Suppofe farr-xx az/rr-xx , and you Naa-2ax+rr √xx+2xx+rr would find the Value of z, proceed thus. Equation farr-xx) az√rr-xx = Vaa-2ax+rr) √zz+2xx+rr\ 2 farr-xx=az√rr—xxx √√ aa—2ax+rr 2ax, &c. √xx+2xx+rr 2xVx2+3 fa √rr—xx × √ zz + 2 xx + rr = az√rr-xxx Jaa-2ax+rr 3√rr-xx 4 fa√zz+2zx+rr = az√ aa—2ax+rr 5fvzx+2xx+rr=z√ aa—2ax+rr 4÷a 502 6 f2 × x2+2xx+rr = z2 × aa—zaz+rr, EXAMPLE. Q. E. I. Hence by Form 14. Table of Theorems for Converging Series, the above Equation may easily be folv'd. Q. E. I. Sometimes Substitution renders the Work more easy, wherein an Equation is involved in Surds, as the following Examples will exhibit. EXAMPLE. √b2+3x2× √c2+3x2 × a = 2x√/b2- b2 +3x2 × c2+3x2 × a2 = 4x2 × b2x2 +8x2 √c2x2 × √ b2x2 + 4+* —x2 4x2 b2c2a2 +3c2a2x2 + 3b2a2x2 + 9a2 x4 = 462x2+4c2x2- 8x4 +8x2 √/c2 —x2 Now for b2c3a2, put m; for 3c2a2 + m+nx2+px4: = 8x2 √√ c2 —x2 × √ b2-9c2 Hence by ordering the Terms you will have the Equation in the eighth Power. EXAMPLE. tions, but differently fet down, this being the old, the other the new Way of Notation, to find the Values of u and ≈ ? Subst. y3, for u and x4 for z. 2 ✔z. Which are the fame Equa Here are two unknown Quantities, and two Equations, by which it will be eafy to find the Value of each by the Rules already laid down, viz. y=x=3. There are a great many Cafes befides, which may by the Judgment of the Algebraift, from what I have laid down, be contracted, or reduced lower, or exterminated by Subftitution, which cannot be brought under any Rule, and can only come by frequent Practice. I think I have faid what is neceffary to enable the Reader, with little Practice, to folve any Equation analytically in the most concife and elegant Manner. I fhall defift giving any more Examples, and make a Tranfition to the other Part, how to folve any adfected Equation into Numbers (after they have been order'd according to our Method aforefaid) by an univerfal Method of Converging Series. ΑΝ ΑΝ UNIVERSAL METHOD OF Converging Series. Handled in a very easy, plain and expeditious Method. DEFINITION. A Series which approaches continually to the Truth, is faid to converge, and which continually goes from it is faid to diverge. COROLLAR Y. Therefore a Series of Fractions continually decreafing are converging, but others whofe Terms continually increase are diverging. Now in all Equations higher than a Quadratic (if adfected) the best Way is to folve the fame by a Recourse had to Infinite or Converging Series, and the common Method, that which I call the most easy, affume m+n for the Value of your unknown Quantity, that is affume m = Root of your Equation, as near as you can (tho' if you affume never fo far from the true Root, yet it will by renewing the Operation converge to it) and affuming +orn for the Deficiency, then it will be m+n, or m―n Root. Therefore for the Usefulness of Difpatch I have raised the following Table to the 8th Power upon the above Affumption, N 2 that |