tremendous simplification that permits us to integrate our differential equations in explicit form. Integration of the differential equations The integration of the differential equations, which in essence is what permits the final results to be obtained in very simple form, is essentially straightforward and will not be carried out in detail here. In short, to obtain the.Vi integrals, we substitute the value of u as given in equation (20) into equation (14) after a simple indexing change. The resulting integrals are all of the well known Lommel type and as such pose no difficulty. The result, after we introduce the dimensionless quantity In a similar way, to calculate the w, integral from equation (15) the value of m2 is required which again is of no difficulty. The expression for w1(/?1) however involves a double summation which makes it somewhat tedious but again the individual integrals are of the Lommel type. It turns out that a simple rearrangement of the integral expression enables us to obtain its value without the necessity of actually performing the indicated integrations in detail. The result is Modified cost function and determination of multipliers To carry out the optimization, our cost function C = C(R0,Rd = a0 • ttR02 + Uj • rrR2 must be modified to allow for constraints on our variables. These are two in number, namely, that the total power transmitted, Wj, is a constant as is the total power collected, Wo. Our variables are yx, y2, y3 . . . yn, and w,; the only one that is fixed at the right hand end when r - Ru is Wh i.e.,
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