CALCULUS. 



whi 

 by 



and 



icKwas iirl giwii t""' vvitlioiit fatisfaftory demonftration, 



De M.)ivit, ill tlie 'IVar.fadions of the Royal Society, 



.•cllanca Analytica," p.S-. The above 



Mil 



form of M. Arbog;ift has many advantages ; it exhibits 

 compcndiouny the law of tlie coctTicitnts by means of the 

 fymbol D ; and, whe-n the operations indicated arc to be 

 adually pcrformtd, the coefficients eafily rtfalt in tcnns of 

 the polynomial quantities. 



The formula tor the funftion, being general, maiiifeftly 

 fcrvcs for the developemcnt of expreffions, inch as 

 ta +;3.v +yx- +, Sic, fin. (* +(3" +>;.- +, .^c.) &C. 



After having exhibited the general form for (p (a + 0.x -j- 

 yx'- -\-, &c.) M. Aibogall. (liews how from one term to de- 

 duce the next fuccetdnig, and likewife how to calculate any 

 term whatever of the developemcnt, indepcr.dently of all the 

 others. The length and intncacy of the calculations render 

 it impoflible for us to give details c f thefe methods. 



In the latter part of the firll article, the autliorapplies his 

 method of derivation to afiign the fum of the powers of the 

 root& of an equation, in terms of the coefficients of the equa- 

 tion ; and the formula which he deduces is remarkable for 

 the finiplicity of the law by which the coefficients are ex- 

 prefled. Vandcrmonde (Memoirs of the Academy, fyi, 

 p. 373.), Euler (Comms. F.it. 15 vol.). La Grange (Memoirs 

 of Berlin, 17O8), and Waring (IVLeditationes Algebraici?, 

 p. ].), having given general formula tor the fum of the 

 powers of the roots of an equation, M. Arbogall compares 

 his own with the demoudration of thofe, and (liews how they 

 follow from it. Whoever v.ill take the trouble of examining 

 thofe feveral formula will find them not only lefs fimply ex- 

 prelTctl than that of the prefent author, but lefs evidently and 

 lefs rigoroufly dcmonltrated. 



The author proceeds to the developement of funftious of 

 two or more polyn(;m]als, arranged according to the powers 

 of the fame letter. Suppote tiie feries a + ix -\- ex' +, 

 &c. and a +iS.v + yx~- +, etc. &c. are to be multiplied to- 

 gether ; then the prod.ft is "'"' jT ^' |- v +, Sec. or making 



b = 'D. a, c = Be', a, &c. /3 = D. a, y = Dc'. a, &c. the 

 coefficients of the terms aftefted with x, x', x^- &c. will be 

 (a. D. a + D. a. a), [a. D, ' a. -{- D. a D. a -\- Dr\ a. «), or 

 D {a. a), Y)r' (a a), and the coefficient of the term aifctted 

 with X" will be D.-", {a a), wh'ch may ealily be developed. 



Hence the form for the produft of any number of feries, 

 arranged according to the powers of x, may be determined. 

 In four feries, for iiitlaiice, of which the firft terms are fl, 

 a', a", a'", the origin of the derivations will be aa', a", a'" ; 

 and tlie coefficient o; the term affedled with x", will be Dc 

 {aa'd'a'"). 



In the developement of the produft of any two funftions 

 whatever of polynomials, for inllancc, of (? (a + bx +, &c.) 

 and if (a -\-0x-\-iLC.), the coefficient of the term affefted with 

 X" will be Y)c'. (?a. 'Pa); which, by foregoing methods, may 



be eafily developed. — Since — and A— i are equivalent ex- 

 A 



preffions, by the foregoing methods, fraftions fuch.as 



bx + ex' +, &c. r 



— — — -, r — ■ T-T may be 



^x -i-yx' +, &c.' (^— .v)'" ia—xy. (A— x)', &c. ' 



converted into feries of the form A -j-Bx + Cx'+ Dx' + &c. 

 M. Arbogaft, having applied his method to fuch fractions 

 as have been already mentioned, ffievvs how to deduce the 

 form for the developement of <p {a-\-bx-\-ex'-\-, &c., u + ^x 

 -^yx' +, &c.) and affignsthe form for the coefficient of the 

 term affefted with x". The law by which thefe forms are 

 regulated is fimple, and eafily comprehended. 



6 



This author's next objeft is the developement of fundlions 

 of one or more polynomials, arranged relatively to the 

 powers and to the produdtsof two or more different letters, 

 into feries arranged in the fame manner. Accordingly he 

 reduces it to the following general problem. " Any func- 

 tion whatever of one or more fimple, double, or triple po- 

 lynomials being given, to write immediately the feries of the 

 developement of thio fundion ; and, moreover, to write im- 

 mediately the developement of any term whatever of this fe- 

 riei, independently of the other terms." H. then proceeds 

 to various applications of derivations to recurring feries, as 

 well fimple as double, or triple, &c. ot any order whatever. 



The authors who have trtatcd of recurring feries are De 

 Movire, in his " Mifcelianet Analytica, and Doftrine of 

 Chances ;" Euler, in his " Introduftio in Analyfinlnfini- 

 toruni ;" La Grange, in " Melanges de Turin," in Me- 

 moirs prefented to the Academy of Paris, and in the Berlin 

 Memoirs ; and La Place, in Memoirs prefented to the Aca- 

 demy of Paris, and in the Memoirs of the Academy. To 

 the fubjeCt of the relearches of thefe great mathematicians, 

 M. Arbogall applies his method of derivations ; and he cer- 

 tainly obtains by it, in our opinion, expreffions very adini-- 

 rable for their limplicity, and for the facility with which 

 they can be expanded. In the methods of the authors 

 above mentioned, in order to find the general term, the de- 

 nominator of the generating fraiftion is refolved into its fac- 

 tors ; which is done by find'ng the roots of the denominator 

 put = ; coufequently, if the denominator exceeds an equa- 

 tion of the fourth degree, the general term cannot be found ; 

 — but, by the method of derivations, the general term isaf- 

 figned in terms of the coeffixicnts of the denominator of the 

 generating fraftioii and other quantities ; thus, if the gene- 



a + S:-: -f yx^ 



rating fradlons be — ■—, the expreffion for the 



a + bx + ex + dx^ 



general term of the refulting recurring feries is Dc" (aa — ' ), 



or a De". ff— '-J-/3. D'«— ' «— ' + y. Df' — '• a—\ (Dc^ a 



Dc^. a. Sec. being = 0). 



In the courfe of this article, M. Arbogaft ffiews how to 

 find at once a part of the general term of a recurring feries, 

 proceeding from feveral equal taftors in the denominator of 



F 

 the generating fraftion : thus, let q be the generating frac- 



p 



tion, and Nx (a — .v)«=Qj in -, put ei for.v, and fuppofe 



P' 



rj/to be what this fraflion becomes: then the required part 



rp/_ii_"71 

 ofthe general terra will be Di-ra — ' - , and Da= — i 



and n + 1 the index of the term. 



M. Arbogall determines the fine and cofine of any mulr 

 tiple angle in cofines of the fimple angle E. His demonllra- 

 tions for the forms expreffing the fines, cofines, &c. are clear 

 and rigorous. He alfo direftly difcuffes the difficult and com- 

 plicated fubjeft of double and triple recurring feries. The 

 next article of his treatife contains applications of the cal. 

 cuius of derivations to the general revcrfion of feries ; and 

 this part of his work is executed with fingular ability. His 

 next objedl is the ufe of derivations in the differential cal- 

 culus, which he confiders as a particular cafe of the calcu.. 

 lus of derivations. The 7th article of the author's work 

 confifts of three divifions : the firft containing the applica- 

 tion of the formula of derivation, to the developement of 

 the functions of polynomials, containing fines, cofines, &c. : 

 tlie fecond gives the application of the derivative calculus to 



the 



