 124 ie 124) § 

= &c.3 or if A reprefent the fecond term, 



k - 19 19 2.19 ; 
B the third, &c. @ = 124 —_—= — —— 1, A mea __ B, &e. 
Foe 124 124)? ey te ae 
™9 
Now aoe == 0.3532255 
ape 9 
- A ==0.0001 
“ ue 94: 
mm 
- = — . B=0.0000006 
124! = 
&c 
Sum == 0 .1535455 
. 124 .cc00co 
Diff. 1238465345 ==, the greateft root of the equation, true tothe Jaft figure. 
The calculotion might have been differently performed, thus: By Simpfon’s Mathematical 
2 2 3 
“Differtations, the value of 2m3, &c. is nearly a Ba a 
Differtations, the value of am -- m3, &c. is nearly equal to ain wherefore Z + at 78 a 



jodie var ge tigre 
&e. is he oe ag eS ey ee 
RE 538 
; a5 == 123 . 8465846 nearly, the fame as before. 
XIV. Example 2. Required, the greateft root of the equation #3 —100x? -+ 273v—0 194.= 0. 

. 2 I 273)? 
Here # = 100, 9 = 273, and r=- 1943 wherefore, « is equal to 1co —— nee 94 vial 
1oo =6rco!?_~—_ goo 8 
&c, — 100 — 3 == 973 and thence the other two routs are found to be x and 2. 
_In thofe cafes where 9, r, s, 2, Sc. are very {mall in comparifon of £, @, will be nearly equal to 
NX 
PD) : 
iN tI dad in other cafes, when the feries converges very flowly, the methods pointed out 
¥ 
by Ms. Stirling * may be employed with fuccefs. From the general feries, too, a number of others, 
of fwifter convergency might be eafily deduced; but this is perhaps unneceflary, as there ere 
many other methods for finding the reots of equations equally general, and requiring much lefs 
labour in their application. The following theorem, however, for quadratic equations, as being ir 
many cafes very accurate and ufeful : 
p7—29 
ety 2 
Let R==r — of? X (Pp)? 
9 (p—20) *-+-7 (p—p) 
(x—22)'-- | nie 
“verdecn, Aug. 1796. B. Cyont. 
EE 
ANSWERS TO THE QUESTIONS THAT HAVE BEEN PROPOSED. 
ais 5 : 
and » == £ X Based, 
PH 

then will == 

Question XXXII.— Anfwered by Mr. Fames Afhion, of Harrington. 
ET AB reprefent the height of the wall, AG the fpout, R the . 
place of the refervoir, and GR a portion of a parabola which the a =, 
water defcribes m its fall, the vortex of which isthe point V. Let v{ IN: 
GFE be perpendicular to AB; in DC take VC=one-tourth of the 
parameter, and let CA be joined, Put AB=a=30, BR=/=10, £ ie 
AG=:=2}, FG=x, and FA=y; then, by the laws of the defcent 
of falling bodies, the velocity with which the water Jeaves the fpout 
at G, is the fame as might be acquired by a perpendicular defcent 
through AF; moreover, a projectile moving in a parabola VGR, : 
hath a velocity, G equal to what may be acquired by a perpendicu- DBH R 
lar defcent through CE; whenee AF=CE, and CA is parallel to DR. 

Oe Wd we ee 

* Methodus Differentialis five Tractatus de Summatione é& Interpolatione Serierum sae 
‘ q OW 
