Vou. IV.3 Mathematical Correfpondence. . si 
m—1 m--2 m—3 
We ee et OM} 
ne Beep kre 
Pe 

MN—T spe 
. mt /py=—t 
. &c 
Hence, by fubftituting , m1, m—2z, m—3, &c. inftead of my the following values are 
Alerived : ity 
a eeyert 
B= — (m—1) 9"? ' 
Eee 
C = (m—z) rf™—3 + (m—2) eae pons 
D == — (m3) —— : im) » g3hm—6 (m—3) h a 297f™—S ry ame ye 
(m—5 5). (m—6) (77) ah — (a—6 6) Raa 
E=(m— 4pm——8 | ( 77 2 pm—7 
ee as ee -(—4) - ata i, 
{ m— 

hk ee (aston) pr-8 4 (mg) 8 
pee siesktie amt pm +. ym gm + 4m +6 &o, Sa pM a ppm? -|- mr fn 3 mm Sie, 
pale = 3 gions 
-- mtpm—s — mupn—6 + mwp?—7 
—m (ma) gripes bm (m—s) qsp™—© — m (m—6) 9-7 
ETS 2M 1 aptarG (ti5) 2, 
OTE RS pe Bitsy abl alt, ap -}-m (m—5)“— Pra 
tm. a rp L1G bE geet (m—6) rsp rT 
Ae 
V. This is exactly the rule given by Waring in his Mifcellanea Analytica, which appears to be 
the moft proper form in which the fum can be exprefled: and from this the Newtonian formule 
may be deduced. 
' Example I. 
Let the given equation be x7—8x-+12=20, In this example # is ==8, and ¢ == 12. 
Wherefore eb = =f == 8 
ms 21. 0? == ft? -— 29 == 64 —24 = 40 
eee 392 + 37 = p3 — 394 = 512 — 288 — 244 
pi — fit — agp? -b 2g? Tila Tw doh 1312 
ee 
ee 
a ra eee 16 = 1312 
&e. 
Example IT. 
Let the equation be »4-— r2x3-+ 49x? — 73x-+-40==0: that is p= 12,9 =49, r= 78, 
==40, ¢=0, v==0, w=0, &e. 
Then a + Bb y+ o== 12 
eet thar mus 840 
EBL + P= — 39/43 ==1728 — 17 ia Fae 788 
LE -- t= f4— 4qp? + arp +29 2 25—= 898 
which may ue ealfily proved, as a, f, y, ¢, are equal to 5, 4,2, 1, refpectively. 
Example III. 
Required, the fum of the 6t® powers of the roots of the equation, 
x5 — 15 x4 aapeeta 274% —120==0. 
By the general theorem 6-4. BO - 76 -f 80 £6 = [6 mn 69/4 $f. Grp3 mem (65 wm 997) fi2 ate 
{6t— 1297) f+ (69s — 293 +- 372). 
But in this example p= 15, 7 = 85, r= 225, $== 274, and ¢== 120, 
Therefore 16 = 11390625, 6743 = 4556250, (997 —6s) /?== 14260725, b9p4 == 25818740, 
(12gr—m 67) £ = aR (273 —= 37? pm 675) == 936635, 
4D2 And 
