A treatise of algebra, in three parts. Containing

261쪽

either case, the series. nvem Miutly that con-1nt of such termx and a se; of the first ternas williive amea value os the roo required. 99. Is a series se vis require sto the propos ed equatiun that hali converge the boner, the les I is in respeet, a to find the first ter of this series, e mali supposie a to vanissi, an extracting the reo os the equation

263쪽

ioo. When it is require t M a serie, si x stat hal onverge oner, the Oreale fis

264쪽

on the suppositionifa being ery litile. Thus, o findis value sor, in the quation-ila in m --γ', o that hali converge thesione die mater , is in respectis supposeis to vanish, amicili remaining term Win. giveo I mi, oris ras . so that when a is vastly great, it appears that πιαί nearly. . Bucto have the value os, more accurately, putra A then

265쪽

whicli series converges the oone the re ter fissupposed tote a centi re in Ofρ. - . ioi. In the solutio of the sir Example thos termi ere always compares in orde to determute, , , c. in i michis and thosequantities, g, , were separaret os sewest dimensions. ut in the second Example tho se term Uere compared in hic and the luantities , , c. ere, leas dimension. 2- parastin Aiathese always suevit proper te nasio bes napare together, ecause thhy ecomevasti greater than the est, in the respuet ive hypotheses. In generat, o deterinius . the rst, . ori asy,

accordinias is suppo sed tot vasti littici, vasti great, in re pectis a.

266쪽

CAAP io. ALGEBRA. et 'Thus to determine the fies term os a converginiseries expressing the value os in the last

sor a: the equation comes; - , -- λwhere their term is of more dimensions hanthe assume term Ux, - and the seconfossewer so that the two firmierm cannot be ne laete in respectis the two last, eithe wheniis ver greatior ver lituri compared with Nor are the term x', Ix, fit to e compared together in orde to obtain the fies terna Daseries sor, so the like reason. Butis may be compares,ith ae x, as sib ab illi a se thacend These motive the fir ter of a series that converges thestone the lest is as Σαω gives the rit term os a series that converges the oone thegreater aris. The la series a give in theprecedin article. The comparingis' uti ci x ives these trio leties,

267쪽

And these series give three values of whena is ver litile the la of hic is it sel also ver litue in that case, ascit appears indeed stoli the quation, that wheno vanishes, in three values of iecome, a, a, and o, Gause, heno animes, the equation becomes ala

S io et I appeam sussicienti stom What,ehave said that when an equatiun is promta involvingis ansa, and the value os xus requires in a converginiseries, the dissiculino fi ingthe rit tir of thes series is reduce to this. et find what ternas assume in Orde to determine a value osci expresse in sonte dimensi sos and ut illiive suctis value os ir, aciuta. stitutia sor it in the other term will mahe themalli in ore dimentions os o alli flat dimensions os a than mose assume ter To determine hi , dra AE and AC at right angle to ach ther, complete the parallel gnam ABCD an dividerit into equat quares,as in the figure. In these quare place theposversis dro A toward C, and thei Mersola rom. A toWard B, and in annother squam place that power of x that is directi belowrit in the line AC, and that power os o that is in aparallel inith it in the line AB so that the index of x in any quare ma expreis it distance sum the sine AB, and the index of a in my, 'u re

268쪽

trices progression in the vertical columes B, orthe horizontal AC, and thei paralles , ut au se in the term taen in an oblique straigheline hatever sor in an such term it is manifes that the indices offandis villie in arithmetica progression The indices o, causethos ternis ni remove equest stoin the line AC,'or approach quali to it, and the indices os

269쪽

he in arithmetical progression, hecause theselemis equali remove ironi, or approach to the line

AB. Thus for exampie, in the termia a a , γ' thes indices of γ decreasin by the

common difference et while the indices of x in- creasse in the progression os thes natura numbers, inexommon ratio of the term is at et Hom tu last observation stat lasanymo te si supposed equat, then ali hester in the fame strato line, illi these terms,ill be equa ' hecause by supposing these No te sequat, the common ratio is supposed tote a ratio ocequali ty, and om his it sollows, statis D substitute every where se x the valuestat arises, it by supposing any two te sequat, expresse in the powers of y the dimentasions os, in ali the term that are found in thesame straight line,illie equale ut the db. mensions os, in the ternis ab e stat line ilibe greater than in tho in stat line,' alid

the dimensions of in the term lesiis thessa id line, ill bestis than ita dimension in that line.'