## advanced algebra

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. This small volume contains what remains of the course in Algebra, after mtltricuIntion, to t,hc stutlents in the Colleges of Civil Engineering, Mines, and Nechallic Arts in the University of California. It is intended as a, contiiuntion of the excellent work on algebra by Nr. John 3. Clarlre, of the Mathematical Department of the University and it is thought it will, in connection with Clarkes Algebra, or with any mork of similar cope, furnish a good and suficient preparatioil for those who intend to pursue tho higher mathematics. The constant aim and endeavor throughout has been so to present the various topics discussed ns to render then1 easy of comprehension by the undergraduate student. ARTICLE . Summation of series of frnctiolls of certnin forms ....... 2- 5 Method of De JIoivre ............................. 6- 7 If a is a root. first member divisible by u and conversely . 9 Eliminatiou ......................................... 10-14 Every equation has at Least one root .................. 17 Eery equation of the nth degree hnb I L roots .......... 18 Composition of Equtltions ............................ 20 Every etyution of an odd degree LRS at least one real root 28 Every equntion of cven degree aud absolute term negntie has trpo real roots of differeut sins . ................ 29 Every equation has an ereu number of real positive roots when absolute term positive an odd number if term is uegntire ...................................... 30 Chnnging sigu of alternate terms changes signs of roots 31 Ues Cartes Rule ................ ..I.. ............. 39 De Guns Criterion .................................. 34 To transform to an equntion hose root8 shall be multiples of formcr roots .............................. 36 To clear of friwtions and yet keep coefficient of highest powerunity .................................... 37 To transform to an equation whose roots shall be reciprocal of the former roots .......................... 39 Recurring Equslions ................................ 42-43 To transform to an equation whose roots shall be squares of tha former roots .............................. 44 To transform to rtn eqlintion where the roota shall be greater or less by n certain quantity ............... 45 Another mode of finding transformed coefficients ........ 46 ................................ Syntheticul Division 47 To transform to an equation wanting a second or any otherparticular term ............................ 49 Derived Polynominls ............................... 50 I vi COXTENTS . ARTICLE . PAOE Relations of Derived Polynomials to the roots of nn equntion ........................................ 51 To discover equal roots. if any ...................... .G 2-53 national Integral Function .......................... 53 45 47-48 50 Any term of such function can be made to contain the sum of a11 which precede or succeed ............... 54 50-51 Enw of Continuity .................................. 56 G1 Limits of Roots. definitions of ....................... 57 52-53 iscLaurinis Limit .................................. 53 Ordinary Superior Limit of Positive Roots ............. 59 54-55 Inferior Limit of Positive Itoots ..................... GO 55-56 Superior Limit of Negative Roots .................... F1 56 Inferior Limit of Negntivo Roots .................... G2 513 Netons Limit ..................................... E3 56 Rudans Test of Imnginary Roots ..................... 64 58 If the sbstitution of p and q give different signs there is ono real root between them ........................ --This text refers to the Paperback edition.

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