THEOREM. 



confequently, when .v in each of thefc = o, we have 



« = -.0 = «^(7^) = =^ 



2 . 3 d.v 



= D; 



therefi 



ore, 



Sec Maclaurin's " Fluxions," and Boucharlat's " Calcul 

 DifFerentiel, &c." 



Moivre's Theorem. See the next article. 



' Myltinom'tal Theorem, fometimes called Moivre's theorem, 

 having been firfl difcovered by that author, ?3 a general 

 expreffion or formula for determining any power or root 

 of a given quantity confiding of any number of terms. 

 This theorem was firil pubhfhed by its author in N° 230. 

 of the Phil. Tranf. 1697 ; but it was afterwards fim- 

 plified by Euler in his " Calcul DifFerentiel," and the 

 fame has alfo been done by Arbogaft in his " Calcul des 

 Derivations." 



The general form of this theorem, as given in Bonny- 

 caftle's Algebra, is as follows : 



(A + A.V + A.\-- + Ax3 + . . . . Ax')" = A" + 



c I -J 3 n 



wAB. Jl + 



. o A ^ 



2m A B" 



a o 



+ 



3 m A B 



+ 



4m A B 



+ (m-i)ABf2A + (2"'-i)ABljc3_ +(3,n_i)AB 



+ {»; - 2) A B 



Where B = A", and B, B, B, &c. are the co-efficients 



00 12 3 



of the terms immediately preceding thofe in which they firft 

 appear ; and the manner of applying this theorem to any 

 particular cafe, is by fubflituting the numbers or letters in 

 the given example for A, A, A, &c. and the numerical 



O 1 3 



value of m for m. It would lead us too far to attempt the 

 demonftration of this theorem in this place, we muft, there- 

 fore, refer the reader for fuch information to the works 

 above-mentioned. 



Ncwtoman Theorem. See Binomial Theorem. 



Taylor's Theorem, an elegant and highly valuable formula, 

 which was firft publifhed by Dr. Brook Taylor in his 

 " Methodus Incrementorum," which is as follows ; wz. 



" If Y reprefent any funftion whatever of the variable 

 quantity x, and if x be increafed by any difference ^ x, the 

 value of Y, viz. Y -H AY, becomes (employing the dif- 

 ferential notation ) 



A»-d"Y Ax'd^Y 



Y + AY=:Y + 



AxAY 



A Y = 



A^-dY 



+ 



1 .2 



A.v"-d 



d.v 



Y 



+ 



1 . 2 . 3dx 



Ax'd3 Y 



+ &c. 



I . 2 . d.t- I . 2 . 3 d.v 

 The demonftration of this celebrated theorem is given very 

 concifely by La Croix, on the following principles. 



Let Y be any funftion of x, and let Y' denote what this 

 funftion becomes, when x becomes x -t- /', we may write 

 Y' = A + B/. + CA^ + D^3 ^ &c. 



in which developemeiit, it is obvious that A, B, C, &c. 

 are funftions of .v. 



If now we difference this equation with h variable and x 

 conftant, we obtain, dividing hy A.h, 



Y' 



y-; = B + 2C/j + 3D/J' + &c. 



Again, differencing with .v vaiiable and h conflant, we have 

 Y' dA dB, dC,^ dD,, , 



J- =-~ + h 4 -r— h' + -— h' + &C. 



Qx doc ax ax ax 



But as X and h enter cxattly in the fame manner, it follows 



Y' Y' 

 that — = -r-^ whence the firil of thefe feries is equal to the 



+ (2 m — 2) A B 

 + (m - 3) A B 



4A 



+ &c. 



focond ; equating, therefore, the co-efficients of the like 

 powers of h, we have 



2 d; 



Now 



B = - — , 



d X 



D = 



3 d .%■' 



&c. 



A = Y, B = 



whence 

 Y' = Y 



dY 



d.v 



,C = 



d-Y 

 I . 2 . d.\ 



l.D = 



d'Y 



dY 



h + 



dT 



I .2 



h^ + 



d'Y 



Or writing 



Y' = 

 we have 



AY ^"''^ 



I . 2 . 3 d .V 

 Y + A Y, and .-« -i- A = .-v -I- A x. 



,3 d: 



//-H&c. 



1 .Ax 



+ 



^d = Y 



, 2 . d .V 



+ 



A.v3d3Y 



+ &c. 



I . 2 . 3 dj 

 See La Croix " Calcul Differentiel," p. 21. 



Trinomial Theorem is only a particular cafe of the MM- 

 nomialTheorem, which fee. 



Wdfon's Theorem is a curious formula relative to prime 

 numbers, publifhed firfl by Waring in his " Meditationes 

 AlgebraicjE," which is as follows. 



" If n be any prime number, then will 



I . 2 . 3 . 4, &c. {« - i) + I 

 be divifible by n." 



This curious theorem was not demonflrated by fir John 

 Wilfon, wlio firfl difcovered it, nor by Waring, by whom it 

 was firfl made pubhc ; it lias, however. Since received 

 different demonflrations from Lagrange, Gaufs, &c. the 

 latter of which is very fimple, and has been adopted by 

 Barlow in his " Theory of Numbers," to which work we 

 beg to refer our readers, as it would require more room 

 than we can allow ourfelves to give it at full length in this 

 place. 



The above include, we believe, all thofe theorems which 

 are known by any particular defignation ; there are, doubt- 

 lefs, many others equally important, and whicli are equally 

 entitled to bear the names of tlieir refpeftivo authors, 

 but cuflom lias not fanftioned the adoption ; and we have, 

 therefore, not introduced them. 



THEO. 



