THEOREM. 



But fince n — n = o, the (« + i)th differences =: o ; and 

 fincc ■a;" ~ ' = w" = I , the nth differences become 



„(„_,)(„_ 2) („- 3) 3.2.1 d." 



It may not be amifs to obferve, tliat we have only em- 

 ployed the firft term of the feveral orders of differences, 

 which however is fufBcient for our purpofe, fince it is ob- 

 vious that the nth difference can have but one term ; for 

 the developcment of {p + d)"give8 n + i terms ; and fince 

 one term vanifhes with every difference, the firfl difference 

 will have 11 terms, the fccond n — i, the third n — 2, &c. ; 

 and confequcntly the nth difference will have « — (" ~ 

 = I term only. See Irifh Tranfactions, vol. xi. or Monthly 

 Review, vol. Ixxiv. 



Coles's Theorem, or Cotejian Theorem. — The geometrical 

 properties of this very interefling theorem are explained 

 under the article Cotesian" Theorem; it will only be necef- 

 fary therefore in this place to ftate the fame analytically. In 

 this cafe, the general enunciation is : 



" All the imaginary roots of the binomial equation x" — 

 1=0, are contained in the general formula x^ — 2 cof. 



2 k;r 



'- .V -I- I = o ; and thofe of x" + I = o, in the formula 



n 



. (2/5+ l)^ ,, . 



X — 2 col. — .V -}- I = o, « bemg any mteger not 



divifible by n, and ^ reprefenting the femi-circumfcrence." 

 See Reciprocal Equations. 



Euler's Theorem is ufed to denote the theorem or formula 

 firft given by this author, for afcertaining the direft integra- 

 bility of differential equations, which is as follows. The 

 equation being reduced to the form 



Md.v + Ndji = o, 



M N 

 where M and N are functions of x and v ; if ^— = -r— , then 



ay ax 



the integration may be obtained by a direft procefs ; but if 



this equality have not place, the integration can then only 



be effefted by indireft means, which frequently involve con- 



fiderable difficulty. 



Fermai's Theorem There are feveral theorems in the 



theory of numbers which are due to this ingenious analyft ; 



but that which is more particularly defigned by Fermat's 



theorem is this; •viz. " Neither the fum nor difference of any 



two integral powers, above the fquare, can be equal to a 



rational power of the fame dimenfion:" or,' which is the 



fame, the equation 



is always impoffible in rational numbers, if « be greater 

 than 2. 



The cafes of n = 3 and n = 4 have been demonftrated ; 

 but notwithftanding the numerous attempts of the 1110ft ce- 

 lebrated anal)'fts of the laft and of the prefent age, the cafe 

 of n = 5, and all the fucceeding values of «, remain with- 

 out demonftration ; and as this is now the only theorem of 

 this author which has not fubmitted to the power of the 

 modern analyfis, the National Inftitute of France has made 

 it the fubjeft of the prize of 3000 francs, to be decided 

 by 1818. 



Under the article Numbbrs, amongft the mifcellaneous 

 propofitions, we have mentioned another theorem of Fermat's, 

 which had not then been demonftrated, but which has fince 

 been effefted by M. Cauchy, correfponding member of the 

 Inftitute. The reader will alfo find fome farther remarks 

 relative to the equation jc" + f = z", under our article 

 Power. 



Gau/s's Theorem is ufed to denote a theorem invented by 

 this diltinguifhcd mathematician, for the folution of certaii; 

 binomial equations. We have feen, in the article Recipro- 

 cal Equations, \n what manner the roots of binomial equations 

 may be obtained by means of a table of fines and cofines ; 

 but Gaufs's theorem is the converfe of this, and fhews in 

 what manner the fines and cofines of certain angles may be 

 obtained, by tlie numerical folution of fuch equations. See 

 Polygon. 



Guldin's Theorem h the fame as the Centrobaryc Method; 

 which fee. 



Lagrange's Theorem is commonly ufed to denote the gene- 

 ral formula affumed by Lagrange as the foundation of his 

 theory of funftions ; which may be thus enunciated. 



" If C X be any funftion whatever of a variable quantity 

 X, and if x changes its value, and becomes x + i, then the 

 <i [x + i) may be reprefented or refolved into a feries of 

 tlie form 



?> [x + i) = f .V + Pi + Qi' + Ri' + &e. 



in which the co-efBcients of the powers of i are new func- 

 tions of .V, derived from the primitive funftion x, indepen- 

 dent of / ; and, moreover, that every co-efficient is derived 

 from the preceding one, in the fame manner as the firll m 

 derived from the original funftion." See Functions. 



Leibnitz's Theorem is a theorem propofed by this author foi- 

 differencing under the i\gnf, and it may be exhibited under 



, , d/Md.v ,dM^ ... Au ^ . 



the form -^^-^j = I — — d x, where JVl := -j— , u being 



ay -^ ay ax 



any funftion of x and y. 



„. d M d « 



omce - — ,— = 



— by the known principles of the 



Q.vdj ay ax 



differential calculus ; if we make u = / M d .■«, we fhall have 



d« ,, d^a dM _,. . ., 



- — = JVl, - — -^— = -r — ; and integrating with regard to x, 

 ax ax ay ay 00 o 



we fhall find 



/• d" a 



.^ d V d X dy *^ d V 



d X dy J dy d x 



This is called by Leibnitz differentiatio decurva in curugm, 

 becaufe in the queftion which he propofed to refolve, he 

 paffed from one curve to another of the fame fpecies, by 

 making one of the conftant quantities variable. See La 

 Croix " Calcul Integral." 



AJaclaurin's Theorem is a formula which we owe to this 

 author for expreffing any funttion y, of a variable quantity x ; 

 •viz. adopting the differential notation, 



x^ -\- &e- 



where {y), [j^\ (-j— ^j, &c. reprefent what thefe feveral 



quantities become when x = o. 



Let jr c= A + B.M + C.v^ + Dk^ + &c. 

 differencing, and dividing by d x, we have 



dx 



ix 



d\y 



dx3 



= B -f- 2C.-C + 



T^.= 2C -I- 



3D.-e' + &c. 

 2 .3 D« + &c. 



+ 2 . 3 D -f &c. 



coufe- 



