If the primary wave be represented by = e-ikx the component rotations in the secondary wave are '1'3= P (- AN y) N r2 ' cwi= r x D y N 'y)' lw2=P (- AD + 6,N z2 - x2 ' D r N r2 where ik3T e-ikr _ P - 4 r The expression for the resultant rotation in the general case would be rather complicated, and is not needed for our purpose.
The Emilian region is one where regular rotations are best observeda common shift being grain, maize, clover, beans and vetches, &c., grain, which has the disadvantage of the grain crops succeeding, each other.
In their cultivation he adopted Tull's practice of drilling and horse-hoeing, and he was also the founder of the Norfolk or four-course system, the first of those rotations which dispense with the necessity of a summer-fallow and provide winter-keep for live-stock (see below, Rotation of Crops).
It seems certain that success in any system involving a more extended growth of leguminous crops in rotations must be dependent on a considerable variation in the description grown.
Although many different rotations of crops are practised, they may for the most part be considered as little more than local adaptations of the system of alternating root-crops and leguminous crops with cereal crops, as exemplified in the old four-course rotation - roots, barley, clover, wheat.
The rotations extending to five, six, seven or more years are, in most cases, only adaptations of the principle to variations of soil, altitude, aspect, climate, markets and other local conditions.
Soc. Trans., p. 1013) considered in detail an octahedral form, and showed how by means of certain simple rotations of his system the formulae of Kekule and Claus could be obtained as projections.
One form, and he isolated a, 13 and y varieties with specific rotations of 105°, 52.5° and 22°.
(b2V2 + n2) (a2 - b 2) = - z It will now be convenient to introduce the quantities a l, a 2', 7731 which express the rotations of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations Ty - 1 = dz ' 3= - dy 2 = dx - In terms of these we obtain from (7), by differentiation and subtraction, (b 2 v 2 + n 2) 7,3 = 0 (b 2 0 2 +n 2) .r i = dZ/dy (b 2 v 2 +n 2)', , 2 = - dZ/dx The first of equations (9) gives 3 = 0 (10) For al we have ?1= 47rb2, f dy e Y tkr dx dy dz
(This solution may be verified in the same manner as Poisson's theorem, in which k=o.) We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of co-ordinates that the rotations are to be estimated.
In most rotations barley is grown after turnips, or some other " cleaning " crop, with or without the interposition of a wheat crop. The roots are fed off by sheep during autumn and early winter, after which the ground is ploughed to a depth of 3 or 4 in.
In these circumstances the number of rotations made by the armature in a given time is proportional to the product of the strength of the current flowing through the armature and that flowing through the field-coils, the former being the current to be measured.
In a direct competitive test the presence of 3.25% of nickel increased nearly sixfold the number of rotations which a steel shaft would endure before breaking.
He loved gardening, experimented enthusiastically in varieties and rotations of crops and kept meteorological tables with diligence.
And it is obvious that the same process applies in all cases in which we deal with quantities which may be regarded as of one directed dimension only, such as distances along a line, rotations about an axis, &c. But it is essential to notice that this is by no means necessarily true of operators.
Also, everything relating to change of systems of axes, as for instance in the kinematics of a rigid system, where we have constantly to consider one set of rotations with regard to axes fixed in space, and another set with regard to axes fixed in the system, is a matter of troublesome complexity by the usual methods.
II); and let J11, Jfs, Jis be the positions in space of the j,~ centres of the rotations by - which the lamina can be brought from the first position to the second, from the second .li_s to the third, and soon.
39~ the effect of three successive positive rotations 2A, 2B, 2C
The composition of finite rotations about parallel axes is, a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.
If AB, AC represent infinitesimal rotations about intersecting axes, the consequent displacement of any point 0 in the plane BAC will be at right angles to this plane, and will be represented by twice the sum of the areas OAB, OAC, taken with proper signs.
It is easily inferred as a limiting case, or proved directly, that two infini tesimal rotations a, j3 about c u parallel axes are equivalent to a ..._ -- - -
If the, A, B rotations are equal and opposite, FIG.
From the equivalence of a small rotation to a localized vector it follows that the rotation ~ will be equivalent to rotations E,ii, ~ about Ox, Oy, Uz, respectively, provided = le, s1 = me, i nc (I) and we note that li+,72+l~Z~i (2)
Thus in the case of fig 36 it may be required to connect, the infinitesimal rotations f, i1, l about OA, OB, OC with the variations of the angular co-ordinates 0, ~, ~.
We will briefly notice the case of two degrees, which involves an interesting generalization of the method (already explained) of compounding rotations about intersecting axes.
We take these as axes of x and y; then if f, n be the component rotations about them, we have -
For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily.
The phenomenon is known as the Eulerian nutalion, since it is supposed to come under the free rotations first discussed by Euler.
If in (21) we imagine that x, y, I denote infinitesimal rotations of a solid free to turn about a fixed point in a given field of force, it appears that the three normal modes consist each of a rotation about one of the three diameters aforesaid, and that the values of in are proportional to the ratios of the lengths of corresponding diameters of the two quadrics.
Epicyclic Trains.The term ep-icyclic train is used by Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates.
The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in ~ 30, 31.