[ 59 1 
VII. Problems concerning Interpolations. By Edward 
Waring, M. D. F. R. S. and of the Injlitute of Bononia, 
Lucafian Profejfor of Mathematics in the Univerjity of 
Cambridge. 
Read Jan. 9, TV /T R. briggs was the firft perfon, I believe* 
J 779- 1.VA that invented a method of differences 
for interpolating logarithms at fmall intervals from each 
other: his principles were followed by Reginald and 
movton in France. Sir Isaac newton, from the fame 
principles, difcovered a general and elegant folution of 
the abovementioned problem : perhaps a ftill more ele- 
gant one on fome accounts has been fince difcovered by 
Meff. Nichole and Stirling. In the following theorems 
the fame problem is refolved and rendered fomewhat 
more general, without having any recourfe to finding the 
fuccefiive differences. 
>• # f N t . y T ; - ' ‘ * -tiff • % 
THEOREM I. 
Affume an equation a+bx+cx*+dx* .... x^-^y, 
. 
in which the co-efficients a , b, c, d> e, See. are invariable ; 
I 1 let 
