The Quarterly Journal of Pure and Applied Mathematics

gen

(so that m'n', ''). The intersections of the curve and Absolute are in this case the points f each twice, and besides 2m-2f points; similarly the common tangents are the tangents at f each twice and besides 2n-2f tangents. Now I remark that the parallel curve, when the radius of the variable circle is 0, reduces itself to the original curve twice, together with the 2n-2f common tangents, and the 2m-2f common points; the order is thus = 2m+ (2n − 2ƒ), and the class = 2n + (2m − 2ƒ): and these are the values in the general case where the radius of the variable circle is not = = 0.

But in the remarkable case where the curve and its evolute have a (1, 2) correspondence, then I correct the formulæ by adding to the expressions for ', ' respectively. We have for the evolute of the parallel curve

viz. assuming 2m+2n4f, this means that the evolute is the evolute of the original curve taken twice.

A very interesting case is when m=n=f: observe that neither m-f nor n-f can be negative, so that the assumed relation m+n2f=0 would imply these two relations. We have here for the parallel curve m' = 2m, n'= 2n, i'=21, K2K; the parallel curve in fact breaking up into two curves such as the given curve. And in this case the formulæ for the evolute assume the very simple form m", n" =ƒ, i" =ƒ, x" = -2f+3K.

Whatever the original curve may be, we have for the parallel curve m'n'f', so that the formula for the evolute of the parallel curve are of the foregoing form m"= x', n"" =ƒ"', ¿"" =ƒ" — ©, к'" - 2ƒ' + 3' - O, which agree with the above values of m' k". In the particular case m=n=f,

we have =0, so that the evolute-formulæ, if originally written down without the terms in O, would still be m"" = 2m", n" = 2n", "" = 2", k" = 2x"; viz. the evolute is here the original evolute taken twice; as already seen, the parallel curve consisted of two curves such as the original curve, and each of these has for its evolute the evolute of the original

curve.

ON THE SPOKE ASYMPTOTES OF RHIZIC CURVES BY WILLIAM WALTON, M.A., Fellow of Trinity Hall. PERHAPS the demonstration of the three following theorems may be interesting to students in the Differential Calculus:

(1) The asymptotes of each of two conjugate rhizic curves pass through a single point and divide plane space equiangularly like the spokes of a wheel.

(2) The common point of the asymptotes of either of two conjugate rhizic curves coincides with that of the asymptotes of the other rhizic curve.

(3) The collective system of asymptotes of both curves divide space equiangularly like spokes.

Let

ƒ{u + v √√(− 1)} = P+Q √(− 1),

where P and Q are possible functions of u and v, P=0 and Q=0 represent conjugate rhizic curves.

Let P, Q, denote the sum of all terms of v dimensions in P, Q, respectively: then

The equations to the asymptotes of P=0 are, by Gregory's formulæ, given by the equations

Hence, by the formulæ (1), the asymptotes of the curve P=0 are determined by the equations

{u + v √(− 1)}" + {u − v √/(− 1)}" = 0,

+ nv' √(− 1) [{u + v √ (− 1)}"−1 — {u — v √(− 1)}"-1]

+ P1 [{u + v √(− 1)}"1 + {u — v √(− 1)}"1]=0;

and those of the curve Q=0 by the equations.

+ nv' [{u + v √(− 1)}" ̄1 + {u − v √(− 1)}"~1]

P1
[{u + v √(− 1)}"−1 — {u — v √√(− 1)}"~1] = 0.
√(−1)

Putting ur cos 0, v=r sine, the equations for the asymptotes of P=0 become

cos no = 0,

nu' cos (n − 1) 0 — nv′ sin (n − 1) 0 + p1 cos (n − 1) 9 = 0.

From these two equations we have

nu' sine-nv' cose+p, sin0=0,

and therefore, λ being an integer,

* Cambridge Mathematical Journal, Vol. IV., p. 42.

In like manner the asymptotes of the curve Q=0 are determined by the equations

sin n0=0,

nu' sin (n − 1) 0 + nv' cos (n − 1) 0 +p, sin (n − 1) 0 = 0,

and therefore by the equations

λπ n

From the equations (2) and (3) we see (1) that the asymptotes of the curve P=0 all pass through the point u' = —

Ρι

v'=0, and divide space equiangularly like the spokes of a carriage wheel, the inclinations of these asymptotes to the axis of u being

(2) that the asymptotes of the curve Q=0 all pass through the same point through which those of P=0 pass, and like them divide space equiangularly, their inclinations to the axis of u being

(3) that also the collective system of the asymptotes of both curves divide space like the spokes of a wheel, the angle

between any two successive spokes being

December 3, 1870.

THE DEDUCTION OF EULER'S EQUATIONS FROM THE LAGRANGE EQUATIONS.

By W. H. BESANT, M.A.

EMPLOYING the ordinary figure (fig. 30), the general coordinates are 0, 4, and 4, and the angular velocities about the principal axes A, B, and C, are given by the equations w=0' sino sino cos,