THEOREM. 



povtaiice, ciiviority, or otlifr coiificlcrations, have retained 

 particular denomiiiatioiis, uiicler which they are frequently 

 referred to bv modern authors : it is therefore neccliary to 

 liaTe them fo clafTed, that a reader may be able to afcertain 

 the principles on which they are founded, the purpofes tliey 

 are intended to anfwer, and the cafes to which they will 

 heft apply. The theorems to wliich we here allude, are 

 Bernoulli's theorem, the Binomial theorem, Cotcs's, Taylor's, 

 Maclaurin's, &o. 



BcrnouUi's Theorem, is a general formula for the de- 

 velopement of any fluent or integral, of any propofed 

 fluxion or diflercntial ; which may be llated as follows : iiiz. 

 X being any function of .v, 



■■c' d'X .v' 



1.2 d x - " I . 2 . 3 



d'X .V* 



- -j— - . , &c. 



d .V ^ 1.2.3.4 



Let us denote by Y, the value of this integral when x =: o, 



X, ^, 414. Sec- bemg alfo denoted by Y', Y", Y'", &c. 

 d .V d .V 



and we (hall have generally. 



/xd.v = x^-'\2i 



-^ I d .V 



/Xd.v = Y + Y' — 



+ Y". 



- + Y'" - 

 2 I 



+ &c. 



1.2 1.2.3 



Now, in order to pafs from the general value of yX d x, 

 which we Ihall reprefent by y, to that which anfwers to tlie 

 cafe of X = o, it is evident that we mufl in Taylor's formula 

 make A = ~ x, which gives 



^=^-di:--T^d~ 



.V 



I . 2 



v-! 



fubftituting in this equation, in the place oi y. 



1.2. 



y 



+ &c. 



d y d*j' 



d X d x' 



, &c. 



their refpeftive values, and taking that of y X d .y, wc fhall 

 have 



rxdx=Y+x. -— . +:t-T- Sec. 



J I dxi.2 dxi.2.3 



the quantity Y being ftill a conftant arbitrary. By inte- 

 grating, we arrive alfo at this developement : thus, if we 

 decompofe the differential X d x into its two faftors X and 

 d X, and integrate the fecond, we (hall have 



/Xdx = Xx-/xdX. 



But 



^dX , X dX , . -d^X 



= / -] . X d X = i X ^ i / X - — — 



.^dx ^ dx ■'.'dx 





d X 



d'X 



dx ' 

 . d'X 



^ d'X 



, ^d^X , ^ 3d3X 



J dx' J Ax' 



&c. 



x' dx= i x" 

 &c. 



And putting fuccelTively for/x d X,/x 

 refpeftive values, there refults, 



/"Xdx 



dx 



d^X 



d.-, 



&c. 

 d' X 



x dX 



A. -; — 



I dx 



2 "*" dx' 



3 



-&c. 





&c. their 



which is the theorem of John Bernoulli, and is the fame 

 with regard to the integral calhilus, as that of Taylor to the 

 differential. 



Binomial Theorem, or Newtonian theorem, ii a (/eiieral 

 formula for the developement of any binomial of the 



fomi (a + x)" ; viz. 



{a -I- x) " = 



" r m f x\ m m — « / X \ ' m m — n 



a- X]i + -{ — ) + -. (_ +- . — 



(. n \ a / n 2 n \a / n 2 n 



m — 2 n / X V „ ? 

 . — -) + &c. \ 



See Binomial Theorem. 



Briggs's Theorem. — There are more than one formula that 

 have received this defignation, but we believe that the fol- 

 lowing is generally underftood to be iipplied ; vi%. 



" The «th differences of any confecutive nth powers, or 

 of any nth powers whofc roots are in arithmetical pro- 

 greffion, are expreffed by the formula 



„(„_,)(„_ 2) („_ 3) (n- 4) . . . i.d» 



d being the common difference of the roots." 



The demonftration of this theorem is commonly made to 

 depend upon principles drawn from the fluxional analyfis j 

 but we prefer giving a fl^etch of that which appeared in 

 vol. xi. of the Irifli Tranfaftions by Mr. Burk, being de- 

 duced from the moft elementary confiderations. 



We know that the expanfion of (^ -|- x)*, is of the form 



p" + nxp'-'+ Cxy-'+ Dx'/)"-' + &C. 



Let 0, p, q, r, s. Sec. be the terms of any decrealing arith- 

 metical progrefTion, of which the difference is d ; then lince 

 o=/ + d, ^ = ^ + d, &c. we have 



0"= {p + d)''=p" + ndp"-+Cd^p'-'+ &c. 

 /."= {q + dY = q" + ndq"-'+Cd'q"-'+ &c. 

 q"={r + d)' = r" + ndr"'+ Ci^r'-'+SiC. 

 Taking the firfl, fecond, third, &c. differences, we have 



Fiijl Differences 



ndp''^'+ Cd'^p"~'+ &c. 

 ndq'-'+ Cd'q'—'+ Sic. 

 nd;-"-' -1- Cd'r'-'-f- &c. 



And fmce />"-'= {q + d)"~' = q"~' + (n — ijdy""- &c. 

 we have for the 



Second Differences 



«d (n — l) dq"-- + &c. 



«d (n- I) dr"—--\- &c. 



nd (n — i) d j"~'-t- &c. 



Third Differences 



nd{tt— l) d (« — 2) dr"-» + &c. 

 nd {« — i) d (« — 2) dj"-"+ &c. 



Fourth Differences 

 nd{n— l)d(n — 2)dj"-'(n — 3) dr"-'+ &c. 

 Whence, by an infallible and obvious deduAion, 



eth D'fferences 



«d (n — ») d (n — 2) (n — n — I ) d'oi" " 



(a + \)th Differences 



nd(n— i)d(n — 2) {n — n)dv~' . 



3 Q 2 But 



