100. To determine it, take the fluxion of SPă, supposing y and s to vary, and we shall have a quantity of the form P'y +P"; we must therefore have V +P'y+P";=Qý + Rz, and consequently V=S {(Q—P) j+(R–P")}; an integral in which there enters only two variables, and which may be had by the preceding method. It is evident that we might have found the integral by means of JQy, or Ś Rz, in the same manner as by / Pc. For example, let the proposed quantity be (2 x y* +4 bz* x*) * + + 3y +24**}} + {*2+2 ** 2+ (yy +22) which satisfies the three conditions necessary to its being integrable; we shall have Po=yo x2 +bzx4, of which the fluxion, taken on the supposition of both - and y being variable, gives P=2y 3*, P = 26% 4. Consequently V = y { y + ✓(yy + zz) } ' the integral of ✓(yy+zz) which presents itself immediately without the assistance of the preceding method, and we obtain V=x++y3 + (yy+zz). Therefore that of the proposed Auxion is zt +y + (yy +zz) +y* ** + bz* :* +C. After what has been shewn, it will not be difficult to ascertain the conditions which must be fulfilled in Auxional expressions containing a greater number of variable quantities, and to integrate them when these conditions take place. These equations are called equations of condition. 101. Thus much premised, we will now proceed to the integration of second fluxions. Let us first take the fluxion of the second order Pä+Qx?, in which P and Q are any functions of the variable If we consider å as a variable y, the proposed Auxion becomes Py +Qyz. Now that this may be integrable, we must have P _Alus (Q. plus (C3) ; but only = and its powers enter into P, and y is y not found in Q. Therefore QY =Q, or = Qi; the condi {t='+ yy+z2 tion necessary that a fluxional expression of the second order Př+ Qir? may be integrable. If this condition is satisfied, we have S(Pö+Q:-)= (Pë+i P)=s Py =Py=P:. Example. The Auxion m **-1+m (m—1) som-em is integrable, because P = m (n-1)“?r=Qr; and the integral is mxats, which being again integrated, gives x" +C, for the primitive Auent. 102. If i has been supposed constant, the fluxion is Qir, of which the integral, (since P=S Qr), is : S Qr+the constant Cé. For example, sia (1rx)=is (e-ara)= (0-1*?)+Cr, and integrating again, we obtain Cr +C'+ xx-t **. 103. Let there be proposed the general fluxion of the second order Pë+Që”; if we take its fluxion, we shall obtain Pë+(°+2Qi) ö+Qxc?. Therefore reciprocally the general fluxion of the third order Rö+söö+T <3 will be integrable, or reducible to a fluxion SÅ of the second order, if -=STx; and the fluent will then be 2 Rë të Str. For example, x? +2x3 +(3xx-1)2}, possesses the condition of integrability, and its integral is t? *+ ** (v5-). 104. If be supposed constant, it is then clear, without any condition, that sT = */Tr+Cr*; the integral of this fluxion is is (* ST)+Cr *+C'*; lastly the integral of this again is sisis ST:+ C**+Cr+C". For example, fumi + Cx2 +Cr +C". (m + 1) (m+2) (m+3) In a similar manner we may find the fluents of higher orders of fluxions, and the conditions of their coefficients. 105. We shall now proceed to the consideration of fluxions of the second order, containing two variable quantities; these may be represented generally by Př+Qi+R*? + Sty+Tj? In order to find the conditions of the coefficients P, Q, R, &c. take the fluxion of de + By, in which A and B are any functions of r and y, and we ob tain Aë+By+Ar+By. Now since A is a function of both o and y, by art. 96, we have Å= +y; consequently the fuxion of y Aë+ Bi B B + y, and y therefore the proposed fluxion is integrable, whenever R = P S= P +, and T = Y 106. These conditions being fulfilled, the integral will be Pë+Qy; and if : has been supposed constant, the integral of QÖ + Rä? + Sty+Ty', will be Qý +ë fRi+C#, (because P=SR:), and the conditions of this integral will be T=Q, and S=Q_flur (SRI)* Y y For example, 6x2 ý +6 xyi* + x3 7, in which 3 is constant, possesses the preceding conditions, and its integral is x3 y + 3x2 yc+ Cã; this result gives by a second integration xy+Cx+C'. In a similar manner we may investigate the conditions required for a greater number of variables. In this last equation of condition the expression (fRx) denotes the partial flexional :oeffi reut of SRi relative to j, APPLICATION OF THE FLUXIONS. The applications of the inverse method of Auxions extend to all parts of the mathematics. But at present we shall confine ourselves to such as are purely geometrical, and which serve as a foundation to all the rest. With this view we shall proceed to investigate formulas for determining the quadrature and rectification of curves, the solidity of bodies in general, as also the formulas peculiar to solids of revolution and their surfaces; and we shall finish with a few examples on the inverse method of tangents. ON THE QUADRATURE OF CURVES. 107. Let AM represent any curve, and let its axis be AP; and suppose PM an ordinate to the point M; to find the quadrature of the space AMP, draw another ordi- 8 nate mp, and the line Mr parallel to Pp; we 2 shall then have the surface of the space Mmp P=MPx Pp+Mmr. Conceive now that the point m approaches towards the point M, it is obvious that the triangle Mmr will diminish more and more, but it will not A become zero till the point m falls upon PP the point M; then M mp P will become the Auxion of the space AMP; Pp will be i, and we shall have flu (AMP)=yè, and consequently AMP=Sys+C, which by Art. 74 Therefore also the space AQM = fxj=C+xy_y* : + 3 ta 24 2.8.ya we کر.1 108. Example 1. Let the ci 'cular arc QMB be a quadrant, described from the 2 centre A, and with the radius a; shall have y=iVaa—xx, and the space AQMP = så vaa — #+C = C + ax 23 1.3 x 1.3.5 2.3a 2.4.5 a 2.4.6.7 as 2.4.6.89a? 1.3.5.7 &c. A P B 2.4.6.8.10 11 a' Make x=0, and we shall have AQMP=0, and consequently C=0. Therefore AQMP=ax23 1 acs 1.3 x7 &c. 3a 2.4 5 až 2.4.6 7 as Ex. II. In the ellipse y = v(aa-xx). (-)-&c.) Ex. III. In the parabola yi=px?, and fy:=fpå szty, or the space APM is two-thirds of the circumscribed ? rectangle. The equation which comprehends 2 parabolas of all degrees is ym in am-*; therefore, mly = nix + ( mn) la, conse Therefore Syž = quently my_ni A PP or min:: yż: xy :: Syz : Sãy :: AMP: AMQ. From which it follows that the space AMP is to the circumscribed rectangle APMQ :: m: m+n. Ex. IV. In the equilateral hyperbola ty=aa, and y: = aa Consequently N fyx=aalx+C. If we wish to compute the spaces from the vertex A, when x=0, the space will =0. Therefore E cs-aa lo, and the space Q’APMN= aa lomaa lo = 0. 2 If x=AD, then Q'ADBN=aa laaa lo; consequently BDPM=aalaq la=aa M or |