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If you then solder the switch back somewhere into this wire, on either side of the resistor, you ll be able to turn on and off the lower-pitched flute sound whenever you like (see Figure 6-10). Other uses for resistors in circuit-bending are lowering the current to cool down audio outputs that are too loud, and to lower current to avoid burning out the LEDs that you ll be using as pilot lights and envelope lamps. Don t worry, I cover this in detail later. It s all very simple stuff. But which resistor Because there are many resistance levels offered by different resistors, as indicated on them by the color-coded bands, the easiest way to experiment with resistors is to use a resistor substitution wheel. This is a small device loaded with resistors inside. You simply clip its two leads into the circuit with which you want to experiment and turn the dial (see Figure 6-11).

N y 1 p p y 1 N z 1 p pz 1 3-186 3-187

Only the English market for long-term government bonds during most of the eighteenth and nineteenth centuries can be well summarized by one series of rates, the yield of consols Gross rates of interest are presented throughout, although these often were gross of tax advantages, tax penalties, redemption or refunding prospects, and other circumstances that might have added to or subtracted from the promised and realized yields In this chapter, the data presented in the preceding chapters are simplified further because in this way some useful generalizations concerning trends and patterns can be derived Only one long-term series and one short-term series for each country are presented in the summary Tables 76 and 77 These rates are presented only as fifty-year minima and.

N z 1 p px y 1 3-188

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N y z 1 p 2 px 2 3-189

Equation 3-189 gives the probability of forming an x-sized polymer molecule by only one of many equally probable pathways. An x-sized polymer molecule can be obtained by couplings of y- and z-sized radicals, z- and y-sized radicals, (y 1)- and z 1 -sized radicals, z 1 - and y 1 -sized radicals, y 2 - and z 2 -sized radicals, z 2 - and y 2 sized radicals, y 1 - and z 1 -sized radicals, z 1 - and y 1 -sized radicals, and so on. There are x 1 possible pathways of producing an x-sized polymer molecule when x is an even number. Each pathway has the same probability that given by Eq. 3-189 and the total probability N x for forming an x-sized polymer molecule is given by

as decennial averages No attempt, therefore, can be made from these tables at cyclical analysis, which requires monthly quotations The trends analyzed in this summary are all secular or suprasecular trends The interest-rate histories summarized in this chapter are those of nine countries: England, France, Holland, Belgium, Germany, Switzerland, Sweden, the United States, and Canada Although all countries at all times have had an interest-rate history, usually unrecorded, modern commerce and, especially, modern finance have become fully developed in only a few nations In these nations, markets for credit instruments have been most fully recorded The mainstream of interest-rate history is here defined, perhaps arbitrarily, as proceeding from ancient Greece and Rome through medieval Florence, Venice, and Genoa to medieval Antwerp and Lyons, thence through the Dutch Republic to England, and thence to most of modern Western Europe and North America.

As you rotate the dial, each click of the wheel connects a different internal resistor between the leads and allows you to see or hear the effect of that resistor on the circuit. Resistor values are printed on the case just outside the dial so that you know which resistor is connected as you turn the wheel. When you hear or see the result you want, stop turning the wheel and look at the resistor value that the dial is pointing to. If it says 500K, you know you need to solder a 500k resistor into the circuit to get the same effect. Resistor substitution wheels are very handy to have around, and one should be part of your bending kit.

N x x 1 1 p 2 px 2 3-190

N x is synonymous with the mole- or number-fraction distribution. When x is an odd number, there are x pathways and the x 1 term in Eq. 3-190 should be replaced by x. For polymerizations yielding high polymer, the difference between x and x 1 is negligible and can be ignored. [Some derivations of size distributions show exponents y and z, respectively, in Eqs. 3-186 and 3-187 instead of y 1 and z 1 , which results in an exponent of x instead of x 1 in Eqs. 3-189 and 3-190). The differences are, again, unimportant for systems that yield high polymer.] Using the same approach as in Secs. 2-7a and 2-7b, the following equations are derived for the weight-fraction distribution, number- and weight-average degrees of polymerization, and X x =X n :

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