Sect. IX.) But as this method is, in many cafes, very laborious, and in others altogether impracticable, espe-.. cially, where feveral furds are concerned in the fame. equation, it may not be amifs to fhew how the method of converging feries's may be alfo extended to these cafes, without any fuch previous reduction. In order to which it will be neceffary to premife, that if A + B represents a compound quantity, confifting of two terms, and the latter (B) be but finall in comparison of the former; then will, All which will appear evident from the general theorem at p. 41: from whence thefe particular equations, or theorems, may be continued at pleafure; the values here exhibited being nothing more than the two first terms of the feries there given. But now, to apply them to the purpose above mentioned, let there be given √ I + x2 + √2 + x2 + √3+x=10, as an example, where, x being about 3, let 3 + be therefore subftituted for x, rejecting all the powers of e above the first, firft, as inconfiderable, and then the given equation will stand thus, V10 + 6e + VII + be + √ 12 + 6e 10: but, by Theorem 2, V10 + 6e will be VIO 3√10xe + nearly; for, in this cafe, A = 10, 13 and B 6e, and therefore A B6, A B + = √10+ 2A X 3V1cxe in like manner is 11+ be = √ II + e 3/11 Xe &c. and confequently ✔10+310x+ 3/11xe 3√12xe 1 2 10 = 10; which, contracted, gives 9.944 + 2.718e = 10; whence 2.718% = .056 and e=.0205; confequently x 3.0205, nearly. Wherefore, to repeat the operation, let 3.0205 +e be now fubftituted for x; then will √10.1234 + 6.041e + √11.12342 + 604 1e + 12.12342 + 6.041e 10; whence, by Theorem 2, =.000447, and therefore x 3.020947; which is true to the laft place. Again, let it be proposed to find the root of the equa x; theu, by proceeding as before, we fhall have + 20 + e × √ 4.05 + 40e = 34: but 2.5 I (by Theorem 3) is nearly = √516 + 45e 1516 45e 1032 x 516 20e and (by Theorem 2) 405 + 40e √ 405 + : which values being fubftituted above, our equation becomes √405 .00192e + 1 √516 1032x516 34, that is, 400 + 20e x .044022 20+ex.804984 + .0398e 34; whence rejectinge, &c. we have 1.713e = .1915; and confequently e.1118. Thirdly, let there be given √1 −x + VI I-2x2+ 13x3 2. Then, if 0.5+e be fubftituted therein for x, it will become Vo.5e + √ c.5—2e + √0.625 — 2.25e = 2; or √o.5-√o.5 xe + √0.5 √0.5 × 2e + √.625- 2,25e = 2; whence 21.625 3.545e.204, e=.057, and x the operation being repeated, the come out = .5516. Laftly, let there be given 1 + x = 6,5. Here, by writing 3 + e 0.557, with which next value of x will ceeding as above, we fhall have 2 + + 10 + I 4 103 X 2e 281+ 281 × 27e = 6.5, that is, 6.455 4 X 28 ×· 1.23e = 6.5; whence e=.036, and x = 3.036. It may be observed that this method, as all the powers of e above the first are rejected, only doubles the number of places, at each operation: but, from what is therein thewn, it is easy to see how it may be extended, fo as to triple, or even quadruple, that number; but then the trouble, in every operation, would be increased in proportion, fo that little, or no advantage could be reaped therefrom. 3 Hitherto Hitherto we have treated of equations which include one unknown quantity, only. If there be two equations given, and as many quantities (x and y) to be determined, one of thofe quantities muft firft be exterminated, and the two equations reduced to one, according to what is fhewn in Sect. 9. But, if this cannot be readily done (which is fometimes the cafe) and the unknown quantities be fo entangled as to render that way impracticable, the following method may be of ufe. Let the values of x and y be affumed pretty near the truth (which, from the nature of the problem, may always be done); and let the values fo affumed be denoted by f, and g, and what they want of truth by s, and t refpectively; that is, let ƒ + s = x, and g +t=y: fubftitute these values in both equations, rejecting (by reafon of their smallness) all the terms wherein more than one fingle dimenfion of the quantities s and tare concerned: let all the terms in the first equation, which are affected by s, be collected under their proper figns, and denoted by As; in like manner, let thofe affected by t, be denoted by Bt; and those affected neither by s nor t, by Q: moreover, let the terms of the second equation, wherein s and t are concerned, be denoted by as, and bt, refpectively; and let the known terms, on the right-hand side of this equation, or those in which neither s, nor t enters, be reprefented by q. Then the equations (be they of what kind they will) will ftand thus, As+ Bt = Q, and as + bt = q. By multiplying the former of which by b, and the latter by B, and then fubtracting the one from the other, we shall have bAs BasbQ Bq; and therefore s - Q-Bq; whence x (= ƒ + s) is given, AbaB Again, by multiplying the former equation by a, and the latter by A, &c. we fhall have aBt Aq, and therefore t= @Q — Aq (=g+) is likewise given. Abt=aQAq-aQ whence 1 It is eafy to fee that this method is alfo applicable, in cafe of three, or four equations, and as many unknown quantities, but as these are cafes that feldom occur in the refolution of problems; and, when they do, are reducible to thofe already confidered, it will be needlefs to take further notice of them here: I fhall, therefore, content myself with giving an example, or two, of the use of what is above laid down. 1. Let there be given *4 + 4 = 10000, and x -ys = 25000; to find x and y. Then, by writing f+s = x, 8+ =y, and proceeding according to the aforegoing directions, we fhall have + 4ƒ3s + 8* + 4g3t 10000, and f3 + 5 ƒts—gs 5gt 25000, or 4f's + 4g3t=10000-fg, and 5f4s-58*t = 25000 + gs-fs: therefore, in this cafe, A = 4ƒ3, B = 4g, Q= 10000-f484, a = 5f4, b=5g, and q = 2500 + g3 —ƒ3. But it appears, from the firft of the two given equations, that a must be fomething less than 10, and from the fecond that y must be less than x : I therefore takeƒ= 9, and g = 8; and then A becomes 2916, B = 2048, Q = 1- 657, a = 32805, b=20480, q = 1281; and therebQ-Bq bA-Ba fore's ( 0.13, and ( Aq bA Ba -0.14; hence x 8.87, and y = 7.86, nearly. Therefore, in order to repeat the operation, let ƒ be now taken = 8.87, and g 7.86; then will A≈2791, B = 1942, Q = 6.76, a = 30950, b = — 19083, bQ- Bq and q 94; confequently (= = .00047, Ab aB .00415; whence x = 8.87047, and y = 7.85585; both which values are true to the laft figure. Example 2. Let there be given 20x + xy}} + 8x1 given equations, by writing f+s for x, and g + t for |