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Page 6
I should like to see the trace and guide chariots on the same line of
rails, one below the other, were this possible without producing the
bad effect of a skew, pull or push.
4. The practical integraph must not have a greater maximum error than
2 per cent. The mathematical calculations, which are correct to five
or six places of decimals, are only a source of danger to the
practical calculator of stresses and strains. They tend to disguise
the important fact that he cannot possibly know the properties of the
material within 2 per cent. error, and therefore there is not only a
waste of time, but a false feeling of accuracy engendered by human and
mechanical calculation which is over-refined for technical purposes.
For comparative purposes I have measured the areas of circles of 1
inch, 2 inches, and 3 inches radius, the guide being taken round the
circumference by means of a "control lineal," first with an ordinary
Amsler's planimeter and then with the integraph. I have obtained the
following results:
---------+------------+-----------+-----------------------------------
| | | By integraph.
Radius | | By |--------+--------+--------+--------
of | Calculated |Planimeter.| | Upper | | Upper
circle. | areas. | |Middle. | end. |Middle. | end.
| | |p=2 in. |p=2 in. |p=4 in. |p=4 in.
---------+------------+-----------+--------+--------+--------+--------
in. | | | | | |
1 | 3.14159 | 3.140 | 3.140 | 3.138 | 3.120 | 3.120
| | | | | |
2 | 12.56636 | 12.55 | 12.36* | 12.546 | 12.568 | 12.552
| | | | | |
3 | 28.27431 | 28.24 | ...... | ...... | 28.280 | 28.288
---------+------------+-----------+--------+--------+--------+--------
* Cross bar had to be moved during tracing.
From this it follows that the error of the planimeter is less than 0.1
per cent. and that of the integraph about 0.5 per cent. Obviously we
could make this error much less if we excluded small areas measured
with large polar distances, or such polar distances that the cross bar
must be shifted. Excluding such cases, we see that the accuracy of the
integraph scarcely falls behind that of the planimeter and is quite
efficient for practical purposes. It must be borne in mind that the
above measurements were made with the "control lineal," an arrangement
which carries the guide round a circle of the exact test area. In most
cases the curve has to be followed by hand, and the error will be
greater--greater probably for the integraph than for the planimeter,
as the former is distinctly hard to guide well.
I think, then, we should be safe in saying that the error of the
integraph is not likely to be greater and is probably less than 2 per
cent., so that in this respect the instrument may be considered a
practical one.
5. A further condition for a good integraph is that it should have a
wide range of polar distances, and that it should be easily set at
those distances.
One of the conditions I gave to the maker of the instrument was that
it should be able to take all polar distances from one to ten
half-inches. This condition he can scarcely be said to have fulfilled.
With polar distances of 1/2 inch and 1 inch, the machine works
unsatisfactorily, which indeed might have been foreseen from the
construction of its sliding bars. It works best from 2.5 inches to 5
inches, and this is the range to which I think we ought to confine the
present type of instrument. As the last conditions I may note that:
6. A practical integraph ought to be easy to read.
7. Draw a good clear curve.
The scale on the present instrument is very inconvenient, as it is
often almost out of sight; the curve it draws, on the other hand, I
consider very satisfactory, when the pencil is loaded, say, with a
planimeter weight. On the whole, I think you will agree with me that
this integraph goes a good way, if not the whole way, toward
fulfilling the conditions of a practical instrument.
I next turn to its construction and the claim it has to be considered
in any way new. Let me briefly remind our members of the process by
which an element Q R of the sum curve (Fig. 1) corresponding to the
point P on the primitive is drawn; P M being the mid-ordinate of L N,
a horizontal element, P B is drawn perpendicular to any vertical line
A B; and O A being a constant distance termed the base or "polar
distance," Q R is drawn between the ordinates of L and W, parallel to
O B. If P' be the point where P M meets Q R, we note the following
relationship of P' to P.
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