next differentiate the equation, λ times with regard to u and μ times with regard to v: then

From this relation we have at once the two systems,

Let F represent P+Q. We proceed to ascertain the nature of what must certainly exist, a minimum value of F. Write u+h, v+k, respectively, for u, v; and let F" be the corresponding value of F. Then

This expression shews that the existence of a minimum. value of F requires that

But, putting in (2), λ = 0 and μ = 1, we see that

dQ dP

hence, by (3), we must have, simultaneously,

and =
dv du

Consequently either P and Q must be simultaneously zero, or

=0, F.

=

du

dv

dP dP dQ d Q must be simultaneously zero. Suppose, du' dv' du' dv " in the first place, that P and Q are not both zero; and choose the latter alternative. Suppose moreover in the expansion, by powers of h, k, for the value of F' F, the lowest order of partial differential coefficients of P, Q, of which at least some are not zero, to be the (2r). It is needless to consider the hypothesis of an odd order of differential coefficients not all vanishing; since, under these circumstances, there could not be any corresponding minimum value of F. We have then

d d

2r

[ 2r (F' − F ) = ( h — 2, + k — )” (P2 + Q°) +...... ..... (4).

du dv

+ multiples of lower partial differential coefficients of P, which are all zero.

In precisely the same way, we have, paying attention to

(1) and (2),

d2r Q

v at a time,

↓ [2r. (F' — F') = P { h2 dr — C-k

By the relations (5), (6), (7), (8), the equation (4) becomes, C denoting the number of combinations of 2r things, taken

V

d2r Q

dur

P dur

--

dur) · [{h + k √(− 1)} + {h − k √(− 1)}”]

(2

-P

we have 2r (F'' – F') = cp2 sin (a + 2r0) + terms of higher powers in p than the (2r)th.

But, by properly varying the value of the arbitrary quantity 0, we may give negative values to the term cp sin(a+2r0). Thus there is no minimum value of F under the present hypothesis. It follows therefore that, the existence of a minimum being obvious à priori, P and Q are capable of being zero simultaneously. This conclusion establishes the proposition that every equation has a root.

I have assumed in the above investigation that the coefficients of the various powers of x in f(x) are possible. This restriction may readily be removed. In fact f(x) may be represented by $ (x) + √(− 1) ↓ (x), where (x) and (x) involve only possible coefficients. Assume that f(x)=0: then ($x)2 = − (†x)”,

This equation, by the proposition we have established, can always have a root u+v √(-1). Hence either

or

¤x + √√(−1) ↓ (x) = 0,

px - √√(− 1) ↓ (x) = 0,

must have a root. Let the former have a root: then, putting u + v √(− 1) in this equation, we have a result of the form P+Q√(-1)+(− 1) {P' + Q′ √(−1)} = 0, P−Q' + √(− 1) (P' + Q) = 0,

or

and therefore

Substitute uv

and we have

P=Q', P'=-Q.

(-1) in the latter of the two equations,

P−Q √(− 1) − √(− 1) {P' — Q' √(− 1)} = 0,

P-Q-√(-1) (P' + Q) = 0,

a true result in virtue of the relations P=Q', P'=—Q. Thus f(x)=0 has a root, whether its coefficients are all possible or not.

September 16, 1869.

אן

ON EVOLUTES AND PARALLEL CURVES.

By Prof. CAYLEY.

abstract geometry we have a conic called the Absolute; lines which are harmonics of each other in regard to the absolute, or, what is the same thing, which are such that each contains the pole of the other in regard to the absolute, are said to be at right angles. Similarly, points which are harmonics of each other in regard to the absolute; or, what is the same thing, which are such that each lies in the polar of the other are said to be quadrantal.

A conic having double contact with the absolute is said to be a circle; the intersection of the two common tangents is the centre of the circle; the line joining the two points of contact, or, chord of contact, is the axis of the circle.

Taking as a definition of equidistance that the points of a circle are equidistant from the centre, we arrive at the notion of distance generally, and we can thence pass down to that of equal circles; but the notion of equal circles may be established descriptively in a more simple manner:

Any two circles have an axis of symmetry, viz. this is the line joining their centres; and they have a centre of homology, viz. this is the intersection of their axes. They intersect in four points, lying in pairs on two lines through the centre of homology: they have also four tangents meeting in pairs in two points on the axis of symmetry. Now if the two lines through the centre of homology are harmonically related to the two axes; or, what is the same thing, if the two points on the axis of symmetry are harmonically related to the two centres, then the circles are equal.

Circles which are equal to the same circle are equal to cach other, and the entire series of circles which are equal to a given circle, are said to be a system of circles of constant magnitude.

Starting from these general considerations, I pass to the question of evolutes and parallel curves: it will be understood that everything-lines at right angles, circles, poles, polars, reciprocal curves, &c.-refers to the absolute.

At any point of a curve we have a normal, viz. this is a line at right angles to the tangent; or, what is the same thing, it is the line joining the point with the pole of the