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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