tesimal portion in which (1 + sin )2 is an infinitesimal of as least as high an order as (n − 1) is. Calling n, for this case, n = 1-i, we shall see that as sin, approaches very near the value 1, the radius of curvature lengthens with sudden and greatly accelerating velocity. The curve thus crosses itself and runs out into branches nearly straight and from the first nearly parallel to the axis. But there are no asymptotes; the tangents, to the infinitely distant part of the curve, pass at an infinite distance from the origin.

Take now n = 1+i, and the radius, as sin approaches-1, will diminish with sudden accelerating velocity. The two sides of the curve, instead of intersecting, recoil from contact. The curve appears to the eye, as before, like a circle resting on a straight tangent; but in reality there

is a breach of continuity at the lowest point; it is an appulse of cusps, not an intersection.

When n is absolute unity, p and p are also unity; the curve is then the circle without the simulated tangent. But the value of Dp shows that, for this effect, we must, when the product (1+sin ) (sec) is an infinitesimal of the xth order, have n— 1 of the (2x+1)st order.

In order to assist the imagination, let us take the meter as unity and put n = 1 — (1%). The curve is now a circle, with the radius of a meter; except that, just at the bottom, the radius would lengthen and the curve intersect itself. Tangents at that point of intersection would make an angle so nearly two right angles, that if a straight line were passed horizontally trough the point, it would not rise one millimeter above the

tangent, until they were each prolonged to 100 kilometers; and the curve would be included between this line and the tangents.

If we put n=1+ (1%)1, the curve would still be, to any observation of the unaided senses, a circle two meters in diameter resting on a straight tangent. There would, however, be a gap at the bottom twentynine hundredths of a millimeter in width, and the curve and the real tangents would be above the horizontal line.

The hirundo may then be described as a circle, whose intellectual law makes it incapable of evolving any forms which are evolved from other circles and in whose own series occurs a sudden break of infinite magnitude, which is, nevertheless, at the point of rupture, not visible to sense; which makes it worth considering in its bearing on the logic of evolution.

SECOND DIFFERENTIALS AND EQUICRESCENT VARIABLES. By Prof. J. BURKITT WEBB, Hoboken, N. J.

[ABSTRACT.]

Ir is thought that text books on the Calculus do not explain with sufficient clearness the differences and relations between the second differentials of different variables as affected by various suppositions as to the equicrescence of the latter. The subject may be illustrated by the equations

x= 4, (y) and z = 0, (y) of a line in space.

2

In Fig. 1 the projections of the line upon the xy and y z planes are shown, these planes being superimposed to make easier a comparison of the projections; the broken lines are projections of a tangent. Calling the projections of the point of tangency P and Q, we will suppose the regions about P and Q to be infinitely magnified and represented in Fig. 2, being also moved vertically to bring them together into one figure. If P', P" and P" (the Ps only being mentioned, though the Qs also are understood) be three equidistant points upon the tangent (it being needless to say that they are "the projections of" points) there will be no difference between' the two differentials of the same name; i.e., d x, dy and d z are constants, which we will suppose belong to the tangent rather than to the curve. The latter will have P' and P' common with the former, but its third consecutive point will be P''' at a second differential distance from P ̧""', this latter point being regarded as an origin of coördinates from which to lay out the second differentials which determine the position of P. Pshould be regarded, to make the conception complete, as lying midway between P' and P. We will call Fig. 2 the first infinite enlargement. Fig. 3 is obtained by magnifying infinitely the regions about P and Q and bringing

them together vertically. In this second infinite enlargement, second differential distances will appear finite and the geometrical connections between these quantities may be traced. The addition of these second differentials to the constant dx, dy and d z of the tangent produces the differentials for the curve. In Fig. 2 the curve and tangent should appear to coincide, while in Fig. 3 they will appear separate but parallel. In the ordinary notation of the Calculus to express the second differential, say, of d2y y on the supposition that x is equicrescent we must write dx2, the dx2 in the denominator having the force of an index, signifying the equicrescence of x, as well as of a divisor, we may not remove it from under d2 y but must multiply by da to neutralize the division; a simpler notation is used in the figure and table whereby dy

=

dy

d x2

A careful comparison of the table and Fig. 3 will show the effect of the different suppositions possible.

1 In this case the curve will become a right line coinciding with the tangeut.

2 p q may be taken anywhere upon the curve within a second differential distance of Po"" Qo".

As an illustration of the geometrical relations evident in Fig. 3 we may produce the ordinary formula for getting the value of the second differential coëfficient of y with respect to z in terms of the second differentials of x and y when neither z nor y is equicrescent. Thus

which acquires the usual form on division by dr and dropping the index

Ir the circle (x − a)2 + y2 = r2 be revolved about the axis of Y, it generates the torus

(x2 + z2 + y2+ a2 — r2) 2 — 4 a2 (x2 + z2).

(1)

If now we cut the torus by a plane parallel to its axis, say by zc, we have as the equation of the section

(x2+y2)2+2 (c2—r2 + a2) y2 + 2 (c2 — r2 — a2) x2 = 2 (a2 r2 + a2 c2 + r2 c2)

=(a+p+c1)

or in general terms

(x2 + y2)*+ A x2 + B y2 = C,

a very general equation of an oval.

If Ꭺ --- - Bin (2), this becomes the equation of a Cassinian Oval. ing AB, we find that c=r, and (2) becomes

(x2 + y2)2+2a2 (y2—x2)=(4r2—a2) a2

(2)

Mak

(3)

=ar, as

By taking c of such value that C-0, which I find to be c= would be expected, (2) becomes

(x2+ y2)+4(a2—a r) y2 - 4ar x2=0

(4)

(using only car), which is the general equation of the Lemniscate. When c= ar, we have only a point.

Now, it is obvious, that we shall have both conditions fulfilled, namely, c = r and c = a―r, when a = 2r; i. e., if the gorge-circle of the torus is equal to the generating circle. And if a plane is passed tangent to the surface at a point of the gorge circle, we have both a Cassinian and a Lemniscate. Substituting in either (3) or (4) we get

(x2 + y2)2+8r 2 (y2 — x2) = 0

which is the Lemniscate of Bernouilli.

NOTE ON TANGENTS TO PLANE CURVES. By Prof. C. M. WOODWARD, St. Louis, Mo.

[ABSTRACT.]

Its

THE following method of writing the equation of a tangent directly from the equation of a curve may not be new, but it is new to me. convenience may justify its statement.

I begin with the simplest case.

1. All the kinds of terms in the general equation of the second degree are represented in the equation a x2+ bxy+cy = 1.

Multiply by 2 and write in this form:

a (xx + xx) + b (xy + xy) + c (y + y)

= 2

Prime one-half the x's and one-half the y's, leaving the equation linear: 2 axx' + b (x y' + x'y) +c (y + y')

is the equation of the tangent at the point x', y'.

2

2. All the kinds of terms in the general equation of the third degree are represented in the equation

ax3 + bx2y + cxy+ey2+ƒx=1

Multiply by 3 and write in this form