£)r. waring on Interpolations. 
DEMONSTRATION. 
Suppofe a+ba.+cod+dod+eod+ Sec. = s“, 
a+bj3+cfi z +d{3 1 + e(3 4 + Sec. = s e , 
a+by+cy*+dy 3 +ey 4 + Sec. = s y , 
a + bS+c^+d^+e^+Sec. =s\ 
a + bz + cP + dp + ee 4 + See. = s € , multiplr 
thefe equations into a, b, c, d, e, Sec. unknown co-effi- 
cients to be inveftigated, and there refult 
axs “- aa + nbx + Acod + nded+neod+Sec. 
Bxs B =Ba+Bb(3 + Bc(5 1 +Bd(3 i -bBe(3*+Sec. 
cxs Y =ca+cby+ccy 1 + cdy 3 +cdy*+Scc. 
vxs^—Da+Dbfi + Dc^+Dd^ + i>d$ 4 +Sec.~ 
E xs ! =Ba+nb£ + BCe z + e^s 3 +e ee 4 + Sec. See. 8cc. 
Now fuppofe AS“+Bs l? +cs y +DS J +ES ! + Sec.=a+bx+cx z 
+dx 3 +ex‘ i +Sec. and the correfpondent parts refpedtively 
equal to each other ; that is, #(a+b+c+d+e+ Sec .) = a 
b (Aa+B/S+cy+D J'+E£+&c.)=^at; a a 2 + b /S’ + c y 1 + D 5 
+ Ef* + Sec. =x z ; a« 3 +b/3 3 + c y 3 + W 3 +E£ 3 + Sic; -x 3 
Aa 4 +B/3 4 +cy 4 +D J' 4 +E£ 4 + Sec. -x\ &c.: But it follows 
from. Theorem i. that (if As a +Bs g +cs'>'+ ds^+es'+8cc. 
z:a+bx+cx'+dx'- 1- * *♦ + &c.) a = El x *E x E! x EI?p , 
K 
Yol. LXIX. 
B 
