Solution for rectangle of plane isothermals.

Hence the general problem of finding u and y has precisely
same statement in terms of §, 7, as that given above, (22), X
and (20), in terms of x, y, with this exception, that we have a
u={}τ(§2+n2), but if f(s, n) denote the function of
which x+y transforms,

u=Tƒ(§, n) for every point of the boundary.

The solution for the curvilineal rectangle

where A, A are to be determined by two equations, obtaine

thus:-Equate the coefficient of sin when ŋ=0 and whe

by Fourier's theorem, § 77. Similarly, Bi, Bí, are determines

from the expansions of ƒ(0, ŋ)

and ƒ(a, ŋ), in series of the form

Of one extremely simple example, very interesting in theory and valuable for practical mechanics, we shall indicate the details

Let

This clearly satisfies (47); and it gives, by (48),

(61.

(62.

The solution may be expressed in a series of sines of multiples

of [on the plan of (37).....(45)] by taking *

πη

α

which, with (54), gives

cos B

It should be noticed that this solution fails for the case of 6 = (21 + 1);

and leaves, as boundary conditions in the solution for v,

Example. Rectangle bounded by two concentric arcs and two

cos (B-27) when =0,

cos B

cos (B—2n)} when έ=a,

(65). radii.

cos B

and

v=0 when ŋ=0, and when y=ß.

The last condition shows that the B and B part of (58) is proper
for expressing v, and the first two determine B; and Bí as usual.
Or when it is best to have the result in series of sines of

and leaves, as boundary conditions in the solution for w,

} when 7=0, and when =ß,

w=0 when =0, and when έ=a.

The last shows that the A, and A part of (58) is proper for w,
and the two first determine A, Aí.

708. St. Venant's treatise abounds in beautiful and instructive graphical illustrations of his results, from which we select

the following:

(1.) Elliptic cylinder. The plain and dotted curvilineal arcs Contour are "contour lines" (coupes topographiques) of the section as mal section

lines of nor

of elliptic cylinder, as warped by torsion: equilateral hyperbolas.

warped by torsion; that is to say, lines in which it is cut by a series of parallel planes, each perpendicular to the axis, or

Contour lines of normal section

of triangular

prism, as warped by

torsion.

Diagram of St. Venant's curvilineal

squares for which torsion problem is solvable.

lines for which y (§ 706) has different constant values. Th.lines are [$ 707 (28)] equilateral hyperbolas in this case. T arrows indicate the direction of rotation in the part of prism above the plane of the diagram.

(3.) This diagram shows the series of lines represented ly (34) of § 707, with the indicated values for a. It is remarka

not equal curvilineal squares (hollow sides and acute angles . one of them turned through half a right angle relatively to the

other. Everything in the diagram outside the larger of these squares is to be cut away as irrelevant to the physical problem; the series of closed curves remaining exhibits figures of prisms, for any one of which the torsion problem is solved algebraically. These figures vary continuously from a circle, inwards to one of the acute-angled squares, and outwards to the other: each, except these extremes, being a continuous closed curve with no angles. The curves for a 04 and a = -0.2 approach remarkably near to the rectilineal squares, partially indicated in the diagram by dotted lines.

lines for St.

(4.) This diagram shows the contour lines, in all respects as Contour in the cases (1.) and (2.), for the case of a prism having for Veuant's

section the figure indicated.

the continuous closed curve are merely indications of mathematical extensions irrelevant to the physical problem.

"étoile à quatre points arron dis."

Contour lines of nor

(5.) This shows, as in the other cases, the contour lines for

mal section the warped section of a square prism under torsion.

of square

prism, as

warped by

torsion.

Elliptic, square, and flat rectangular bars twisted.

(6.), (7.), (8.) These are shaded drawings, showing the appearances presented by elliptic, square, and flat rectangular

bars under exaggerated torsion, as may be realized with such

a substance as India rubber.