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This will give the correct value of d when r = 1, the value of L being obtained from equation 25, the data necessary being obtained from the first pair of pulleys D, and d1.

The equations needed in solving any problem of cone pulleys have been collected and are given in this paper in the order in which it is most convenient to make the calculations.

These formula being based upon the graphical method already explained, the accuracy of the former as well as the latter will be seen by comparing a few examples, calculated from them (equations 1 to 8). But it was also thought desirable to know how these would compare with some of the previously existing formulæ. For this purpose I have selected formulæ from two authors, the only ones which I have seen, so far, that have a general and practical form, being dependent upon the velocity ratio. I will repeat them here for convenience and transpose them to the notation of this paper.

The approximate formulæ given by J. Howard Cromwell in his "Treatise on Belts and Pulleys," page 55, are as follows:

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But these equations are in an unsolved condition, and inconvenient to apply as they are. Combining equations 28 and 29, and solving for d we have:

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The following approximate formulæ have been taken from "Elements of Machine Design," by W. Cawthorne Unwin.

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The relative accuracy of these formulæ can be seen by comparing the following tables, which were calculated from them. The tables will explain themselves. Great care was taken to have these tables accurate by the use of the logarithmic tables of Baron Von Vega.

EXAMPLE 1.

AS CALCULATED BY THE NEW METHOD, FORMULÆ 1 TO 8.
Center Distance, C = 32′′.

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AS CALCULATED BY J. H. CROMWELL'S FORMULA, EQUATION 30.

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AS CALCULATED BY UNWIN'S FORMULÆ, EQUATION 32 to 34.
Center Distance, C = 32′′.

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Maximum difference in the length of belt = 4.6746".

EXAMPLE 2.

AS CALCULATED BY THE NEW METHOD, FORMULÆ 1 to 8.

Center Distance, C = 100".

This example was taken from Cromwell's "Treatise on Belts and Pulleys."

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Maximum difference in the length of belt = .0097".

EXAMPLE 2.

00

AS CALCULATED BY J. H. CROMWELL IN HIS TREATISE ON BELTS AND PULLEYS,

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Maximum difference in the length of belt =.0938′′.

EXAMPLE 2.

AS CALCULATED BY UNWIN'S FORMULÆ, EQUATIONS 32 to 34.
Center Distance, C= 100".

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Maximum difference in the length of belt = .3155".

EXAMPLE 3.

AS CALCULATED BY THE NEW METHOD, FORMULÆ 1 TO 8.

Center distance, C= 41.625".

This example was taken from a machine upon which a crossed and an open belt were used on the same cones at different times. The diameters taken from the drawings are

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It will be seen that these are correct for crossed belt. Under the conditions under which the pulleys were used, it would have been better to have taken the mean between the open belt diameters and those of the crossed belt.

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Maximum difference in the length of belt = .0072".

EXAMPLE 4.

AS CALCULATED BY THE NEW METHOD, FORMULÆ 1 to 8.

Center distance, C = 13.5".

This example was taken from the countershafts of a machine, and the cones were laid out by a young man according to the writer's former graphical method, illustrated in Fig. 49. The following are the diameters taken from the drawings:

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First Cone 2.", 3.75", 5.5", 7.125", 8.625", 10.125". Second Cone 13.", 11.875", 10.625", 9.25", 7.75′′,

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7.75", 6.125".

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3

5.5005 10.6283 1.9318 52.8237.311 +.12987 11°, 17', 20' 0°, 20', 37" 10°, 56', 43":

10.9453°

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Maximum difference in the length of belt = .0195".

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