TO FIND AXES OF CYLINDROID.

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ob; let oa be parallel to the second axis, and i. op the direction of the translation which together with a spin about oa is equivalent to a twist about that axis. If Ρ be the pitch of the second screw, h the distance of its axis from o, tan aop=p: h. Then the problem is to find an ellipse (or hyperbola) having oa, ob for conjugate diameters, and also op, oq. Or rather, having given that these are the directions of two pair of conjugate diameters, it is necessary to find the relative magnitudes of one pair.

For this purpose we observe that if p, q are points on the conic, on : om= mp nq, or the areas ong, omp are equal. Let po meet qn in q'; then

om* : on2 =

area omp: area onq = om2: on3

since they are similar. But ong ong = nq ng so that ng ng'. Given q, this determines p, so that the ratio op, oq is known. A conic described on these as semi-conjugate diameters is similar to the pitch-conic. Screws parallel to its axes compounded of the two given screws will be the oX, oY of the cylindroid.

The analytical solution is as follows'. Let p, q be the pitches, k,, k, the distances from o the centre of the cylindroid, 7, m the inclinations to oX, of the two screws, h their distance and the angle between them. Then from the equations

p=a cos2l+b sin2 7, k1 = (a - b) sin 21,

q=a cos2 m +b sin2 m, k2 = (a - b) sin 2m,

we have to find a, b, l, m, k,, k, in terms of p, q, h, and 0.

and

Therefore

p+q=a+b+1 (a− b) cos (21+ cos 2m)

(q − p) cot 0 = k1 + k2

p− q = h tan (l + m) = h tan (21 — 0) ;

whereby a + b, k1± k ̧, l and m are expressed in terms of P, q, λ, and 0.

MOMENTS.

When a straight line moves as a rigid body, the component of velocity along the line of every point on it is the same. For consider two points, a, b; the rate of change of the distance ab is the difference of the resolved parts of the velocities of a and b along ab. If therefore the length ab does not change, this difference is zero. This component of velocity of any point on the line may be called the lengthwise velocity of the line.

n

k

The lengthwise velocity of a line due to a given twist is called the moment of the twist about the line. Let lm, = k, be the shortest distance between the axis In of the twist and the straight line mr. It will be sufficient to determine the velocity of m along mr. Now m has the velocity ko perpendicular to the plane mln, and po parallel to In, if a be the magnitude and p the pitch of the twist. Let be the angle between mr and ln, then the resolved parts of these components along mr are ko sin 0 and + po cos 0. Thus the moment of the twist about the line is a (p cos 0 -k sin 0).

The moment of a screw about a straight line is the moment of a unit twist on that screw about the line. Thus p cos -k sin 0 is the moment of a screw of pitch p about a line at distance k making an angle ✪ with its axis.

COMPLEX OF A SCREW.

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All the straight lines in regard to which a given screw has no moment, are said to form a complex of lines belonging to that screw. When a line belonging to the complex is moved by a twist about the screw, every point in it moves at right angles to the line.

All the lines of the complex which pass through a given point lie in a given plane, namely, the plane through the point perpendicular to its direction of motion due to a twist about the screw. This plane passes through the perpendicular from the point on the axis, and makes with the axis an angle 0, such that tan 0=p: k.

Conversely, all the lines of the complex which lie in a given plane pass through a certain point, at a distance p cote from the axis along a straight line in the plane perpendicular to it. If any other line in the plane belonged to the complex, every point in the plane would move perpendicularly to the plane, and the twist would reduce to a spin about some line in the plane.

In the case when p=0, or the twist reduces itself to a spin about its axis, the moment becomes k sin e, and can only vanish if the line meet the axis (k=0), or is parallel to it (sin = 0), which is the same as meeting it at an infinite distance. Hence the complex reduces itself to all the lines which meet the given axis.

All the lines of the complex which meet a given straight line, not itself belonging to the complex, meet also another straight line. For, suppose the cylindroid constructed, which contains the given screw and the given straight line, considered as a screw of pitch 0. Then the pitch-conic must be a hyperbola, since there is one screw with pitch 0; this is parallel to one asymptote, and there must be another parallel to the other asymptote. Hence every twist may be resolved into two spins, the axis of one of which is any arbitrary straight line, not belonging to its complex. Now, since the two spins are equivalent to the twist, the lengthwise velocity of any line due to the twist. is the sum of its lengthwise velocities due to the two spins; or the moment of the twist is the sum of the moments of

the two spins. If then a straight line belong to the complex and meet the axis of one spin, the moments of the twist and one spin are zero, consequently the moment of the other spin is zero, or its axis meets the line. Therefore a straight line of the complex which meets the axis of one spin, meets also the axis of the other.

If however the axis of one spin belong to the complex, that of the other spin must meet it, since the moment of the twist about it is zero; but in that case it must also coincide with it, since otherwise the pitches of all screws on the cylindroid would be zero. We have then the case noticed above, in which the pitch-conic reduces to two parallel lines.

From the symmetry of the expression - k sin 0 in regard to the two straight lines concerned, we perceive that the lengthwise velocity of a line A due to a unit spin about a line B is equal to the lengthwise velocity of B due to a unit spin about A. Hence we may speak of this quantity as the moment of the two lines, or of either in regard to the other. We shall also define the moment of two spins as the product of their magnitudes into the moment of their axes. If one of the axes goes away parallel to itself to an infinite distance, and at the same time the angular velocity w about it diminishes indefinitely, so that ko the spin becomes a translation-velocity v perpendicular to that axis, making, therefore, an angle 4,7-0, with the other axis; and the moment becomes vo' cos p, if w' is the magnitude of the finite spin. In the same way we may speak of the moment of a twist and a spin, meaning the magnitude of the spin multiplied by the moment of the twist about its axis.

= v,

Suppose the twist resolved into two spins A, B; then its moment in regard to the spin C will be the sum of the moments of the component spins. Let us combine with C a spin D, making a second twist; then the sum of the moments of the twist A+B in regard to C and D will be equal to (AC) + (BC) + (AD) + (CD), (where (A C) means the moment of A in regard to C), that is, it will be the sum of the moments of the twist C+D in regard to A

RESOLUTION OF TWIST INTO TWO SPINS.

135

and B. Therefore it is independent of the way in which the second twist is resolved into two spins.

Consider then two twists a, B, whose pitches are p, q. The moment of the first in regard to the rotation of the second is aß (p cos 0 − k sin 0), and in regard to the translation it is a. Bq cos 0. Thus the whole moment is

aß [(p + q) cos 0 -k sin 0].

The quantity (p + q) cos 0 - k sin 0 is called the moment of the two screws, or of either in regard to the other. It may be thus defined:-Let a unit twist about one screw be resolved into two spins, and let the magnitude of each of these be multiplied by the lengthwise velocity of its axis due to a unit twist about the other screw. sum of the products is the moment of the two screws.

The

Hence, by making the two twists coincide, we find that the moment of a twist in regard to itself is the square of its magnitude, multiplied by twice its pitch. Since then the moment of a spin in regard to itself is zero, the moment of a twist A+B is twice the moment of the spins A, B; and this is therefore the same, whatever two spins the twist is resolved into.

Now the moment of two spins in regard to one another is six times the volume of the tetrahedron which has the lines representing the spins

for opposite edges. Let ab, cd be the representative lines; since each may be slid along its axis without altering the spin, let them be so placed that the shortest distance fg bisects them both. Draw through f, a'b' equal and parallel to ab, bisected by f; and through g, c'd equal and parallel to cd, bisected by g. Then a'db'c,

ad' be are equal parallelograms, and the volume of the parallelepiped, of which they are opposite faces, is

fg.ab.cd sin 0,