Cor. 2. The rectangles MK × KN, ML × LN, MK × ML ad KNX NL, are all equal.

Cor. 3. The subtangent FR OR, the distance from the centre. Cor. 4. FD touches the adjacent hyperbola in D.

PROP. LXIII. Theorem. If a straight line which its the hyperbola, or the opposite hyperbolas, meets he asymptotes, the rectangle contained by the segents of it between a point in the hyperbola and the symptotes, is equal to the square of the semidiameter arallel to it.

Let MN (see last figure) cut the hyperbola in K, and meet he asymptotes in M and Ñ, and let DO be the semidiameter arallel to it. Draw OE the diameter conjugate to DO, it isects KL; draw also FEG parallel to MN, it touches the yperbola, and EGOD. But OE2: OD EG :: OP2 PM2, and also OE2: OD2 :: OP2 — OE2 : PK2, therefore DE2: OD2:: OE2: PM2 - PK2 = MK × KN, therefore

)D2 = MKX KN.

=

Again, let KW cut the opposite hyperbolas, and meet the symptotes in T and Y, and be parallel to the diameter OE, nd let OD be its diameter which meets it in S, and let FH e the tangent parallel to it. Then OD2: DF2 =OE2:: OS: ST2, and also OD2: OE2 :: OD2+OS2: KS2, thereore OD2: OE :: OD2: KS2-ST2 TK x KY, therefore )D2=TKXKY.

Cor. The rectangles under segments of parallels between oints in the hyperbola and the asymptotes are equal.

PROP. LXIV. Theorem. The rectangle contained by any two straight lines BD, BE, drawn from a point B in the hyperbola to the asymptotes, is equal to the rectangle contained by other two lines FG, FH, parallel to them, drawn to the same asymptotes from any point F of the four conjugate hyperbolas. Through B and F draw any

two parallels BKL and MFP. Then the triangles DBK, FGM, are similar, and also the triangles BEL and FHP, and therefore BK: BD:: MF: FG, and BL: BE:: FP: FH. Wherefore BK X BL: BDX BE:: MFX FP: GFX FH, and BK × BL=MF × FP; therefore BD × BEGFXFH.

L

T

RXR

B

D

Cor. 1. If BD, FG, be parallel to the asymptote, the reet angle DBX DO=OG × GF, and if BE, FH, be also parallel to the asymptote, the parallelogram DE = HG.

Cor. 2. If AR be the line which joins the vertices of the axes, and C the focus, ARRO OC; therefore the rectangle OD x BD = AR2 = 1 OC2.

=

Cor. 3. If the hyperbolas be equilateral, or have their are equal, the rectangle OD x DB=OA (OA being the semiaxis.)

SECTION III.

OF VARIABLE QUANTITIES.

QUANTITIES which alter their values are called variable quantities. These are often so related to one another, that when one of them is increased, the others are increased or diminished according to a constant rule. Thus, if a body moves uniformly, the space it describes increases in the same ratio with the time; that is, if T and t be two times, and S and s the spaces run over in these times, then T:t::S: s. This proportion is expressed generally thus, T c S, and read, the time is as the space.

If the quantities S, T, V be so related to one another, that when S is increased, both T and V are increased, so that their product has a constant ratio to S, then S c TV, read, S is as T and V jointly.

If these quantities be so related, that when V is increased, S is increased, and T diminished, so that their quotient has a

constant ratio to V, then VV is as S directly, and as

T

T inversely. In this case, if S be constant, V. These are called general proportions, and if the values of the variable quantities can be determined at a given period of their increase or decrease, they can be reduced to determined proportions. Thus, if S becomes m at the same time that T becomes n, then S: T::m:n; and the particular value of S, corresponding to a given value of T, is given.

PROP. LXV. If TV, then ST SV, and AT AV, for t:v:: T: V:: ST: SV:: AT: AV PROP. LXVI. If S T, and V X, then SV TX

te

ors: t::S: T, and v:x:: V: X; therefore s v. tx:: SV: TX.

Cor. Hence, if Sa T, and SV, then S2 α TV, or
√TV.

PROP. LXVII. If ST,
V, for t: T::s: S::v: V;

CV and SxTV.

and S V, then ST therefore s: S::tv:

Cor. If TV, then T+V TV, for t:v::T: V and +v:t-v::T+V: T-V; therefore T+V T-V.

PROP. LXVIII. If (V+T)2 ≈ (V — T)2, then V2+ r2 ∞ VT. For (VT)2 + (V − T)2 ∞ (V + T)2 — VT)2; that is, V2+T2 ∞ VT.

-

SECTION IV.

LIMITS OF QUANTITIES AND RATIOS.

A CONSTANT quantity or ratio is said to be the limit of a variable one, when this latter can be altered, so as continually to approach to the constant one, and at length to come nearer to it than any other given quantity or ratio, but never to be equal to it.

A regular polygon is always less than the circle containing it, but increases as the number of its sides increase, and at length comes nearer to an equality with the circle than by any given difference. The circle is therefore said to be the limit of the polygon.

Also the perpendicular from the centre of the circle upon the side of the inscribed polygon is always less than the radius, but continually increases with the number of sides, and at length comes to be more nearly equal to the radius than by any given difference. The radius is therefore the limit of the perpendicular.

The tangent of an arc of a circle is always greater than its sine; but the ratio of the one to the other continually diminishes as the arc becomes less, and at length comes nearer to a ratio of equality than by any given difference. This ratio of equality is therefore the limit of that of the tangent to the sine.

PROP. LXIX. Let a and b be constant quantities, always greater than the variable quantities and y, but

let x and y be capable of increase, so that a—x shall be less than any given quantity; and also, that b-g shall become less than any given quantity; the ratio of a to b is the limit of the ratio of x to y.

Let a be always to z as x to y. If the ratio of x to y be constant, then z is constant. If zb, then y may be taken z. But xy::a: z, and xa, therefore yZz; and it is alsoz, which is impossible, therefore a:b::x:y.

If the ratio of a to y be variable, then z is variable; but its limit is constant, and cannot be less than b, as was proved be fore, neither can it be greater; for then if y:x::b:v, v would be less than a, which may be shown to be impossible as before; therefore the ratio of a to b is the limit of that of x to y.

In like manner, if x and y be always greater than a and b, but decrease so that xa and y. b become less than any given quantities, it may be shown that the ratio of a to bis the limit of the ratio of x to y.

Let x+y=2a to find the limit of xy, suppose x the greater =a+v, then y=a—v, and xy=a2-v2, as x or y ap proaches to = a, v becomes continually less, and is ultimately =0, therefore the limit of xy=a2.

=t+a a

In like manner, if x- -y=2a by making x = y=ta, the limit of is a2. xy

and

Let x+y=2a to find the limit of x2+y. By proceeding as before, x2+y2 = (a+v)2 + (a — v)2 = 2a2 -2v, and as v continually diminishes the ultimate value of x2+y2 = 2a2; and the same will be the case if x-y=2a.

Let t be the increment of x to find the ratio of the limit of ax to that of x. When a increases to x+t, then ax increases to ax+at, and subtracting the first quantity ax, the incre ment is at; the ratio then is that of at to t, or of a to 1, which is independent of the value of x.

To find the ratio of the limit of the increment of ax2 to that of x; when x becomes x+t, then ax2 becomes a (x+t)2 = ax2+2axt+at2, and subtracting ax2, the increment is 2art +at2, which is to t as 2ax+at to 1; and when t becomes= O, the limiting ratio is 2ax to 1.

PROP. LXX. Let t be any increment of x, and v the corresponding increment of y. It is required to determine the limit of the ratio of the increments of the rectangles ay and ax.

When a becomes x+t, then y becomes y+v, and the rectangles become (x+t) × (y+v), and a × (x+t), and therefore

heir increments are (x+1)x(y+v) — xy = xv+yt+tv, nd at. Let x be always to s as t to v, so that xv= st, then v+yt+tv: at:: st+yt+vt: at, or::s+y+v: a; and as v continually diminishing, and at length becomes less than any iven quantity, therefore the limit of the ratio of the increents is s+y: a, or if x:y be the limit of t: v, it will be sx+

r:ax, or since sx=xy, it will be xy+yx: ax.

Cor. 1. If a decreases while y increases, the limiting ratio ill be xy — yx : ax.

If we divide the limit by the quantity, we get

x

y

xy yx
+

xy

ух y

for the limiting ratio of the quantity. If x decreases, it

y x

Here x and y are the ultimate values of the increments of and y.

Cor. 2. The limiting ratio of the increment of xyz: a2x is

For let yz=v, then xyz=xv, and the limit xv+vx: a2x, ut v=yż+zy, and substituting xyz+xzy + zyx : a2x. The ratio of the limit of the increment of xyz to the quan

++ In the same manner, the ratio of xyzv=

+-+

y

&c.

PROP. LXXI. To find the limit of the ratio of the ncrement of x2 to ax.

Suppose xy, then x2 = xy, and the limiting ratio is y+yx: ax, or 2xx: ax.

In like manner, it may be shown that the limit of the ratio f the increment of x3 to a2x is 3x2x: a2x, and that of x1: a3 x 8 4x3x to a3x, and so on; therefore the limit of the increnent of a to that of an-1x is nxn-1x to an-1x. 1x

If the quantities x, y, z, v, (70. Cor. 2.) be equal, the ratio