Equation of the varied movement of heat in a ring

101. WE might form the general equations for the movement of beat in solid bodies of any shape.

But this would involve avoidable very complicated calculations.

Instead of this, I base my heat equations on the appropriate conditions.

102. I have already considered the uniform movement of heat in a thin prismatic bar whose extremity is immersed in a constant source of heat.

The next problem is the variable state of a solid ring whose points have received arbitrary initial temperatures entirely .

The solid ring is generated by the revolution of a rectangular section around an axis perpendicular to the plane of the ring (see figure 3).

• I is the perimeter of the section
• S is the area as the coefficient & measures the external conducibility
• K the internal conducibility,
• C is the specific capacity for heat
• D is density.

The line o.xxx" represents the mean circumference of the armlet, or that line which passes through the centres of figure of all the sections; the distance of a section from the origin o is measured by the arc whose length is ; R is the radius of the mean circumference.

The smallness and the circular form makes the temperature at the different points equal.

103. Initial arbitrary temperatures are given to the different sections of the ring.

It is then exposed to air maintained at the temperature 0 and displaced with a constant velocity.

The system of temperatures will continually vary.

Heat will be propagated within the ring, and dispersed at the surface.

What will be the state of the solid at any instant?

Let v be the temperature which the section situated at distance will have acquired after a lapse of time t.

v is a certain function of x and 4, into which all the initial temperatures also must enter: this is the function which is to be discovered.

104. What is the movement of beat in an infinitely small slice, enclosed between a section made at distance x and another section made at distance + dr?

The state of this slice for the duration of one instant is that of an infinite solid termi- nated by two parallel planes maintained at unequal temperatures; thus the quantity of heat which flows during this instant de across the first section, and passes in this way from the part of the solid which precedes the slice into the slice itself, is measured according to the principles established in the introduction, by the product of four factors, that is to say, the conducibility K, the area of the dv section S, the ratio and the duration of the instant; its expression is KSdt. To determine the quantity of heat

da' dv de