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QUESTIONS AND DISCUSSIONS. 85

Solution by Benjamin F. Yannet, College of Wooster, Wooster, Ohio.

Let the digits in the initial order be

a», 1„_i, • • •, Oi.
Then by hypothesis,

a„10»- 1 + o„_il0»- 2 4 + a 2 10 + ai s (mod p).

Multiply each member of the congruence by 10, remembering that 10" = 1 (mod p), and
place the digit a n in units place. We thus secure one cyclic permutation. Another cyclic permuta-
tion is secured by multiplying again by 10; and so on. This completes the proof.

It will be observed that the theorem may be generalized by multiplying all the terms to the
left of any specified term by 10*™, in the case of any cyclic permutation, where k is any positive
integer. Thus, in the example given, 480001, 800000014, and 140000000008 are also each mul-
tiples of 37. By successive application of this method, we may obtain different types of cyclic
permutations. Thus, 400000080001 is a multiple of 37. We may have other than cyclic permuta-
tions, with ciphers, by multiplying any one or more terms of the above congruence by 10"™,
where k can have a different value for each term multiplied. Thus, 80401 is also a multiple of 37.
It is interesting to note in this more general application that no two integers of the original
number can ever collide.

Also solved by L. C. Mathewson, Philip Fkanklin, W. R. Ransom, Feank
Iewin, Paul Capbon, Hoeace Olson, and C. C. Yen.

262 (Number Theory). Proposed by C. N. SCHMALL, New York City.

If x, y, z, are 3 integers, consecutive among the integers prime to 3, show that

x(x - 2y) - z(z - 2y) = ± 3.

Solution by Edwabd H. Vance, Graduate Student, Ithaca, N. Y.

Let v — 1 be any number divisible by 3, then any set of three integers consecutive among
the numbers prime to 3 may be represented by one of the following sets:

v — 2, v, v + 1; v, v + 1, v + 3.

Substituting v — 2, v, v + 1 for x, y, z, respectively, in the lefthand side of the given equation
we have

x{x - 2y) - z(z - 2y) = 3.

Substituting v, v + 1, v + 3 for x, y, z, respectively, we have

x(x - 2y) - ziz - 2y) = - 3.

Also solved by Paul Capbon, N. P. Pandya, Louis O'Shaughnessy, Lewis
Clabk, E. F. Canaday, Geobge W. Haetwell, J. L. Riley, Albebt G. Rau,
Hebbebt N. Cableton, Hobace Olson, and V. M. Spunab.

QUESTIONS AND DISCUSSIONS.

Send all communications to U. G. Mitchell, University of Kansas.
DISCUSSIONS.

I. On Making Mathematical Results Mobe Available fob Engineebs.

By Willis Whited, Harrisburg, Pennsylvania.

Some time ago I received a circular from the Mathematical Association of
America regarding the Annals of Mathematics. I like very much the idea of a

86 QUESTIONS AND DISCUSSIONS.

series of articles setting forth the "state of the art" of the different branches of
mathematics in a form that would be intelligible to people who are not specialists
in the respective branches.

I am an engineer and know that there are numerous unsolved problems
in engineering science which are chiefly mathematical. The engineer studies
mathematics primarily for its value as a tool in solving his problems, however
fond he may be of the subject for its own sake. Very few engineers find time,
in the course of an ordinary lifetime, to acquire a reasonably complete knowledge
of all the pure mathematics that they can use to advantage in following up the
latest advances in their respective specialties and in doing the research work that
devolves upon them. It not infrequently happens that work which appears at
the time to be little more than mathematical gymnastics is subsequently de-
veloped into something quite useful; but years elapse before the people who need
the mathematics learn of its existence. The investigating, engineer and the
mathematician must keep in closer touch with each other in the future than they
have in the past. America must take a larger place in the advancement of science.

The engineering investigator who encounters difficult mathematical problems
must have better facilities for acquiring the knowledge he needs of the many
powerful methods of mathematical analysis which have been developed within
the memory of men now living. Works on advanced mathematics are prac-
tically all intended for professional mathematicians. Their contents are almost
wholly academic in character and they are beyond the reach of the engineer.
Articles in mathematical periodicals are seldom intelligible to any but a very
few specialists. This is doubtless unavoidable and perfectly proper, but I would
urge that occasional articles be written bringing various branches of the subject
down to date, omitting, perhaps, mucl/of the purely academic work and express-
ing the whole, if possible, in terms that can be understood by the engineer who
has kept up his collegiate mathematics.

From what little I know of modern mathematics, I would imagine that prog-
ress useful to the engineer has been or soon may be attained in the following
branches (among others): differential equations, calculus of finite differences,
vector analysis, successive integration, elliptic and hyperelliptic functions, tran-
scendental equations and analytical geometry.

Most of the modern writers on advanced analytical geometry use homo-
geneous coordinates. This method has some advantages in certain kinds of
work, but it is rarely taught to undergraduates in engineering and, moreover,
most of the engineer's problems are metrical, so that Cartesian coordinates are
better adapted to their solution. Many theorems in projective geometry could
be used by the engineer who employs graphical solutions if the theorems were put
in such form that he could acquire a knowledge of them in a reasonable time.

Most of the fundamental principles of those branches of science which aspire
to become exact can best be expressed in the form of differential equations.
Many of these equations have not, thus far, been solved. Approximate solutions
are better than none. Hence, I would urge that methods of approximate solu-

QUESTIONS AND DISCUSSIONS. 87

tions be so developed as to make them, so far as practicable, accessible to the
engineer. In the practical applications of mathematics to engineering and,
probably, to other sciences, the solutions of problems are often not exact. Graph-
ical solutions are subject to a very considerable margin of error and arithmetical
solutions almost always involve the multiplication or division of decimals in
which only a certain number of decimal places are retained. Transcendental
functions and radicals are only given approximately in the tables and it may well
happen that a solution in a rapidly converging series is just as convenient as
an exact solution. If a solution is in the form of a series with general expressions
for coefficients, it may be almost as satisfactory as any other kind of a formula.
In that case, if a similar problem occurs again, it will only be necessary to sub-
stitute the proper values for the constant terms in the coefficients, which can be
done by an assistant who is not familiar with differential equations. I therefore
hope that mathematicians will publish freely their methods for approximate
solutions of differential equations and other problems, preferably in a form that
will not compel the busy engineer to search through a multitude of monographs,
many of which are in foreign languages and some of which can not be readily
obtained, before he can get an adequate idea of the nature of the solution.

Elliptic integrals are met with occasionally and if they merely have to be
integrated once approximate methods are available. If successive integration
is required, it is apt to be " another story."

It may be that all problems that can be solved by vector analysis can also
be solved by the older methods, but this method is often so much simpler that
the subject is worthy thorough study.

The engineer often meets with transcendental equations and they usually
have to be solved as individual problems. If more general methods, even if only
approximate, have been developed, they should be more generally known.

Complex variables are occasionally encountered, chiefly in connection with
differential equations. If a practical knowledge of the subject could be imparted
without requiring the reader to toil through ponderous tomes in an effort to find
an explanation, it would be helpful.

The modern theory of functions is a subject which is very interesting to one
who is fond of mathematics for its own sake; but can not some way be found by
which the student can get at the pith of the matter in a reasonable time? The
subject is chiefly academic, but is very attractive.

II. Relating to New Remainder Tebms for Certain Integration Formulae.

By S. A. Cobey, Albia, Iowa.

In the June, 1917, number of the Monthly Professor Daniell notes the fact
that at least one of the remainder terms of the integration formulae which I
gave in the June-July, 1912, number of the Monthly is needlessly large. I also
observe that the remainder term to my formula 25s which he gives is too small,
as he has tacitly made the unwarranted assumption that the signs of his Si and St

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