water-spout. This is generally first formed above, in the form of a cloud, shaped like a funnel or inverted cone. As there is less resistance to the motions in the upper strata than near the earth's surface, the rapid gyratory motion commences there first, when the upper strata of the agitated portion of atmosphere have a tendency to assume somewhat the form of the strata in the case of no resistance, as represented in Fig. 3. This draws down the strata of cold air above, which, coming in contact with the warm and moist atmosphere ascending in the middle of the tornado, condenses the vapor and forms the funnel-shaped cloud. As the gyratory motion becomes more violent, it gradually overcomes the resistances nearer the surface of the sea, and the vertex of the funnel-shaped cloud gradually descends lower, and the imperfect vacuum of the centre of the tornado reaches the sea, up which the water has a tendency to ascend to a certain height, and thence the rapidly ascending spiral motion of the atmosphere carries the spray upward, until it joins the cloud above, when the waterspout is complete. The upper part of a water-spout is frequently formed in tornadoes on land.

When tornadoes happen on sandy plains, instead of waterspouts they produce the moving pillars of sand which are often seen on sandy deserts.

70. The routes of cyclones in all parts of the world, which have been traced throughout

their whole extent, have been

S
Fig. 7.

found to be somewhat of the form of a parabola, as represented in Fig. 7. Commencing generally near the equator, the cyclone at first

moves in a direction only a little north or south of west, according to the hemisphere, when its route is gradually recurvated towards the east, having its vertex in the latitude of the tropical calm belt, as represented in the figure. This motion of a cyclone may be accounted for by means of what has been demonstrated in (§ 31), which is, that if any body, whether fluid or solid, gyrates from right to left, it has a tendency to move toward the north, but if from left to right, towards the south. Hence the interior and most violent portion of a cyclone, always gyrating from right to left in the northern hemisphere, and the contrary in the southern, must always gradually move towards the pole of the hemisphere in which it is. While between the equator and the tropical calm belt, it is carried westward by the general westward motion of the atmosphere there, but after passing the tropical calm belt, the general motion of the atmosphere carries it eastward, and hence the parabolic form of its route is the resultant of the general motions of the atmosphere, and of its gradual motion toward the pole.

It may be seen from equation (52), that the tendency of a gyrating mass to move towards the pole is as sin y, or the cosine of the latitude, and the square of the diameter of the gyrating mass. Hence, near the equator, where the dimensions of the cyclone are always small, it moves slowly toward the pole, but as it gradually increases its dimensions, after passing its vertex, its motion towards the pole, and also its eastward motion, are both increased, and hence its progressive motion in its route or orbit is then accelerated, in accordance with the observations of REDFIELD,

71. By comparing equations (27) and (44), it is seen that they are very similar, and consequently the motions which satisfy them must be also similar. Hence the general motions of the atmosphere are similar to those of a cyclone. For the general motions of the atmosphere in each hemisphere, form a grand cyclone having the

pole for its centre, and the equatorial calm belt for its limit. But the denser portion of the atmosphere in this case being in the middle instead of the more rare, instead of ascending it descends at the pole or centre of the cyclone.

The southern cyclone having the more rapid motions on account of the resistances from the earth's surface being less, causes a greater depression of the atmosphere there than in the northern cyclone, and throws the calm belt a little north of the equator, as has been explained.

The tendency of the smaller local cyclones, as has been seen, is to run into the centres of the grand hemispherical cyclones, and thus to be swallowed up and become a part of them.

=

10. If a and ẞ have the same direction, the quaternion qẞ÷ a degenerates into a real and positive number, expressing the numerical ratio of the length of ẞ to that of a, and is then called a Tensor. If, in this case, a = 1, the tensor q B1 expresses the length of the line ẞ, and is called the tensor of the line B. The tensor of ß is written Tẞ. The algebraic sum of two or more tensors is evidently a tensor; and, by §7, a tensor may be applied to any line in space without regard to its direction. Tensors, then, satisfy the condition of § 9, and are commutative in combination with any quater

nion.

11. If Ta= TB, the quaternion qßa degenerates into the single operation of turning the line a around some axis till it

VOL. II.

13

coincides with B, and is then called a Versor. An axis perpendicular to both a and ẞ, and such that rotation around it from the positive direction of a to that of ẞ is positive," is called the Axis of the versor.

β

The angle measured positively is called the Angle of the versor;

α

and is equal, as usual, to the angle (— “).

12. The lines a and ẞ being given, let y be a line having the same direction as a, and let Ty TB. These conditions completely determine 7. Then, by § 9, as a, ß, and y are co-planar,

=

(B÷y) (y ÷ a) = (y ÷ a) (ß ÷ 7) ;

γ

but ya is a tensor, and ß 7 is a versor. Any given quaternion, then, may be resolved into a product of two other determinate quaternions, one of which is a tensor and the other a versor; in this case the former is called the tensor of the given quaternion, and the latter its versor. The tensor of a quaternion, q, is written Tq, and its versor, Uq; thus

(1)

q= Tq. U q = Uq. Tq.

The axis and the angle of Uq may be written Ax. Ug and Uq; or simply Ax.q and 9, and called the axis and the angle of q. If the axes of three or more quaternions are co-planar, it follows that their planes intersect in a common line. If B = =a, then q= a ÷ a = 1, and Tq = 1, Uq + 13. If two or more quaternions are § 12, be equal, and also their versors. we have

but evidently

1.

equal, their tensors must, by As tensors are commutative,

* Either direction of rotation may be arbitrarily assumed as the positive one.

We see, however, from §8, that we do not have in general T = ΣT; and since U must depend on the values of the tensors of the quaternions under the sign 2, while Ug does not depend on these values, it is also evident that we do not have in general U Σ = Συ.

14. To determine Ax. q, that is, to fix the direction of a line in space, requires two independent elements (such as latitude and longitude, altitude and azimuth, &c.). A quaternion, therefore, involves four independent elements, -two to fix its axis, and two more for its angle and tensor; and from this fact its name, quaternion, is derived. If two quaternions have their four elements equal each to each, these quaternions must be equal.

=

15. If in the equations of § 9 we make 7-ẞ, we have p=-1, and pq-q; whence

(3) T(−q)=Tq, Ax. (−q) = Ax. q, <(−q)= 180° +<q. If now we consider one axis the negative of another, when it has the opposite direction, a positive rotation around Ax. q is equivalent to an equal amount of negative rotation around instead of (3) we may have

Ax. q; therefore

(3′) T(−q)=Tq, Ax. (−q)=— Ax.q, <(−q) = 180° — <q.

and by substituting K q for q in (1) we have, since, by (4), U K = K U,

(6)

Kq=TKq.UKq= Tq. KUq;