APPENDIX.

It has been my object, in the above paper, to reduce it to the most simple form possible without sacrificing any of the accuracy of the methods presented. . Since writing the paper I have noticed that the mathematical part-formula 1 to 8-can be simplified still more. That is, the number of the equations necessary can be reduced as follows:

If we substitute the value of R, from equation 2 (a) in equation 4, we get

Now, substituting the value of sin A, from equation 1, or 21, we

sin Eo = sin Bo (cos A, - ("1+1)d1); or, since (r1 + 1) d1 =

4 a C

By the use of this equation we can dispense with equations 2 (a, b, c, and d). Reproducing, then, the equations necessary to solve any problem of open belt cone pulleys, we shall have the following

[ (b)

41. Drd.

A = B° + E° when sin E° is negative.

d

2 C sin A

=

r-1

d = 0.3183 (L - 2 C') when A = 0 and r = 1.

42. L = 2 C cos A+ .01745 d { 180 + (r−1) (90 + A) }

Now all that we need to bear in mind, in the use of these formulæ, is to give to a, in equations 37 and 38, the value .314 as long as the belt angle, A, is less than 18°, and the value .298 when A lies between 18° and 30°. Equation 36 is used only once for any pair of cones, to obtain the constant cos A, (by the aid of tables of sines and cosines) for use in equation 38.

DISCUSSION.

Prof. J. E. Sweet.-As my name has been used in the paper now before us for discussion, I feel justified in stating as a matter of history, that the method illustrated on page 13 of Mr. Smith's paper was, so far as I know, first used by myself in repairing a foot-lathe for Prof. Wing in 1873 or 1874, and published with illustration in the American Artisan of the same year. I would also like to state as a matter of fact that in that lathe, which was of the ordinary sort, the actual variation in the length of belt, when the size of pulleys was determined by my method, was only of an inch from the length when determined mathematically. This variation of of an inch in a belt of some 9 feet in length is entirely within practical limits, and, as a foot-lathe is almost as extreme a case of great variation in sizes and angles as is usually met with in practice, I believe the method to be a practical one.

While it is true that the method does not hold good in an imaginary extreme case, if no one wants to use the extreme case, or so seldom as to be the exception, is it worth while lumbering up our minds with so much to get the two or three fractions of an inch, when the workman will cut two or three inches off the belt if he discovers it a little loose?

The case appears to me to be similar to the Watts parallel motion. It is true that motion is not absolutely correct, but, in the case where it is used in the Richards indicator, it is so nearly correct, that any possible error arising from its imperfection is infinitely less than dozens of other errors occurring in the use of the instrument.

The peculiarity of the parallel motion is that it is very nearly

correct for quite a distance and then diverges very abruptly, and it appears to be so with this system of determining the size of pulleys; it is practically correct within quite the ordinary limits, and only runs astray where one has very little use for it.

This simple diagram, Figs. 86 and 87, or that of Fig. 48 of Mr. Smith's paper, is, for one who has his drawing tools at hand, one of the most ready ways to determine the length of a belt. The length of the belt in the case shown, and necessarily for any pair of pulleys,

is simply twice the distance E F, and the circumference of a circle of which B G is the radius.

I wish to add in conclusion this question: If this simple method is sufficiently accurate for all practical purposes, is it best to hamper it with additional refinements, when the result of these additions is sure to cause those who might profit by it to throw it aside altogether?

Prof. J. E. Denton.-Passing over the point which the professor raises, which unquestionably stands by itself, that when we know the error of any method to be less than a variation that is of any consequence in practice, practice does not ask anybody to go any farther, there is a certain credit to be given to any one who desires to improve it, and who, speaking from the ranks of practice, as Mr. Smith does, thinks that he finds a return for his labor in giving us the refinement. Now, assuming therefore that he desires to add a refinement, I wish to put this remark in the discussion to do simple justice to those who before him have also sought to give us the last refinement, and having that thought in mind I criticise the paper for not mentioning what has gone before with regard to exact solutions of this problem. I would mention that two cases come to mind which, it seems to me, ought to have been quoted. One was Prof. Klein's effort, made some years ago, in which he published a table and a graphical method, which were very simple to use and absolutely exact; also, an approximation by Prof. Sponkler, which was exceedingly simple. I make this remark so that Mr. Smith may reply, as undoubtedly he will, in an able manner, showing what, if anything, is the objection to those as exact solutions, and why, therefore, he offers something which is beyond all that practice wants, as Prof. Sweet has explained.

Mr. Henry Leon Binsse.*—I wish to present a note by Mr. A. J. Frith, C. E., on Rankine's formula and its use, and I have found that for ordinary practice it is quicker than any graphical method, while it is easier to remember. It is also accurate enough.

Without reflecting on the beauty of the method presented to us, we are inclined to think that a perusal of Mr. Smith's paper is apt to give an erroneous impression of the value of Rankine's formula. In our experience we have not found that a slight variation from desired ratios was of any practical moment, and with this allowance Rankine's formula has been not only perfectly applicable, but very

* Added after adjournment, under the rules.

simple, rapid, and beneath our notice. is as follows:

accurate, in. the examples that have come The form in which we have used the formula

Do the diameter of equal steps, and C distance between centers in inches; D1 + D2, the diameter of unequal steps. It is

(DD)) represents one-half

noticed that the last expression (12.56

the difference between the sums of the diameters of the equal and unequal steps.

If, then, we had a set of cones to design, the extreme diameter of which including thickness of belts-were 40" and 10", and the ratio desired 4, 3, 2 and 1, we would make a table as follows, C being equal to 100":

The trial diameters being chosen to give the sum of 50" and the desired ratio, from the differences of these trial diameters,

obtain the value of (D

12.56.

D0)2). The difference between each of

these and the largest add to each diameter, thus leaving the difference unchanged, and we very rapidly obtain values which approximate closely to the ratio chosen, and which give equal belt tensions.

The example given in the paper of 37" cones, 30" apart, ratio 1 to 7, is such an extreme one, that it gives an exaggerated impres sion of the differences of different methods. We tried Rankine on this example, but had to use the result of our calculations to obtain new differences; not because the results were incorrect, but because they varied somewhat from the ratios determined. The sum of the diameters we obtained differed but of an inch in the most extreme case from the result of Mr. Smith, and gener