By P. G. Tait

Excerpt from An undemanding Treatise on Quaternions

To the 1st variation of this paintings, released in 1867, the subsequent used to be prefixed: -

'The current paintings used to be started in 1859, whereas i used to be a Professor of arithmetic, and much extra prepared at Quaternion research than i will be able to now fake to be. Had it been then accomplished I must have had technique of checking out its instructing functions, and of bettering it, sooner than ebook, the place came upon poor in that respect.

'The tasks of one other Chair, and Sir W. Hamilton's want that my quantity are usually not seem until after the e-book of his components, interrupted my already large arrangements. I had labored out approximately all of the examples of Analytical Geometry in Todhunter's assortment, and that i had made quite a few actual purposes of the Calculus, particularly to Crystallography, to Geometrical Optics, and to the Induction of Currents, as well as these on Kinematics, Electrodynamics, Fresnel's Wave floor, &c., that are reprinted within the current paintings from the Quarterly Mathematical magazine and the court cases of the Royal Society of Edinburgh.

'Sir W. Hamilton, while I observed him yet a number of days earlier than his loss of life, steered me to arrange my paintings once attainable, his being nearly prepared for booklet. He then expressed, extra strongly maybe than he had ever performed sooner than, his profound conviction of the significance of Quaternions to the growth of actual technological know-how; and his wish quite simple treatise at the topic should still quickly be published.

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Each slice is a disc, a thin cylinder. Its volume is πr 2h, where h is very small. If we add up all the discs, taking the limits properly, we get the volume. We will take the same region in six different problems, rotating this region differently six times and getting six different volumes. Example 5— Find the volume if the region R is rotated about the x-axis. The volume of each disc is πr2 h: h = ∆x; r = y. So r2 = y2 = x, and x goes from 0 to 9. The integrals are almost always easy. Once you understand the picture, all will be easy.

We use x = a sin u (dx = a cos u du). and the square root is gone. Here are the other two cases. Type 2 (a2 + x2)1/2; we use x = a tan u (dx = a sec2 u du) Type 3 (x2 - a2 )1/2; we use x = a sec u (dx = a tan u sec u du) I have demonstrated each of the three types. However, it is essential that you know by sight what the answer is without substituting. Otherwise, the problems will take forever. " Example 12— We didn't start with u; we started with x. We must draw a triangle with x = 4 sin u. sin u = x/4.

6θ = π. θ = π/6. So the integral is Note I know this last trick is one almost no one will use. But I've left it in because my original editor, David Beckwith, was so great. This is one of his favorite tricks. Thanks, David. Example 11— Find the area of r = 4 cos 6θ. We know we can slide the curve y = f(x) + a units to the right by replacing x by x - a. In the same way, we can rotate r = f(θ) through a counterclockwise angle +α by replacing θ by θ -α. Thus, by rotating our curve by 15º = π/12 radians, r = 4 cos 6(θ - π/12) = 4 cos (6θ - π/2) = 4 sin 6θ, which is exactly the curve in Example 10!!!!!!

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An elementary treatise on quaternions by P. G. Tait

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