DIF 
ing pale tha’ the ae mut terminate when any of the 
difkrences d”, d”’, &c. become = o. 
xm 
, &c, of 
It is alfo evident, that the coefficients ~ - 
the differences, are the uncie of the binomial Goan 
uppofe it were required to find the log. — 
or 5’, 1”, 2066, 
2 
5/3 1's seo 
Take out the log. tangents to feveral minutes — ‘econds, 
and take their firft and fecond differences, thu 
so ees . , 
I 3 
5x p6414r7 | TP! | 49 | ag, 
52 7-2055821 | Tyee | 47 
5 & G1O79179 | 4397 
ere A = 9.164147 3 «2 = $253 d = 144043 and the 
mean fecond difference d’ = — 45. Hence 
‘ » ~ - - 71641417 
xa! : : : . 2977 
x X—'I ” 
1 I ¢ ° - 
Therefore, the tangent of 5’, 1”, 12”, 24’, is lhe 
method may be deduced from the foregoing expreffion, 
terms o aferies. For if 
we imagine a new feries, whereof the firft term fhall be = 0, 
the fecond = the , the fourth = 
+ C, the fifth = A + D, and fo on, it is plain 
that the affigning one term of this feries i : oe the fum 
of all the terms, A, B, C, - - Now e thofe terms 
are the differences of the fum Ane ‘A +B+ 
+ B-+C + D, and that oy the pene es fome of the 
differences of A , &c. are = O; it follows that fomeo 
the differences of the fone will alfo be = 0; and that whereas 
in the feries A + xd’ + a d", &c. by which a term 
was affigned, A seoeitaiee ie firft term, d’ the firft of 
the firft differences, and that x i bean the interval 
between the firft term and the >» we are to write o 
inftead of A, A inftead of d’', a! inftead of d", d” in- 
fiead of d”, &c. and x +1 inftead of x, which _being 
x+I 
done, the feries expreffing the fums will be o + 
Co Sop stt 
I. 
Ayiti 
sl =e i 4, we ore I 
Rex —mIl.x—2 
2iZe4 > 
Or, again, if the real number of terms of the lines be 
At ed 4 ae 
d", xe 
called z, that is, if == + 1, or 3 —1= x, we fhall have 
Z—-1.2—2 
253 
d' + zt .e—2.% ee &c. See De Moivre, 
3. 
Dot. of Cinace. Pp. ao 60. Mifc. Analyt. p. 153. Simp- 
fon’s Eff. 
For juflarice, Tees it be required to find the fum of fix terms 
of a feries of the {quares . the natural numbers 1 + 4+ 9 
16425 +36. Thus 
DIF 
Terms. d’ a" dit 
I 
‘ 3 
5 ce) 
9 2 
° 
16 z 
a6, &e. > 
H d' = 3,d"=2,d"’=0,and2=6. The 
fum confequently will be 
=e x 1b Ko 
24.3 
exp dhr dy ecae he 
3 
a 
_ It$2z+22% %.I+s2.14+2% 6.7.93 
= x 3 ; =—E— = 91. 
This eafy example will be fofficient to fhew the applica= 
tions of this rule. Thofe who are — of fecing its ufe- 
in queftions of chance, may confult oivre’s Doc. 
_ Various eects fet of 
m 
P- as 
baie differential method, it is to be obferved, that 
though | fir Ifaac and o others have treated it as a method of 
See 
Thi ingeni- 
ous pace hes treated Fally of the cf eee cea and 
' fhewn its ufe in the folution of fome very difficult problems, 
See = ERIES. 
ERENTIAL fealey in Algebra, i is ufed for the {cale of 
lata fubtracted from u Recurring Serres 
DIFF Te, from forma, thape, i isa ae ufed 
in oppofitien to uniform; om oe fina there is no regu= 
larity in the form, or appear of a thing, 
‘The botanifts ufe it as a diftingtion a the flowers of feveral 
{pecies of plants. 
DIFFRACTION of Licur, is the bending of the rays 
of light, which is occafioned by their pafling by other nae 
Thus, if the light of the fun be admitted through a 
room, otherwife darkened or clofed, the image of che fun 
within the room will be found to be larger than it ought to 
ride if the 2 rays aaeaeias in ftraight lines from the fun to the 
c m. 
* DIFFUSE, Dirrusive, is chiefly ufed for a proliz 
manner of writing, &c. 
iGtionary Cae well be too diffufive: for a man is 
never 
