to radius 1 and tang. √/==, which is the first form of the fluent in n°. XI. 5. And, for the latter form of the fluent in the same n°; because the coefficient of the former of these, viz, is na' na 2 double of the coefficient of the latter, therefore the arc in the latter case, must be double the arc in the former. But the cosine of double an arc, to radius 1 and tangent t, is In by the former case, this substi 1-12 1412 ; and because t2 = tuted for t in the cosine a 1-12 1+12) it becomes as in the latter case of the 11th form. 6. Again, for the first case of the fluent in the 13th form. By art. 61 vol. 2, the fluxion of the circular arc z, to radius » to the radius 1. hence (1- y2) = 1 √ (1 − = ) = √ — × √ (a−x”), and j =√ — × \nx In-1 then these two being substituted in the value of %, give ż is = 2, and therefore its fluent isz, that is a x arc to sine√ as in the table of forms, for the first case of form XIII. 7. And, as the coefficient, in the latter case of the said 2 form, is the half of the coefficient in the former case, 92 therefore the arc in the latter case must be double of the arc in the former. But, by trigonometry, the versed sine of double an arc, to sine y and radius 1, is 292, and, by the former case, 2y2 = -; therefore x arc to the versed sine is the fluent, as in the 2d case of form XIII. 8. Again 8. Again, for the first case of fluent in the 15th form. By art. 61 vol. 2, the fluxion of the circular arc %, to radius r then these two being substituted in the value of ż, give ż or √TM, as in the table of forms, for the first x arc to secant √ case of form xv. 9. And, as the coefficient 2 na the coefficient of the former said form, is the half of case, therefore the arc in the latter case must be double the arc in the former. But, by trigonometry, the cosine of the double arc, to secant s and radius 1, is -1; and, by the 2a- 2 nNax arc to cosine 24 is the fluent, as in the 2d case of form xv. 2 na Or, the same fluent will be x arc to cosine, because the cosine of an arc, is the reciprocal of its secant. 10. It has been just above remarked, that several of the tabular forms of fluents are easily shown to be true, by taking the fluxions of those forms, and finding they come out the same as the given fluxions. But they may also be determined in a more direct manner, by the transformation of the given fluxions to another form. Thus, omitting the first form, as too evident to need any explanation, the 2d form is x = (a + x”)m−1"-x, where the exponent (n-1) of the unknown quantity without the vinculum, is 1 less than (n) that under the same. Here, putting y = the compound quantity a +": then is y = nx”—1 x, and ¿ = ут (a+xn) m mn = mn 11. By the above example it appears, that such form of Auxion admits of a fluent in finite terms, when the index (n-1) of the variable quantity (x) without the vinculum, is less by 1 than n, the index of the same quantity under the vinculum. But it will also be found, by a like process, that the same thing takes place in such forms as (a+x")TMxn−1x, where the exponent (cn 1) without the vinculum, is 1 less than any multiple (c) of that (n) under the vinculum. And further, that the fluent, in each case, will consist of as many terms as are denoted by the integer number c; viz, of one term when c = 1, of two terms when c = 2, of three terms when c3, and so on. 12. Thus, in the general form, ż = {a + x»)” x‹−1ƒ‚ putting as before, a + x" = y; then is "y-a, and its fluxion nx-1=j, or r"-1; = ў and ren- or xn-n n 1 n x"−1 ; = — ( y − a)c−1ÿ;'; also (a + .x")" = ym: these values being now substituted in the general form proposed, give x = · (y — aɣc—1ymj. Now, if the compound quantity (ya)- be expanded by the binomial theorem, and each term multiplied by yy, that fluxion becomes &c); then the fluent of every term, being taken by art. 36, it is ym+c c-1 aym+c-1 c-1 c-2 4zym %= = 2 m+c-2 c-1.c-2 (12 + m+c-1 d-2 2y2 2 - &c), c-1.c-2.c-3 Q3TM &c), putting d = m+c, for the general form of the fluent; where, c being a whole number, the multipliers c— 1, c−2, c-3, &c, will become equal to nothing, after the first c terms, and therefore the series will then terminate, and exhibit the fluent in that number of terms; viz, there will be only the first term when c=1, but the first two terms when c = 2, and the first three terms when c3, and so on.Except however the cases in which m is some negative number equal to or less than c; in which cases the divisors, m +€, m + c − 1, m+c 2, &c, becoming equal to nothing, before the multipliers c-1, c-2, &c, the corresponding terms of the series, being divided by 0, will be infinite: and then the fluent is said to fail, as in such case nothing can be determined from it. 13. Besides this form of the fluent, there are other me thods of proceeding, by which other forms of fluents are VOL. III. derived, of the given fluxion &= (a + x")TM xc# −1, which are of use when the foregoing form fails, or runs into an infinite series; some results of which are given both by Mr. Simpson and Mr. Landen. The two following processes are after the manner of the former author. 14. The given fluxion being (a + x”)”xn−1; its fluent may be assumed equal to (a + ")" + multiplied by a general series, in terms of the powers of r combined with assumed unknown coefficients, which series may be either ascending or descending, that is, having the indices either increasing or decreasing; I 35 viz, (a + ")+1× (A.x" + Ex2-s+ C.x2-25 + Dt'—35 + &c), or (a + x")" +1× (Ax" + Bx' +'3 + cx2 + 25 + Dx” + 3o + &c). And first, for the former of these, take its fluxion in the usual way, which put equal to the given fluxion (a + xn)m -x, then divide the whole equation by the factors that may be common to all the terms; after which, by comparing the like indices and the coefficients of the like terms, the values of the assumed indices and coefficients will be determined, and consequently the whole fluent. Thus, the former assumed series in fluxions is, n(m + 1)x"1x(a + x")" × (Ax2 + Bx2- ' + C2-25 &c)+ (a+x")+1× (r'Ax ̃−1+(r−s) Bx ̃—♪~1 + (r−2s) c.r” — 25 — &c); this being put equal to the given fluxion (a + xo1)mxcn=1ƒ, and the whole equation divided by (a+"), there results n(m+1).x′′ × (Ax2 +B.x2- + cx2-25+ D.x2-35+ &c) ܨܐ 1511 -S +(a+x")× (rax” + (r− s)вxTM-s+(r−2s)cx2-2&c) ( Hence, by actually multiplying, and collecting the coefficients of the like powers of x, there results +r n(m + 1)} cn -I +r-25 +(r—s)aBx*—3 &c Here, by comparing the greatest indices of r, in the first and second terms, it gives r + n = cn, and r + n − s = r; which give r = (c 1)n, and n = s. Then these values being substituted in the last series, it becomes - (c+m)naxTM+(c+m−1)nâxTM — n +(c+m- 2) ncx -x+(c−1)n@AxTM”—”+ (c−2)naBxTM — &c Now, comparing the coefficients of the like terms, and ting c+m=d, there result these equalities: == &c; which values of A, B, C, &c, with those of r and s, being now substituted in the first assumed fluent, it becomes &c, the true fluent of (a + ")-1, exactly agreeing with the first value of the 19th form in the table of fluents in my Dictionary. Which fluent therefore, when c is a whole positive number, will always terminate in that number of terms; subject to the same exception as in the former case. Thus, if c = 2, or the given fluxion be (a + x")” x2−1 x ; then, c+m or d being = m + 2, the fluent becomes 312 And if c = 3, or the given fluxion be (a + x")" x3n-1 x ; then m + c or d being = m + 3, the fluent becomes ;). 2a2 + And so on, when c is other m+3.m+2 m+3.m+2.m + 1 I 15. Again, for the latter or ascending form, (a+x")" + 1× (A.x2 + BX'+s+ cx2+25 + Dxr+ 35 + &c), by making its fluxion equal to the proposed one, and dividing, &c, as before, equating the two least indices, &c, the fluent will be obtained in a different form, which will be useful in many cases, when the foregoing one fails, or runs into an infinite series. Thus, if r+s, r+ 2s, &c, be written instead of r ́— ́s, r — 2s, &c, respectively, in the general equation in the last case, and taking the first term of the 2d line into the first line, there results 25 +raAx+(r+s)aв.x*+3+ (r+2s)ac.x*+ &c }· = 0. Here, comparing the two least pairs of exponents, and the coefficients, we have r=cn, and s=n; then A= 1 1 = (c + 1)cną3 3 C= Therefore, denoting e+m by d, as before, the fluent of the same fluxion (a + x)mxcn. *, will also be truly expressed by agreeing with the 2d value of the fluent of the 19th form in |