's III. If now we make P—Qy=O, this equation will fulfil all the conditions proposed but the last ; and that may be fulfilled in an infinite number of ways, by properly determining the value of Q.
0 For, in the first place, since the values of P and Q, put equal to 0, have no common root, they have no common divisor, and therefore the equation can not by division be reduced to two ; it is therefore simple, and expresses one simple and continued curve, which is not composed of others. This is the first condition.
Next, a curve of this kind will cut the axis CAC' in all the points E, G, I, E', G', &c. and only in these. For it can only cut the axis in those points in which y=0, and it will cut it in all these. Besides, when y=0, Qy=O, and since P—Qy=0, therefore P=0, which can only happen when z is one of the roots of the equation P=0 ; that is, as we have al ready shown, only in the points E, G, I, &c. or E', G', &c. Wherefore the quantity y vanishes, and the curve cuts the axis only in these points. That the curve will cut it in these points, is also clear froM this, that in each of them P=0, therefore also Q y=0, and it is not Q=0, for there is no common root of the equations P=0 and Q=0 ; therefore it must be y=0, and consequently the curve meets the axis : which fulfils the second condition.' Besides, since P—Qy=0, y=— Q ; and if any `ab scissa x be given, z is also given, and therefore P and Q are single and determined, and therefore y is also single and determined. To every absciss x, therefore, there is a corresponding ordinate y, and only one. This is the third condition.
Again, whether x be assumed positive or negative, while it is of the same length, the value of z=s2. is al ways the same ; and therefore the values of P, Q, and consequently of y, must be the same. So that if equal abscisses be taken on each side of A, the corresponding ordinates will be equal : which is the fourth condition..
If x be diminished in infinitum, whether it be po sitive or negative, z will also be diminished in infini tum, and will be an infinitesimal of the second order. Wherefore in the value of P; all the terms will de crease in infinitum, f only excepted, because all the rest besides it are multiplied by z; and thus the va lue of P will be as yet'finite. But the value of Q, which involves the formula drawn entirely into z, will diminish in infinitum, and will become an infinitesimal of the second order. Therefore will increase in infinitum, and becomes an infinitely great quantity of the second order. Wherefore the curve will have for its asymptote the straight lineAB ; and the area, I3AED will increase in infinitum, and if the positive ordinates y are taken towards the parts AB, and ex press the repulsive forces, the asymptotic arc ED will be towards the same parts AB: which was the fifth condition.
It is clear, .then, that however Q be assumed with the given conditions, the first five requisites will be fulfilled. Now the value of Q may be varied in an infinite number of ways, so as still to fulfil the con ditions with which it was assumed. And therefore , the arc of the curve intercepted between the in tersections may be varied in infinite ways, so that the first five conditions may be fulfilled. It may
therefore be varied so as to fulfil the sixth condition. For if there be given however many, and. whatever arches of whatever curves, providing they be such that they recede always from the asymptote AB, and thus no right line parallel to that asymptote cut these arches in more than one point, and in them let there be taken as many points as you please, and as near one another ' • it will be easy to assume such a value of P, that the curve may pass through all these points, and the same may be varied infinitely; so that still the curve will pass through all the same points. Let the number of points assumed be what you please =r, and from every one of such points, let right' lines.be drawn parallel to AB, as far as the axis CAC'', which must be ordinates of the curve that is sought ; and let'the abscisses from A to the said or dinates be called M', M', &c., and the ordinates N', N', &c. Let there now be taken a certain quantity Gz, and let this quantity be supposed equal to R. Then let another such quantity T be assumed, so that z vanishing, any term of it may vanish, and so that there be no com mon divisor of the value of P, and of the value of R + T, which may be easily done, seeing all the di visors of the qtiantity P are known. Let it now be made Q=R+T, and then the equation of the curve will be P—Ry—Ty=0. After this, let there be put in the equation •s/P, successively for x, and N', N', &c. for y ; we shall have r equations, each containing values of A, B . C G, of one dimension, besides the given values of &c. N', N', &c. and the arbitrary values, which in T are the coefficients of z.' By, these equations, which are in number r, it will be easy to determine the values of A, B, C, . G, which are likewise in number r, assuming in the first equation, according to the usual method, the value A, and substituting it in all the following equations, by which means the equations will become r-1. These, again, by throwing out the value B, will be reduced to r-2, and so on, until we come to one only, in which 'the value Q being determined,' by means of that, in a retrograde order, all the preceding values will be determined, one by each equation. The values A, B, C, . G being in this manner deter mined in the equation P—Ry—Ty=0, or P—Qy=O, it is clear that the values M', M', &c. being suc cessively put for x, the values of the ordinate y must successively.be N', &c. • and therefore that the curve must pass through these given points in those given curves, and still the value Q will'have all the preceding conditions. For z being lessened be yond whatever limits ; seeing all the terms of the va lue of T are lessened which were thus assumed, and likewise the terms of the value R' are lessened, which are all multiplied by z, and this there will be no common divisor of the quantities P and Q, seeing there is none of the quantity P and R+ P.