## R Fig. 10.13 Contour lines of a truncated conical shell

In a grazing incidence telescope, the slope i of a mirror is necessarily small; assuming that o will be small and neglecting the x2-term in the expansion of 1/a2(x), we obtain t(X) „ to a2(X) a2

hence the condition of constancy for this ratio is satisfied if the expression into the brackets is equal to zero,

For instance, if the inner angle slope is with i = 2° and the mirror thickness-ratio is t0/a0 = 1/20, the outer slope is with o = 2.210°; For an outer surface which is homothetic to the inner surface, we would have o = 2.105°.

10.3.3 Linear Product Law - Flexure-Thickness Relation

We assume that the truncated conical shell is with a small slope angle i, say i < 5° typically, and that t0/a0 < 1/10. For practicable reasons, we also assume that the external load q is always perfectly uniform. Returning to (10.63) and using the above expression of a(x), after substitution of i + o, this relation becomes

Similarly to (10.25), let us introduce a dimensionless thickness and flexure, but now with respect to a0,

ao qao where C is a constant. From (10.66), we obtain a relation between the dimensionless thickness and flexure. This is the following linear product law (Lemaitre )

^ If a truncated weakly conical shell is radially submitted to a uniform load q applied all over the outer or inner surface, and if no discrete circle-force is applied except the small axial reaction Rq to the load, then the product thickness-flexure TW is a linear function of the axial coordinate %.

This law is of fundamental importance for the aspherization of an X-ray mirror because the shell is with both free edges. Thus no bending moment or radial force is needed at the boundaries except the small axial reaction Rq which comes from the axial components of load q.

• Avoiding poles in the linear product law : For instance, let us assume that the flexure function W is represented by (1o.62) from which the dimensioned flexure is w =

1E Wa0. Then, from the linear product law, the associated thickness is

0 0