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THE MESSENGER OF MATHEMATICS.
o
• • • •
• • •
• • • • *
•: A ":
o • • •
• • • • •
• ■ 9 •
THE
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MESSENGER- •OI'^'-MATHEMATICS,
BDITBS BT
W. ALLEN WHITWORTH, M.A.,
FBLLOW or n. JOHM'8 oollxob, oambkidob.
C. TATLOE, M.A.,
FBLLOW OF IT. JOHN'S COLLBOB, CAMBRIDOB.
R. PENDLEBUET, M.A.,
FBLLOW OF 8T. JOHN'b OOLLBOB, CAMBKIDOB.
«
J. W. L. GLAISHEE, M.A., F.E.S.,
FBLLOW OF TBINITT OOLLBOB, CAMBRIDOB.
VOL. VII.
[Mat, 1877— Apeil, 1878].
MACMILLAN AND 00.,
'Eonbon anO ^ambritfge.
BDINBUROH: EDHONSTON * CO. OLASOOW: JAMES UACLEHOSE.
DVBLIN : HODOES, FOSTER & CO. OXFORD : JOHN HENRY AND J. PARKER.
1878.
% «
• • 9
• ••
• • • •
• » • •
* • •
• •
« • • • •
• • • •
• »
• •
• a
• •• •
a • • •
• • « «
• ••
• •••
« . •
• • .
• •• •
• • • •
• « % • •
« • * •
• • •
* • •
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• •••••• •
• <••-•• •
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• • • • • «
1 37289
CAMBRIDGE :
FBINTED BT W. MBTOALFB AND SON, TRIKITT STREET.
A
CONTENTS.
PUEB GEOMBTRT.
PAOI
On the Poriflm of the Bing of Ciicles tonching two Circles. Bj JET. M.
Taylor . . . , . . ... .148
On the Bing of Circles tonching two Circles, and Kindred Porisms. B j
W. W. Taylor 167
ANALYTICAL GEOMETBY OF TWO AND THBBB DIMENSIONS.
On the Theory of Envelopes. By R. W. Genese , . . .61
Note on the above. By Professor Cayley ..... 62
On the Wave Surface. By Professor J.. ifannAem . • . • 100
ARITHMETIO, ALGEBRA, AND TRIGONOMETRY.
Note on a System of Algebraical Equations. By Professor CayUy • 17
On Certain Series in Trigonometry. By H. M, Taylor . • .22
Computation of the Cube Root of 2. By Artemaa Martin . . 60
Yerification and Extension of the value of the Cube Root of 2. By
Peter Gray ........ 51
On a Pair of Algebraical Equations. By Professor P. Mansion • • 57
On the Determination of the Sign of any Term of a Determinant. By
Professor W, W. Johnson . . • • • .59
Bumii^tion of a Series. By /. M, Croker . • • • . 60
Solution of a Cubic Equation. By R. W. Genese . • . .61
An Arithmetical Theorem. By Professor H, W, L, Tanner • . 63
Theorems relating to the difference between the sums of the even and
uneven divisors of a number. By J. W. L. Glaisher . . .66
On some Continued Fractions. By /. W, L. Glaisher ... 67
An Identity. By Professor Cayley . . . • . .69
On the Resolution of the Product of two sums of Eight Squares into the
sum of Eight Squares. By J, J, Thomson .... 73
VI CONTENTS.
PAGE
Series and Products for ir and Powers of it. By /. W. L. Glatsher . . 75
On an Arithmetioal Theorem of Professor Smith's. By Prof essor P. i/afwum 81
Cube Boots of Primes to 81 Places. BjS.M.Drach . . .86
On Long Suooessions of Composite Numbers. By /. W. L, Glatsher 102, 171
On Two Related Quadrio Functions. By Professor Cayfey . . .116
Pour Algebraical Theorems. By J, W, L. Glaishm- ... 119
Elementary Proof of a Theorem m Functional Determinants. By Professor
E.J,Nan$(m ........ 120
Proof of the Formula for 1» + 2» + 8»...-fn». By /. M. Croker . . 122
Expansion of Products of Cosines and Sines. By jR. Verdon . . .122
A Trigonometrical Identity. By Professor Cayky .... 124
Extract from a Letter from Professor Cayley . . . . .125
Sub-Factorial N, By W, A. Whitworth 145
Arithmetical Note. By Professor S. W. L, Tanner ... 157
On a Class of Determinants. By /. W. L, Glatsher . . . .160
Formula inrolying the Seventh Boots of Unity. By Professor Cayley irr
On the Literpretation of a Passage in Mersenne's Works. By JB, Lucas 185
A Problem in Partitions. By Professor Cayley .... 187
Arithmetical Note (on Circulating Decimals). By /. W. L, Glatsher . 190
Buler's Formula in Trigonometry. By J, W. L. Glatsher . . 191
Numerical value of a Series. By /. W, L. Glatsher , . • .192
DIFFERENTIAL AND INTEGRAL CALCULUS.
On Spherical Harmonics. By W. D. Niven . . . . 1, 131
On Special Methods of Interpolating. By W, D, Niven • . .35
On the Product l«.2«.3»...n». By /. W, L. Glatsher . . • 48
On a Discontinuous Series. By Professor P. Mansion . . .58
On a Class of Definite Integrals. By B, Ratoson .... 65
On the Development of ( . _ ^ in a Series. By J?. Lucas . . 82
On the Successive Summations of 1*» + 2** + 3*»...+ «*•. By JB, Lucas . 84
Suggestion of a Mechanical Integrator for the Calculation of f{Xdx + Tdy)
along an Arbitrary Path. By Professor Cayley . . * 92
On Development in Series. By E, Lucas ..... 116
On a Theorem due to Rodrigues. By W. M, H, Hudson . . . 117
Expansion derived from Lagrange's Series. By J. (T. I>. Glaisher . .118
On Eulerian Numbers. By E, Lucas ..... 139
An Extension of Arbogast's Method of Derivations. By J, J, Thomson • 142
Note on the Calculus of Functions. By Professor H, W, L. Tanner • 156
Note on Arbogast's Method of Derivations. By Professor Cayley . . 158
CONTENTS. VU
DIFFERENTIAL EQUATIONS.
PAQB
On Cognate Biccatian Equations. By R, Eawson . . . . 69
On Certain Partial Differential Equations of the Second Order which hare a
General First Integral. By Professor H, W. L. Tanner . . 89
Note on a Differential Equation. By Professor S, W L. Tanner . . 107
On Linear Partial Differential Equations of the First Order. By|Professor
E» J, Nanson ........ Ill
New Demonstration of the fundamental Property of Linear Differential
Equations. By Professor P. Mansion ..... 188
THEORY OF ELLIPTIC FUNCTIONS.
On the Occurrence of the Higher Transcendents in Certain Mechanical
Problems. By W. S^ L, Russell ..... 18, 136
Reduction of some Integrals to Elliptic Forms. By Artemas Martin . 24
An Illustration of the Theory of the O-Functicms. By Professor Cayley . 27
On the Triple Theta-Functions. By Professor CayUy ... 48
On a Formula in Elliptic Functions. By /. W* L, Glaisher , . . 144
An Elliptic Function Identity. By M, M, U. Wilkinson . • * 156
KINEMATICS, MECHANICS, HYDROMECHANICS, OPTICS,
AND ASTRONOMY.
On the Irregular Flight of a Tennis-Ball. Bj Lord Rayhigh . . 14
On the Potential of an Elliptic Cylinder. By Professor H, Lamb . 33
Note on a Theorem in Hydrodynamics. By Professor S, Lamb . . 41
A Simple Proof of a Theorem relating to the Potential. By Lord Rayleigh 69
On a Simple Proof of Lambert's Theorem. By Professor J, C, Adams . 97
Proof of the Principle of the Composition of Couples in Statics. By Professor
CNiven . 118
Theorem in Elinematics. By C, Leudesdorf .... 125
On Equiyalent Lenses. By jR. Pendlebury ..... 129
A Theorem in Areas, including Holditch's, with its Analogue in Three
Dimensions. By E, B. EllioU • . ^ • . . 150
Note on Mr. Leudesdorf's Theorem in Kinematics. "BjA^B^Kempe • 165
Note on Hydrodynamics. By Professor £./. iViafMon , . , 182
On Sylvester's Kinematic Paradox. By H, Mart . • i • 189
A Theorem in Kinematics. By A, B» Kempe . • i • 190
TRANSACTIONS OF SOCIETIES.
London Mathematical Society. By R, Tueker . . 12, 55, 127, 159, 192
The Meeting of the British Association at Plymouth. By /. W, L, Glaisher 96
«
[NoTB. Pp. 1 — 16 were published in May, 1877; pp. 17 — 32 in June,
1877; ...pp. 177—192 in April, 1878 f so that the month of publication of any
paper may be readily ascertained.]
f
MESSENGER OF MATHEMAHOS.
ox SPHERICAL HARMOXIC&
B7 W. D. Xkem, If. A , Trimtj College.
The object In view in these artides Is to give a method
of dednciiig from what is known as the zonal harmonic of
any degree the most general harmonic of the same d^ree.
As the method was suggested bj Clerk Maxwell^s treat-
ment of the subject, and as the results here proved have a
close alliance with those contained in his work on Electricity
and Magnetism^ the notation there used has be^i as far as
posnble adopted here.
After establishing a certain theorem on which the method
above referred to depends, we will employ it in finding the
most general expression for a harmonic referred to its poles
as axes. We will then show its application in the case of
symmetrical harmonics other than zonaL The last part of
the paper will be taken up with a sketch of Professor
Maxwell^s proof of Laplaoe^s expansion modified to some
extent, especially by the insertion of the same theorem.
§ 1. The general expresnon given in the Electricity and
Magnetism for a harmonic of the i^ degree is
where any operator -^ Is the same as I -. — V in -7- + n — .
"* ^ ah ax ay dz
VOL. VII. B
2 MR. NIVEN, OK SPHBEICAL HARMONICS.
It will thus be seen that the expression contains 2i+l
independent constants, viz. one from Mi and 2t from the i
arbitrary directions of the A — axes, each set of direction-
cosines being of course connected by a relation of the form
f + m' + n'—l. It may, conversely, be proved that any
harmonic of the t''' degree can be thrown into the above form
in only one way, bo as to have all the quantities I, m, n real ;
in other words, that the i poles of the harmonic may be real
points.*
If ail the axes coincide, we have
and it is easy to prove that this is equal to
where Qi is a certain fanction of the polar coordinate 6, The
quantity Qi is called the zonal harmonic of the f"" degree,
and may be easily obtained in various forms, two of which as
being suitable to our purpose, we here quote
u being pnt for cosd and v for sind.
will be convement to have an abbreviation for the
tor beloBgiiig to the general harmonic, we therefore put
dh^...dhi~ [dhy
irmony with the above relation between a harmonic and
corresponding spherical harmonic when it is zonal, ire
put
^ ' \dh\ r r"' '
quantity 1' ia in general a function of the two polar
iinates 6 and <^, it is the quantity we want to determine.
1 "Sphericftl Haroiomca," by
MB. NIVEN, ON SPHERICAL HARMONICS. 3
§2. We will now prove the theorem expressed by the
equation
(-»'[l]'i=[i]V*)-
Let Q (fig. 1) be any point on a sphere of radius a. and
let P be any other point ; r, r' the distances of P from 0, Q ;
also let PQ = TT - 0. Then
Let us now perform the operation (~ 1) 37 on both sides
of this equation. If a, b, c be the coordinates of Qj we may
write aj = a + a;', &c. Hence
d -. d d d
dh dx ay dz
dx'
+ m
d
dy'
d
^""dz'
d
~ dh"
•
i
-\\
r '^T.
therefore (-
Now r'^Qn is a homogeneous function of the n**^ degree of the
coordinates x\ y\ z'. Hence, the effect of the operation
■=77 upon the first i terms is to differentiate them
down to zero. If after the differentiations have been per-
formed on the other terms, we also put x\ y\ z' zero, we snail
have only one term left, so that when r' = 0, or r = a,
(-')[sTJ=i-LsT"-«''-
Hence, dropping the dashes on the right-hand side, we get .
L!^- [I] (»-«•)'
B2
' MM. XITEX, OS STHEUCAL HAKHONICS.
§3. la mamg thti theorem to find the most general
esfntMorao for a hwroonic, since the choice of coordinateB
u ariMtrarr, we shall appose that the axia of z passes
thnwri tlw point Q (fig. §2). In that case, using the B
formula for Q^ ve get • -»
whew S=x-i-s,;
becomes 'je+"'j-+«^.
"f rfi? (& *
wbov ( " / + jn^,
the anglocrW'rtthotwoen two axes f?., Bi.nl f; «, «\
^» ■ - 2 4 n,n^
tt« R«««M«l oporator oa Q.r-, maj bo written
!■ i. » (\ll,«li™ ,,f th, I'. ,„d «•, derived by picking out
tWi'telllrfQjj (J'j from tho fcclora
iVlI'Iu '.'".llT',1,""' ""!L°'' i'l """ »f I' «ni "-oin
»1 « m », Mill llio imiiibDr of Iho terms will ho
(L«)-
HR. NIVEN, ON SPHERICAL HARMONICS. 6
Now the symmetry of V «how8 us that It is possible to
express it in a series of terms of n products of the form
where p and q may be any numbers less than n. But if
we have 2w quantities, the number of factors that we can
form of them, the sum of two of the quantities constituting
A factor, taking n faictors at a time is
equal to/suppose*
If now we arrange in lines the results of multiplying out
(each of these combinations of n factors we shall have 2*
individual terras on the right in each line. Hence, the sum
of all the combinations of factors will produce 2y individual
I2w
terms, i,e. k=. Hence, since the number of terms in V
In
is this number, divided by In, it is clear that the symmetry
is such that each individual term in the addition referred to is
repeated In times. It follows that In V may be found by
taking the combinations of factors as described. But we
proved
Ip^g + Igta^ = 2 {fi^g - n^Tig) ;
therefore [n F= 2** {(/*,, - w,nj...(/i,^„^, - n^n^J + &c.},
where all the changes must be rung among the factors.
All the changes amongst the n's in the general expression
must also be taken.
The first three terms of the spherical harmonic of the
t. degree are
The most general spherical harmonics of the 2nd, 3rd,
and 4th degrees are easily derived by the direct application
of the above method, but they may be deduced from the
general formula now found. They are given by equating
2 F,, [3 Fj, [4 Y^ respectively to
3 (5w,w,W3 - /A„n3 - fi^^n^ - /a^w,),
3 {SSWjWjWgW^ - 5 {n^%fi^ + 5 similar terms)
+ /*i2/^84 + ^ similar terms}.
8 MR. NIVEN, ON SPHERICAL HARMONICS.
Ro • ^^ *^ example of the use of the general theorem of
9 2, m cases where the axes of coordmates are not arbitrary,
we may take the case when a of the axes of differentiation
are m the equator and t-o- coincide with the axes of z.
rrofessor Maxwell points out that in this case^
B]'-(ir{(i)-^©i-
Tiet (a, fi) be the polar coordinates of the point Q
on the sphere (fig. i, §2). Then
^ a cos 5 cosa + sina sin 9 cos {<f} - fi).
Hence, taking the first expression quoted for the zonal
harmonic
"L^ ^ 2'[n \i^n\i^2n
X |cosa.;5 + —^ [^e'fi + ve^l*^ («* -f fiy)"] .
Picking out the terms «^^ and «'"V) and performing
the required operation on them, we get
C08C7/3 sin<^a ^2i-2n[(^-Q; _^.,^
5«:S=i S(- 1) . j-; :-; ^r COS tt
-[I 2---- n " 2(2.'-i) '^' « +-} •
If we take
* If, in factorizmg any homogeneous operator, quadratio factors aie introdttoed
of the form
/ £ ^£,Q£y,( A + ft— £?
\ dx dy dz) . \ dx dy dz) '
we see by throwing this into impossible factors that A + /a, &c. must be pro-
portional to direction cosines. Hence Aa + Bb + Cc = 0. If, therefore, we put
K = JM« + £*+ C*), k = 4{a^ + A« + <J«), and suppose ^> ife, the operator becomes
K^ ('r-s) + ^ (tp ) » where ^ , t« denote <Hfferentiations with regard to two
lines at right angles. Hence the operator becomes, by Laplace's equation,
and the two real poles belonging to it are thtis discovered.
ME. NIVEN, ON SPHERICAL HARMONICS. 7
as operator, where A and B are arbitrary constants, we thus
arrive at Laplace's formula, given in Thomson and Tait,
p, 149, (37), (38).
§ 5. In some of the expansions which occur it is necessary
to find
\dh\ r«"^''
In these cases It may be useful to observe that
1U.5...P.- D) [I]' ^ -(- ly^. ^]' {f ^-) ,
a theorem which can be proved in a similar manner to that
contained in §2, It is, however, quite as convenient to
deduce , "^ from the expansion of (a'* + 2ar/A + /**)" ^ by
picking out the coefficient of ix* as to differentiate Q^^i.
Laplace's expansion.
§ 6. Consider a solid globe of homogeneous matter, the
particles of which attract an outside particle with a force
varying inversely as the square of the distance. Then, if the
centre of the globe is taken for origin of coordinates, the
M
potential at any outside point is , where Jf is the mass of
the globe and r the distance of the point from its centre.
If another globe of equal dimensions and mass, but whose
particles repel according to the inverse square of the distance,
have its centre at a small distance A, from the origin along
the line whose direction cosines are Z„ w^, n^ ; then the poten-
tial due to this globe is
If we now imagine the two globes to coexist, then so far as
attraction or repulsion is concerned we may suppose all that
part of each which occupies common space to be annihilated.
We thus arrive at the conception of a shell of matter, part of
which attracts and part repels. The potential due to it is
Let the mass M now become infinitely large whilst the thick-
8 MR. NIVEN, ON SPHERICAL HARMONICS.
nesa h^ becomes infinitely small, the quantity M\ remaining
finite and equal to 3/^. The potential due to the shell is then
' dn^ r
Moreover^ each of the spheres attracts or repels external
matter as if its mass were collected at its centre. It is clear
therefore that the shell will possess the same property. In
Professor Maxwell's nomenclature this centre is then a
compound point of the first degree. In accordance with
this description we may call the shell a compound shell of the
first degree.
Let us now suppose a compound shell of the first degree to
be placed with its. centre at the origin and let its potential be
If we place with its centre at the point [IJi^^ ^a\? %K)^ *
shell exactly equal in all respects except that it repels where
the other attracts and vice versd^ its potential, if h^ is small,
will be
" ' \d\ r ^« dh^ d\ r) •
Let the two shells co-exist and let Mfi be ultimately M^'i
Jf, becoming infinite and h^ zero ; then the potential due to
a compound shell of the second degree is
M. ^
' (fAj dh^ r *
Pursuing this reasoning, we might show that for a compound
shell of the V^ degree, the potential at outside points is
{? 1
^ ' * dh^dh^,,.dhir'
We have thus arrived at a physical meaning of a spherical
harmonic, viz. that it is the potential of an infinitely thin
shell of matter, part of which attracts and part repels ac-
cording to a complicated arrangement determined by the
positions of 2' equal, spheres whose centres ultimately coincide
with the origin.
Since the action of the shell on outside matter is the
same as that of a compound point at its centre, conversely
outside matter will attract or repel the shell as if it were
MR. NIVEN, ON SPHERICAL HARMONICS. 9
concentrated at its centre. Hence, if it be placed in a field
of force whose potential at the centre is F, the potential
energy of the shell will be
'' [i] y^
(see Clerk Maxwell's Electricity^ vol. I., p. 168).
For example, let this shell be surrounded by another
which is concentric with it and produces a potential at inside
points equal to
©'*•
The potential energy of the inner shell is then
' \dh\ ^^''
being estimated on the compound point at the centre, and,
therefore, we are to put r = after differentiation. But in
those circumstances it is obvious the result is zero except
when 1=^, and we have proved, §2, that in that case the
result will be
.Mi
L- a
V)
where Yij is the value of Y. at the pole of the harmonics.
But if <Ti is the symbol for the density of the inner shell
at any point, then the expression for the potential energy
estimated over the surface is
jjfTiQjds.
This therefore will be zero except when i =«y, and then it is
a
Let us now draw a line from the centre to the pole of
the zonal harmonics, and let" it cut the inner shell at j.
Then, by the ordinary surface condition, viz.
dV dV ,
we get, if ,<r,- be the density at j,
10 MB. NIVEN, ON SPHERICAL HARMONICA.
therefore (2i + 1) //o-,- QidS = 47ry<r ..
Now let there be any infinitely thin distribution whatever
in the place of the inner shell. We may suppose that it
consists of a series of compound shells of degrees 0, 1, 2, &c.
Let the density at any point be therefore given by
o- = o-Q 4- <r, +-..+ 0-,.+....
Then since JJo'iQjdS vanishes, except when t = /, it is
obvious we may put, in place of the last written equation,
Hence 4.7r<r=^fJ<Td8+...+ (2t + 1) JJ(TQid8 -{-... ,
which is Laplace^s expansion,
§7. The theorem of §2 admits of a physical interpre-
tation. Let a sphere of radius less than OQ be described
with Q jas centre, and let matter be distributed over it so
as to produce at all points of it the same potential as a solid
homogeneous globe of unit mass with its centre at 0. The
potential of the solid globe is - , and of the distribution
1 / , r *
- - ^ ^,+...+ (-1)*^ Qi+... ,
and it is obvious that the distribution consists of a series of
r d~\*
compound shells. By performing the operation (—1) -^1
on these two expressions, we arrive in one case at the
potential due to a compound shell of degree i with its centre
at 0, and in the other at a complicated distribution on the
sphere whose centre is Qj producing at all points inside of
it the same potential as the (?-shell, viz. equal to
^. [|]r'(2, + &c,
at the point Q only the first of these terms exists. It will be
seen that this manner of looking at the subject is practically
the same thing as finding the distribution of electricity
induced in a spherical conductor due to an electrified point
outside of it, and suggests an extension of the method to the
general case of a spherical conductor placed in a field of
electricity.
Let y be the potential due to the field at the centre of
MB. NIVEN, ON SPHERICAL HARMONICS. 11
the conductor. Then, by Taylor's theorem, the potential
due to the distribution inside of it must be
d d d
^^ dx ' dy de y
where the centre is taken for origin, and the differential
coefficients of V belong to the centre. This may be written
where, after differentiation, we are to understand the dif-
ferential coefficients as belonging to the centre.
The outside potential due to the distribution is
-s — eP ^P V.
P
Hence the density is given by
47ro- = - T- + 2 -J- J /^ F.
In the common case of a point of unit electricity placed
at distance /from the centre,
/I c d\ 'T-l
d_
dp,
^ _ /I 2 ^ V
"" \c dp J r
cr' '
where r is the distance from the electrified point to the point
on the surface under consideration.
§8. It may be remarked that the method now given of
expanding the potential furnishes a very easy proof of the
theorem due to Gauss, that the average potential over any
sphere not cutting through any attracting matter is equal to
tne potential at its centre. The average potential is
47rc
12 TRANSACTIONS OF SOCIETIES.
If we expand the exponential, we have a series of terms
of the form
I f d^ d d^ y
\i\ dx ^ dy dz)
Now this is a solid harmonic, since it satisfies Laplace's
equation ; we may therefore write it
The average potential is therefore
All the terms are zero except the first, and the result is Fl
(To be continued.)
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, FebruarySth, 1877.— 0. W. Merrifield, Esq., F.R.S., Vice-Presidentf
in the chair. Mr. G. W. Von Tunzelmann was admitted into the Society. The
following communications were made. (1) "On the area of the quadrangle
formed by the four points of intersection of two conies," Mr. 0. Leudesdorf, B.A..
Pembroke College, Oxford; (2) "On the numerical value of a certain series,"
Mr. J. W. L. Glaisher, F.R.S. j (3) "On the general differential equation
dx dy
jj + jp=. 0, where JT, F are the same quartic functions of a, y respectively,"
Professor Cayley, F.R.S. : (4) " On the classification of loci and on a theorem in
residuation," Professor Clifford, F.R.S.: (1) The four points of intersection are
all realj so that the discriminant ot the cubic JPA + iT'e + iTO' + A' =
la positive; also it is supposed that the quadrangle formed by the points
is non-reentrant. The condition for this may bo found from the con-
sideration that no real ellipse or parabola can be drawn round a
reentrant quadrangle, and is v' — 4CC' > (where, with the usual notation,
C=ab- A«, C = a'b^- A'«, v = ab' + ba''- 2hh') . The author first investigates the
simpler case where the conies both represent pairs of right lines and then takes the
general case. (2) The series considered is the following l—i+^ — -^+-^—Ac,
The series •nJ~Si + c;i~7;i+*c-» i^ ♦^be uneven, admits of finite expression
as a series of terms involving the first n Bemoullian numbers and having ir** as a
factor, but when n is even there is no such formula. The most troublesome case
to calculate directly is that of n = 2, since for this value of n the series converges
very slowly ; and the calculation is not very easy even when recourse is had to
Eulefs formula
Su. = constant + fu^ _ i„. + ^^ ^ - J^^ -^ + 4c.
TRANSACTIONS OF SOCIETIES. 13
Mr. Glaiflher obtains the yalue of the series to 20 places of decimals. Use is
made of a paper communicated by the author at the January meeting of the
Society. (3) In this paper Professor Cayley investigated the connexion between
_ dx dy
Euler's solution of the equation j^ "^ If ~ ^ *°^ ^^^ given by Abel's theorem.
Thursday, March 8th, 1877.— C. W. Merrifield, Esq , F.R.S., Viee-Preiident, in
the chair. Mr. B. F. Davis, B.A., was admitted into the Society, and Mr. Charles
Pendlebury, B.A., was proposed for election.
The following communications were made : "On a new view of the Pascal
form," Mr. T. Cotterill, M.A. (the paper turned on dividing the 46 Pascal points
into triads ; 1°. 15 self -conjugate triads, 2*^. 15 diagonal triangles, 3°. 60 triangles,
corresponding intersections of two inscribed triangles ; 4°. 60 Pascal lines, each
Pascal line corresponding to a triangle of 3°. The Pascal points were denoted by
Greek letters, thus (a, a', a") a conjugate triad, so that only 15 Greek letters had to
be used. Forming triangles of those points, the author easilv obtained the Steiner
and Eirkman points as well as other points and properties. "On a class of integers
expressible as the sum of two integral squares," Mr. T. Muir, M.A. (the class of
inte^rs considered included those whose square root when expressed as a
contmued fraction has two middle terms in the cycle of partial denominators. A
general expression was given for all such integers, and an equivalent expression in
the form of the sums of two squares). "Some properties of the double theta
functions," Prof. Cayley, F.R.S., (the investigation was founded on papers by
Gtoepel and Bosenhain).
Thursday, April 12th, 1877.— Lord Rayleigh, F.B.S., Prmefeftf, in the chair.
Mr. Charles Pendlebury, B.A., was elected a member. The following communi-
cations were made : " On Hesse's ternary operator and applications," Mr. J J".
Walker, M.A. ; " Geometrical illustration of a theorem relating to an irrational
function of an imaginary variable, and on the general differential equation
dx dy
-jy + -Ty = 0> where JT, F, are the same quartic functions oi Xy. y respectively,**
Prof. Cayley, F.B.S. j (Profs* H. J. S. Smith and Henrici took part in a discussion
on these papers). Mr. C. W. Merrifield, F.R.S., Vice-President, having taken the
chair, Mr. Hairy Hart, M.A., deduced some cases of parallel motion from the
consideration that the contra-parallelogram represents the motion of two equal
ellipses rolling upon each other, and that of these {i.e. parallel motions) two
espedally were very simple, inasmuch as the motion was obtained in either case by
the use of five bars only, and was, moreover, perfectly continuous. Mr. Tucker read
an abstract of a paper by Prof. H. W. Lloyd Tanner, on a method of solving
partial differential equations which have a general first integral, applied to
equations of the third order with two independent variables. The following
results were stated to have been obtained : When an equation of the third order
admits of a general first integral of the 8(toumed form, the latter is determined by
a system of ten linear homogeneous equations of the first order; which are
however equivalent to three, and only three, independent equations. The
coefficients of this auxiliary system are of two kinds. Those of the first group
can be expressed directly in terms of the coefficients of the given equation.
Those of the second ^up are derived indirectly from the coefficients of the given
equation ; and most smiply by expressing that of the ten auxiliary equations only
three are mutually independent. The coefficients of the second group may have
one, two, or three sets of values, to which correspond one two or thi«e auxiliary
systems, each of which serves to determine a fii^t integral. It will be found that
in some cases these auxiliary systems are sufficient to determine not only the first,
but also the second and even the third and final integrals of the given equation ;
analogous results are obtained when the method is applied to the equation of the
n-th order with m independent variables. The i^est of the paper is devoted to a
classification of equations of the third order, and to the solution of some examples.
It may be noticed that in these examples the author succeeds in solving equations
of the second order which involve an arbitary function in their expression. Also
some theorems are indicated which may serve as the basis of a general method of
solving equations of the second order analogous to those employed in the case of
equations of the first order,
B. TucKBR, M.A., Hon, S«e.
{ u )
ON THE lEREGULAR FLIGHT OF A
• TENNIS-BALL.
By Lord Bayleigh, F,Il,S.
It 18 well known to tennis players that a rapidly rotating
ball in moving through the air will often deviate consider-
ably from the vertical plane. There is no difficulty in so
projecting a ball against a vertical wall, that after rebounding
obliquely it shall come back in the air and strike the same
wall again. It is sometimes supposed that this phenomena
is to be explained as a sort of frictional rolling of the rotating
ball on the air condensed in front of it, but the actual devia-
tion is in the opposite direction to that which this explanation
supposes. A ball projected horizontally and rotating about
a vertical axis, deviates from the vertical plane, as if it were
rolling on the air behind it. The true ejqplanation was given
in general terms many years ago by rrof. Magnus, in a
paper "On the Deviation of Projectiles," published in the
Memoirs of the Berlin Academy ^ 1852, and translated in
Taylor's Scientific Memoirs^ 1853, p. 210. Instead of sup-
Sosing the ball to remove through air, which at a sufficient
istance remains undisturbed, it is rather more convenient to
transfer the motion to the air, so that a uniform stream
impinges on a ball whose centre maintains its position in
space, a change not affecting the relative motion on which
alone the mutual forces can depend. Under these circum-
stances, if there be no rotation, the action of the stream,
whether there be friction or not, can only give rise to a force
in the direction of the stream, having no lateral component.
But if the ball rotate, the friction between the solid surface
and the adjacent air will generate a sort of whirlpool of
rotating air, whose effect may be to modify the force due to
the stream. If the rotation take place about an axis per-
pendicular to the stream, the superposition of the two
* motions gives rise on the one side to an augmented, and on
the other to a diminished velocity, and consequently to a
lateral force urging the ball towards that side on which
the motions conspire.
LORD RATLEIGH, ON THE PLIGHT OF A TENNIS-BALL. 15
The only weak place in this argument is in the last step,
in whieh it is assumed that the pressure is greatest on the
side where the velocity is least. The law that a diminished
pressure accompanies an increased velocity is only generally
true, on the assumption that the fluid is frictionless and
unacted on by external forces ; whereas, in the present case,
friction is the immediate cause of the whirlpool motion. The
actual mode of generation of the lateral force will be perhaps
better understood, if we suppose small vertical blades to pro-
ject from the surface of the ball. On that side of the ball
where the motion of the blades is up stream, their anterior
faces are in part exposed to the pressure due to the aug-
mented relative velocity, which pressure necessarily operates
also on the contiguous spherical surface of the ball. On the
other side the relative motion, and therefore also the lateral
pressure is less; and thus an uncompensated lateral force
remains over.
The principal object of the present note is to propose and
solve a problem which has sufficient relation to practice to be
of interest, while its mathematical conditions are simple
enough to allow of an exact solution being obtained. For this
purpose I take the case of a cylinder round which a perfect
fluid circulates without molecular rotation. At a great
distance from the cylinder the fluid is supposed to move with
uniform velocity, and the whole motion is in two dimensions.
On these suppositions the stream function, on whid the whole
motion depends, is of the form
-^safl — ijr sin^ + iS logr,
where r, are the polar coordinates of any point of the fluid,
measured from the centre of the cylinder, and the direction
of the stream, as pole and initial line respectively, a is the
radius of the cylinder, and a, /9 are constant coefficients
proportional respectively to the velocity of the general
current and the velocity of circulation round the cylinder.
When r^a^ yjr is constant, shewing that the surface of the
cylinder is a stream-line. The radial velocity at any point
is given by
S = «('-?)^'^5
SO that, when r = oo and ^ = 0, the radial velocity is a, which
is therefore the general velocity of the stream.
16 LORD RAYLEIGH, ON THE FLIGHT OF A TENNia-BALL.
At the surface of the cylinder there is no radial velocity,
and the magnitude of the tangential velocity is given by
at a
Hence, if p^ be the pressure at a distance, and p the
pressure at any point on the surface,
2(p-j)J = a'-(2asm^ + |y,
the density of the fluid being taken as unity. Thus the
lateral force
(^— ^)a sin^c?^= — Tra^ ;
it is therefore proportional both to the velocity of the motion
of circulation, and also to the velocity with which the
cylinder moves relatively to the fluid at a distance.
If the velocity of circulation depending on /8 be small,
the character of the stream lines diners but little from that
given by
/:
Vr = «^l-^jr sin^,
corresponding to a simple stream ; but when attains a certain
point of njagnitude, the stream-lines in the neigbourhood of
the cylinder become re-entrant.
Sir. W. Thomson has proved that, if in an infinite mass
of otherwise quiescent fluia there exist irrotational circulation
round a moveable cylinder, the amount of the circulation
cannot be changed by any forces applied to the cylinder.*
Hence, if the cylinder receive an impulse, it will afterwards
move in a circle, and the direction of revolution will be
opposite to that of the circulation of the fluid.
It must not be forgotten that the motion of an actual
fluid would differ materially from that supposed in the
preceding calculation in consequence of the unwillingness
of stream-lines to close in at the stem of an obstacle, but
this circumstance would have more bearing on the force
in the direction of motion than on the lateral component.
* Yortez Motion, Edinburgh TrantaetionSf 1868.
( 17 )
NOTE ON A SYSTEM OF ALGEBRAICAL
EQUATIONS.
By Professor Cayhy,
AflSUME • X +y -f« =P,
xyz s=jB,
A=iX {nyz + ^) — w;* [mx + P),
B^y[nzx'\' Q) - u^ {my + Pjj
C^z [nxy -\'Q)-w^{mz-{' P),
Then (wj« + P)-B-(my + P) (7
= [myz + Py) {nzx -{- Q) — [myz + Pz) {nxy + Q)
= myz {nzx + ^ — wiry — C) + Pnxyz -f P^y
— Pnxyz - P^«
= mnxyz {z — y)- PQ {z - y)
= f« -y) [mnxyz-- PQ] = (y- «) 0,
whence, identicallj
(m«+P)P-(7ny + P) (7=(y- «) 0,
(wo? + P) (7 - (w« + P) -4 = (« - a) 0,
{my + P)^ - (7waj + P) P = (aj -y) 0.
Hence any two of the equationa -4 = 0, P«=0, (7=0 imply
the third equation.
We have
-4 = a? {(ri + 1) y« + «a? + xy) — «?' {(wi + 1) a? + (y + «)}
= («'- t(?') (y + «)-a;[(w+ 1) tt;*-(n4 l)yz\
VOL. VII. C
18 MB. RUSSELL, ON THE HIGHER TRANSCENDENTS
and similarly for B and C. The three equations therefore
are
X y4 z
y __ z + x
z ar + y
z* — w* (w + 1) w?* — (w + 1) iry '
and any two of these equations imply the third equation.
ON THE OCCURRENCE OF THE HIGHER
TRANSCENDENTS IN CERl^AIN MECHANICAL
PROBLEMS.
By W. n, L. Russell, F.R.S.
Professor Schellbach, in his Lehre von den ElUptischen
Integralen^ p. 315, has considered the following problem :
**To determine the motion of a particle on the surface of
an ellipsoid, subject to the influence of a force varying as
the distance, situated in the centre of the ellipsoid." He
has shown that the solution depends upon an hyperelliptic
differential equation. In endeavouring to extend this theorem
I found that mechanical problems might in a great number
of cases be resolved by means of hyperelliptic functions and
the higher transcendents. I propose to give the results of
my investigations in the present paper. These Integral
Transcendents have, as is well known, been the subject of
investigation by Gopel, Rosenhain, Weierstrass, Riemann,
Clebsch, and many other mathematicians. But their re-
searches give rather certain properties of the integrals than
the integrals themselves, and do not, as it seems to me,
afford methods by which the actual values of the integrals
might be found. I have, therefore, endeavoured to find a simple
process for their evaluations which I hope to communicate at
the end of these contributions.
IN MECHANICAL PBOBLEMS. 19
(1) A tube of small bore in the form of a circle revolves
about one of its diameters as a fixed vertical axis, and
contains a smooth particle, to determine the motion.
Let the centre of the circle be the origin of coordinates,
(a) the radius of the circle, a:, y, z the coordinates of the
particle where the fixed vertical axis is the axis of [z)^
6 the angle which the circular tube makes with the plane
of {xz)^ <f) the angle which the radius vector of particle
makes with the axis of {z)j P and R reactions on particle
in the plane of the tube, and perpendicular to that plane;
then, if M and n are masses of tube and particle,
cPO ^ Basin(f>
cPx
»i -7^ = — J2 sin^ — Psin^ costf,
cPy
m -~ = J2 cos^ - P sin^ sin 5,
cPz ^
w-^ = -Pcos</»-^w,
also 0? = a sin^ cosd, *y^a sin0 sin d, z = a cos^ ;
wherefore after an elimination which presents no difficulty,
we have
,_ /r^r^u;I ^^" + ^^' »i^V I*
r - a V W ja(p 1^^ _ 2^^ ^^^ ^j ^j^^, _^ ^^, g.^.^^ _ ^,^ ,
where c and Cj are two constants introduced by the integration,
the result is an hyperelliptic function, and may be reduced
to the algebraical form by putting cos^ = w.
This question was proposed many years ago in the Senate-
House. 1 observe that the number of unknown pressures,
and therefore of geometrical equations, are less by two than
the number of geometrical variables. The same is true of
Schellbach's problem just mentioned. In the great majority
of cases we have to consider the number of geometrical
equations is less by one than the number of geometrical
variables.
(2) Two particles P and P' whose masses are {m) and
{m) are connected by a straight rod, and compelled to move
C2
20
MR. RUSSELL, ON THE HIGHER TRANSCENDENTS
in two grooves j4P, AP' In one vertical plane to determine
the motion.
Let A be the origin of coordinates, the arcs horizontal
and vertical at -4, a;, y coordinates of P, oc\ y' (where for
convenience x' is essentially negative) of P\ inclination of
rod to the horizon, (a) its length, P, F' reactions of grooves,
T tension of rod. Then if AP^ AP' are inclined to the
horizon at angles 7 and /8, the equations of motion are
d*x
♦w ^ = -Psin7 +rcos^,
d'v
w ^ = PCOS7 +rsintf-5r7w,
m' ^ =-P'sin/3+rcos^,
w'-^ = P' cos;3-rsin^-^7w,
a? f aj' =s a cos^, y - y' = a sin^, t/ = x tan7, y' = x' tan/S,
we immediately find from these equations
fdS V(^ sip' g + P sin g cos g 4- C cos* g)
" •' V(^' sin ^ + B'co&e + C) '
, J ^ wag* 8in*)8 + 7/20' 8iD*7
cos^/S cos* 7 '
^ _ 2?na* tan/9 2 w V tan 7
cos'' 7 cos*^ '
^_ ma* m'a*
cos'' 7 cos^'^S '
A'ss^ 2ga {m tan7 - m' tan^) (tan7 4- tan^)|
JB'=i^ '2ffa {m + Tw') tan/S tan 7 (tan 7 4- tan/S),
and C is to be determined by the conditions of motion.
This integral may of course be changed into an hyperelliptic
function by putting u — ihu^d.
(3) A sphere rolls inside a hollow rough cylinder with its
axis horizontal. The centre of gravity of the sphere does
not coincide with the centre of the sphere, but these two
IN MECHANICAL PROBLEMS. 21
centres are always in one vertical plane. To determine the
motion.
Let (a) be the radius of the cylinder, (r) the radius of the
sphere, o the distance between centre of sphere and centre of
gravity of the sphere, let the intersection of the axis of
cylinder with given vertical plane be the origin, the coor-
dmate axes, horizontal and vertical with the axis of (y)
downwards, ^ the angle with the line joining the centres of
sphere and cylinder makes with the vertical, the angle
which any radius of sphere makes with its initial position, J?
the re-action between the two surfaces, and F the friction ;
then we have from the known properties of roulettes the
equation = 0. Then the equations of motion are
m ~j^ -= — ¥ cos(f) — R sin <^,
m -~ = Fsin^ — B cos^+ mg^
mW^^ ^{r-ccos(^ + </»)} -^csin(^-f </»),
x—[a — r) sin^ — c sin^,
y = (a - r) cos<^ + c cos^,
we shall suppose (a) a multiple of (r) and = wr, then elimi-
nating, we find
n — 1 fj
c* 4- A;* 4- »•' — 2cr cosw0
Cj + (n - 1) r COS0 + c cos(n — 1) ^
9
where the integral may be easily reduced to an hyperelliptic
function. This is in fact Euler's Pendulum, the length of
the isochronous simple pendulum will be found in Walton.
(To be continued.) .
( 22 )
ON CERTAIN SERIES IN TRIGONOMETRY.
By H. M. Taylor, 3i,A.
The following method of establishing some formulae in
Trigonometry is perhaps worthy of remark. It can easily be
•bewn that
co»a?7r cosa^r - sino^Tr sinaTr = cos (a? + a) 7r
\ 1 - 2a A 3 - 2a A 5 - 2a;
\ H-2aA 8 + 2aA 5 + 2a; ^ ^
If we equate the coefficients of x in the two sides of this
6<|UAtion, we obtain
(2 2
— ;- '^ -^ — t: &c.
1 - 2a 3 + 2a
2 2 ^ \
+ IT2^ + 8T2-a-^H'
or l,rtana7r-^-_,^--^^^^ + 3-^-— -+&C.
(2),
^ . 1 1 1 p
or r— tanaTr « ^ - , + - — --^ + -r — rn + &c.,
8a 1 - 4a' 9 - 4a' 25 - 4a" '
which may bo written
Also from equation (1), by equating coefficients of a;*, we
obtain — ^tt" cosaTr
■ 4 cosaTT X product of terms in (2) taken two and two together}
whence
L 4. ^ -L ^ J. ^ + &c
(I - 2a/ ^ (1-f 2a)« ^ (3 - 2a)» ^ (3 H- 2a)« ^ '^''*
9 S V
a= -- tan'aTT + -~ = — sec^aTr,
«i4 4 4
MR. TAYLOR, SERIES IN TRIGONOMETRY. 23
tt' ,a7r 1 1 1 1 p
or -J- sec —- = J- r^ + /, . ., -»■ 7^ xf + ,^ , v M +&C.
4 2 (I — a) (l+«) (3 — a) (3+ a)
(*).
Similarly, from the equation
sinarTT cosaTr -f cosxtt sinaTT = sin {x-\'a)ir
" (' + I) (' + ifa) (• + 2-fa) *= • <»)•
by equating coefficients of a;, we obtain
/I 1 1 V
TT COSaTT =ft smaTT - + 1- \- cfcc.
\a 1 + a 2 + a
1-a 2-a 3-a j'
11 1 1 1 P //.N
or TT cot air = - - ^ + — + +&c... (6),
a l-al+a 2-a 2+a ^ ''
or
= 2a (- 5 + ,r5 J + -^i 5 + &C. ) ,
a \1 — a 2 —a* 3 —a /'
1 - aTT COtaTT 1 1 1 p /-v
Also by equating the coefficients of a;' in equation (5), we
obtain ^tt* sinaTr
= sinaTT X product of terms in (6) taken two and two together ;
whence
1 1 1 1 1 a
a« ■*■ (1 -a)' "^ (I +a)- "^ (2-a)' "*" (2 + a)" ■^'^^'
= 7r' cot*a7r + TT* = TT* cosec*a7r (8j.
( 24 )
REDUCTION OF SOME INTEGRALS TO
ELLIPTIC FORMS.
By Arteuuu Martin.
(Concluded from Vol. ti., p. 29).
11. Pot
12. Put J„=f{a?-<^i{a'-a?)*dx.
Let a*-a?=^, then oj = (a* - y*)*, <^ = r-r§-^> »nd
13. Put •^t»=j(a,»-c»)i (a" -»«)*•
liCt <^ — a^ = y*, and we have
14. Put ■'i«=°j(a;''-c7 (a* -»»)♦•
Let c?^a?=j^, and we have
0* (i' - a:')*
'|3a''& V - 4 (g' + &') a;* + Sa;'} dx
(a''-a!")*(6'-«")*
MB. MAKTIN, INTEQBALS TO ELLIPTIC FORMS. 25
15. Put ^.s^J- (y -«:>)* •
Decomposing,
, f x* {a* - a^)* dx fa?{a*b*-{a' + V) x' + x*]dx
"*j (o* -»•)»(&" -a:")*
16. Put
'W
[{a?-h')idx
"j (aj»-a')* •
Let X — ^j._y«ji »
tnen *"-(a''_y)i(5''-y»)i' (as'-a")* y»
and/. =5(a - * ) J („. _ y.)i (j" _ ^.)f " j j (a''-2,»)*(6»-/)*
17. Put ^=j(a^_o.)i(a,«_5»)*-
and we get /„ = |^^— ^^L__ = [i^].
26 MB. MARTIN, INTEGRALS TO ELLIPTIC FORMS.
18. Put -^i. = j (c« _ a,») (6» - a:*)* (a* - ««)* *
Decomposing,
- _ / ta''-a:*)*<to , , , ^,^ f <fo
^ ;(c» - 0^) (6' - «")» (a* - »«)»
19. Put /„=/[^] diB. Integrating by parts,
'x[^+^{a*-a?)i{b*-x*)i+i (a'-J") log{(o''-a!»)»+ (J'-x*)*}.
20. Put 7^ «=pa! [E] dx. Integrating by parts,
21. Put /^j =/[-^ ^i»« Integrating by parts
I,,^x[F]-Sxd[F]
= a; [2?^ + log {(a* - a;")* + (i' - a?f].
I will now conclude this series of papers with a few
unreduced examples for the exercise of those who may wish
to try their skill upon them.
_ _ C a?dx _ _ ^a? — a*)* dx
"~j(a!«- a")* (»'-&*)*' "~j {x*-by '
fa^jx^-bydx J _ f x'ja^-a^Ydx
""J (x'-a*)* ' "'} (x'-by '
J^ =/(a? - a»)* (a!* - 5"]* <fo, /„ =/x'' (x* - a*)* {a^ - i*)* dx,
Erie, Pa., U.S.
January 31, 1877.
( 27 )
AN ILLUSTRATION OF THE THEORY OF THE
^-FUNCTIONS.
By Professor Caylet.
If X be a given quartic function of a;, and if m, or for
convenience a constant multiple au, be the value of the
integral 1 . y.. taken from a given inferior limit to the
superior limit x] then, conversely, x is expressible as a
function of w, viz. it is expressible in terms of ^functions of
w, where 3w, or say S (w, :ff) {jf a parameter upon which the
function depends), is given by definition as the sum of a series
of exponentials of u] and it is possible from the assumed
f dx
equation au— \ /.yv ; and the definition of .9m, to obtain by
general theory the actual formulae for the determination of x
as such a function of u,
I propose here to obtain these formulae, in the case 'where
X is a product of real factors, in a less scientific manner, by
connecting the function Su (as given by such definition) with
C dx
Jacobi's function 0, and by reducing the integral i ..y. by
a linear substitution to the form of an elliptic integral ; the
object being merely to obtain for the case in question the
actual formulae for the expression of a? in terms of 3-functions
of w.
The definition of Su (or, when the parameter is expressed
S (m, JF) is
^M = 2 (-) VJ^^'^,
where s has all positive or negative integer values, zero
included, from — go to -t go (that is from — /8 to + /S, S = oo ) ;
the parameter JF, or (if imaginary) its real part must be
positive.
Su is an even function .5 (— m) = Su, Moreover, it Is at
once seen that we have
^(w+ 7r)= Suj
^ (m + tiF) = - dF-2t«^M,
whence also S{u + mir + wtdF),
where m and n are any positive or negative integers, is the
product of ^u into an exponential factor, or say simply that
it is a multiple of 3u.
28 PROP. CAYLEY, ILLUSTRATION OP THE
Writing u = - it^^p, we have ^ (- lijf) = S (Je JF), that Is
and therefore also .5 {mir + (n + ^) ijf] = 0.
The above properties are general, but if iF be real, then
A, Z| ^', J being as in Jacobi (consequently k being real,
positive, and less than 1, and JTand JT' real and positive), then
assuming Jf = ~ j or, what is the same thing,
irK'
the function 3 is given in terms of Jacobi's by the equation
~j ; or, what is the same thing, 0m = ^ f^\ .
We hence at once obtain expressions of the elliptic func-
tions sn Uj cu t(, dn u in terms of .5, viz. these are
Consider now the integral
r ^ ^r
where a, 5, o, c7 are taken to be real, and in the order of
increasing magnitude, viz. it is assumed that J — a, c—a^ d—ay
c — bj d-bjd'-c are all positive ; x considered as the variable
under the integral sign is always real ; when it is between
a and b or between c and d^ X is positive, and we assume
that V(-^) denotes the positive value of the radical; but if a?
is between b and c, X is negative, and we assume that the
sign of V(-^) IS taken so that ..yj is equal to a positive
multiple of i, and this being so tne integral is taken from
the inferior limit a to the superior limit a?, which is real.
Take x a linear function of y, such that for
x=^ajbj Cj d
y = 0, 1, pj , oo respectively.
dx
suppose.
THEORY OF THE .5-PUNCTIONS. 29
80 that, X increasing continuously from a\o d^y will increase
continuously from to oo .
,« 6 — a,d ^ c
We have
^b—d x—a
^'^b-a x-d^
^d — a cc — &
"^^b-a x-d^
l-iV =
d^a oj —
__, •
c- a x — d^
and, thence,
where . /( j is taken to be positive: and the sign of
^s/iX) being fixed as above ; then for y between and 1 or
> rj , y . 1 — y .1 — ^V ^^ ^^ positive, and V(y'l-y«l""^"y)
will also be positive ; but y being between 1 and p ,
y.l - y.l — i*y will be negative, and the sign of the radical
is such that -j-. — -* — p-r is a positive multiple of i.
We have moreover
y a— a , , ,. dx
and therefore
dy ifjT ^ dx
where is/{d-'b,c^a) is positive ; or, say.
Hence, writing y^z^^ sn*2^ we have
and it is to be further noticed that to
x = a^ bj c, d^
30 . PROF. CAY LEY, ILLUSTRATION OF THE
or we may say
Writing for shortness
= a,
we have au
_ r dx
^ a
and moreover aK=-\ -ttv^;
or If for a moment we write / . • ,,^. =-4, &c., then these
equations are
a{K + iK')=C''A,
a{2K'\-iK') = D-A.
Hence 5+ C-2A=D-Aj that is ^-5- C + 2) = 0,
or ^— A^D" C^ that is
where observe as before that a? = a to a; = J, or a: = c to a: = rf,
X is positive, and the radical ^/{X) is taken to be positive.
We have also
where, as before, from b to c, X is negative, and the sign of
the radical is such that ,ttF\ ^^ ^ positive multiple of t; the
last formula may be more conveniently written
dx
'^■'L
TFT)'
THEORY OP THE 5-FUNCTIONS. 31
where, from J to c, -X is positive, aud ^{-X) is also
taken to be positive.
Collecting the results, we have
/.
VIZ)"""' ^"^/{d-b.c-a)' d-b.c-a
and also k'^= , — , ,
and then conversely
__ a (rf— J) + rf (i - a) snV ^
^- ld^b)-t{b-'a)sn'u '
or, what is the same thing,
b — d.x — a
sn'tt
V
\A/ •
•*/
v»
~b-
a.
X-
-d'
cn'e*
d-
a
,x
-b
~b-
a.
,x-
-d'
dn*w
d-
~b-
a,
.X
,x-
-e
where, in place of the elliptic functions we are to substitute
their 5-vaiues; it will be recollected that :ff the parameter
of the ^functions has the value
/ 7rIC\ _ f" dx _ r"* dx
and, as before, K= - 1 - ,, ^, .
Hence, finally, a, kj k\ K^ Jf denoting given functions
of a, by c, dj if as above
dx
we have conversely
/.
b — d.X-a 1 -iJF+-J7i7 na /tTW 1 .^\ n^'^'U
b ^a.x— d k
iir«
rf-a.a;-5 k' HiJF+^
= 7-e
-3-(=n-+j«)**S,
b — a.x — d k
which are the formulae in question.
32 PROP. CAYLEY, THEORY OP THE 5-PUKCTIONS.
The problem is to obtain them (and that in the more
general case where a, 5, c, d have anj given imaginary
values) directly from the assumed equation
/.
dx
and from the foregoing definition of the function 5.
It may be recalled that the function ^u is a doubly
infinite product
m and n positive or negative integers from — oo to +ao ;
I purposely omit all further explanations as to limits; or,
what IS the same thing,
u
2mK+{2n-\-l)iK\
and consequently that, disregarding constant and exponential
factors, the foregoing expressions of
b — d.x— a d^a.x — b d—a.x — c
h — a.X'-d^ b — a.x — d^ b-a.x-d
X Y Z
are the squares of expressions ^, jj?^, ^, where X, F, Z, W
are respectively of the form
unn{i + -!^l, nnji + ^^l,
^^{l + -J^}, nn|i4-^l,
I [m, n)) \ {rn, n)) '
where (w, n) = 2mK-\- 2niK\ and the stroke over the m or
the n denotes that the 2m or the 2n (as the case may be)
is to be changed into 2m + 1 or 2n + 1. But this is a
transformation which has apparently no application to the
3-functions of more than one variable.
( 33 )
ON THE POTENTIAL OP AN ELLIPTIC
CYLINDER.
By Professor JT. Lamh^ 3f,A , late Fellow of Trinity College, Cambridge.
The potential U of an infinitely long homogeneous elliptic
cylinder, of unit density, at any external point P, is given,
save as to an infinite additive constant, by tne formula
1- ^ y
^=^"*/x V(a-tA.y-hX) ^^- ••(^)'
where a?, y denote the coordinates of P with respect to the
principal axes of the section of the cylinder made by a plane
through P perpendicular to its length, 2a, 2b the lengths of
these axes, and where the lower limit of the integral is the
positive root of
1 — . •^ =s
a +\ a +\
Let us transform (1) by assuming
where c* = a*'- J*, so that we have also
2~>/ •
We thus find
80 that the Integral in (Ij becomes
J'4X = c'(*
or, on reduction,
r 4d5*e"»» ^y^er^ "1 , .
yOL. VII. D
34 PROF. LAMB, ON THE POTENTIAL OF A CYLINDER.
taken between the proper limits. Now let us write
x + ii/==c sin(|' + ty),
or a; = c8inf
y = ccos|
2
(4).
We see at once that the limits of integration in (2) are
rj' and oo , so that (3) becomes on reduction and omission of
additive constants
- 2rj' + e-2ii' cos2f '.
Hence, if V be the function conjugate to Z7, we have both
U and V given by the formula
Z7+tF=7raJ{2t(| + ;i7) + e^*^^+*'''^} (5).
The chief advantage of the above transformation is that
it enables us to find without much difficulty the forms of the
equipotential curves and lines of force due to the cylinder.
I had occasion hpropos of certain hydrodynamical problems*
to draw the curves oy which the first and second terms of
the expression on the right-hand side of (5) are separately
represented. The curves ?7= const, F= const, are found at
once bv combining these according to Maxwell's well-known
methoci.
A few of the equipotential curves are drawn roughly in
fig. 2. One of the ellipses confocal with the section of the
cylinder is dotted in for comparison.
It may be remarked that (5) contains implicitly the proof
of Maclaurin's theorem on the attraction of ellipsoids, as
applied to our particular case. For c is the same for all
confocal cylinders ; it follows then from (4) and (5) that the
values of U for two such cylinders are at any external point
proportional to the values of the product abj that is to the
masses of equal lengths of the cylinders.
Adelaide,
March 6, 1877.
Quarterly Journal of Mathematics, December, 1875.
( 35 )
ON SPECIAL METHODS OF INTERPOLATING.
By TT. D, Niven, M.A., Trinity College, Cambridge.
The simplest formulae of interpolation are expressed in
terms of the values of the function at equi-diiferent values of
the independent variable. It may, perhaps, be of advantage
to consider what modifications are required in formulae of
this kind to render them serviceable when the values of the
function are given for vulues of the independent variable
which do not proceed by equal intervals. It is obvious there
would be a ^ain in the methods of approximate calculation if
there should be found to be no considerable addition to the
practical difficulties in using the formula when so modified.
1. We begin with remarking that, in the well-known
formula of Lagrange, viz.
[x-b){x- c)...(x'-k) [x — a)(x — b){x — c),..
^*"^« (a-J)(a-c)...(a-i) '^'"'^ ^* {k~^){k^b){k^^d)Z. '
we should have a result which is true, and, for practical pur-
poses, in some cases almost as simple, if in the coefficients of
w^, w„ ..., w^, we write (f>{x)j ^ (^)> •••) 4>{k) instead of
a;, a, ..., A; where (f) {x) is any function of x] provided always
that none of the quantities 4> (^)) ^ {^)j &c., becomes infinite
within the limits of interpolation.
This suggests an examination of interpolation formulae,
in which the data are the values of the function at successive
equal intervals of ^ {x).
2. We shall in the first instance suppose, that the suc-
eessive intervals of <f> {x) differ by unity, and that we know
the values of an unknown function f(x) at each of those
intervals. The question is then to find an interpolation
formulae for determining approximately the values of /(a;).
Let (f) {x) be put equal to z^ and suppose that x is accord-
ingly equal to -^ («). Then /(a?) =y^(«), and this, by
Taylor's theorem, may be expanded in powers of «, viz.
y^ (0) + ^ ^/^ (0) + ^ QV^ (0) +... .
02
36 UR. NITEN, SPECIAL METHODS OF INTERPOLATING.
If the values of/(a;) when « = 0, 1, 2, 3, ... be w^, w,, u,j w^, ...
we have
fi^ (0) = u„,
A/V^ (0) or Au^ = u^ - u„,
AX = w^ - 2u^ + w^,
A'w^ = M^- 3v^ + 3m, - u^y
&c.
The above expansion may then be written
M. + ^(x).log(l + A)«,+ ffil.{log(l + A)}'«,+...,
the general term being
in which {log(l + A)}" must be expanded in powers of A, and
any difference of w^, say A^^w^, is to be found according to
the above scheme.
3. We have here two remarks to make. In the first
place, the formula of the last article is estimated for dif-
ferences of unity in ^(cc). If, however, the successive
differences of <f> (x) be a constant other than u^ity, say c,
the formula will run thus :
«.+ ^^ log(l + A)«,+ i |*i^y {log(l. + A)r«.+....
In the next place it is often convenient that, while using the
differences above defined, viz. the differences of the values of
/{x) at the points corresponding to values of z equal to
0, c, 2c, &c., we should have the expansion, not in powers of
<t> (a?), but in powers of (f) {x) - nc where n is an integer, this
in fact amounting to changing the origin of z over n intervals
without however changing the origin of x. In those circum-
stances it is easy to see that the proper expansion is
or, what is the same thing,
(l + Ar«,+ {^-n|log(l + A)(l + A)- «.+...,
MR. NIVEN, SPECIAL METHODS OP INTERPOLATINa. 37
the general tenn being
» |*^)_„|'{log(l + A)}'(l+A)-«..
4. One of the immediate uses of interpolation formnlss is
to determine approximate values of definite integrals. To
show the application of the expansion of §2 to this use, it
will suffice to take onlj three given values of the function
f[x). Geometrically, this is the same as knowing the ordi-
nates w^, u^^ u^ corresponding to three points -4, B^ C on the
axis of X whose abscissae are a, h^ c. The formula becomes
f[x) = u, + (zl - ^) u^.4>[x) ^'\u^{4> [x)]\
Let us now multiply both sides of this equation by -p c£c,
and integrate between the values a and c. The correspond-
ing values of <f> {x) are and 2. We get
5. As regards $ (a;), we can determine it as a quadratic
function in a:, by Lagrange's formula, the values at Aj -B, C
being 0, 1, 2 ; thus
_ (a?-c)(a?-a) {x-a){x-b)
Now let /(a;) ^^^^F{x), and let the values of F{x) at the
points -4, -B, C be r„, r„ r*; then taking the above value
of ^ (oj), we have
f a— c a^b 1
f c-g _2_ _2_) _
38 MR. NIVEN, SPECIAL METHODS OF INTERPOLATIKG.
If we now substitute these values of u^^ w^, Uc in terms of
^o) ^*? ^tf ^° *he formula of the last article, viz.
we shall have an approximate value of the integral.
The same method of treatment may obviously be employed
whatever be the number of points -4, B^ (7, ..., the only diffi-
culty being the reduction of the equations connecting the
i^^s and v's.
6. One of the simplest cases of interpolation, other than
the common case with the equi-different abscissae, is when the
logarithms of the abscissae are equi-different, or when the
distances OA^ OB^ 00 ... are in geometrical progression.
Let there be (2w-f 1) of these distances, and let | be the
{n-\-iy\ then the general expression for/(a?) may be taken
(l + zl)"u„ + log|(l + Ariog(l + zl)u„+...
p(log|y(l + Ar{log(X + A)}X+-.
In this formula the diflferences are estimated for diflferences
of unity in the logarithm. If the latter diflferences are not
, . X . OB
unity, we must divide log - wherever it occurs by log yta •
In like manner we might establish a formula when the number
of distances OAj OB^ ... is even.
7. There is a peculiarity about the case of three points
which makes it worthy of special discussion. The three
points being -4, j9, (7, let AB=^ a and BG= )8, in which ^ is
greater than a. Then if we choose the origin 0, so that
OA = K'l OB^ f , 00=^^6% we shall have
c = log-,
and f =
+
Now making ase of the formula of § 6, we get
log-
log
a/3
,«,f
MR. NIVEN, SPECIAL METHODS OF INTERPOL ATI NG. 39
The peculiarity of the case of three given values is, that B
may be anywhere between A and (7, subject to the condition
fi>a^ and therefore we are able to select the origin. This is
of course in our power for other forms of (f) (x) besides logar,
for we have in general as an equation to determine the dis-
tance of the origin from B^
In the ordinary system of equidistant abscissae, this equation
makes the value of f infinite.
It is to be observed, that the form of the result given
above for y (a;) is true for the ordinary system of logarithms.
8. Pursuing the method of § 4, we might prove that the
value of any definite integral
/.
F{x) dxj
between the limits of interpolation, is approximately
logff-logg a%a + 4a/8?;^ + fi\
/3-a 3
9. The method of interpolating which has been developed
in the preceding articles was used in finding, from Mr.
Bashforth's tables for the resistance of shot, an approximate
representation of the law of the resistance in terms of a
simple power of the velocity. As this seems an interesting
application of the method we will now show how it may
be made use of for that purpose. We will employ an
interpolation in logarithms taking the values for five velocities.
The formula applicable will first be stated.
Let the five abscissae points be -4, B^ (7, -D, E so situated
that their distances from are in a geometrical ratio e'^
from 0. Also let 0(7= f. Then
f{x)^u, + -^ '^ ' - log I
6 \'^^ e.
(.o«|)'
2u^ - 8m, + 12Mj — 8m^ 4 2w,
(.».|)
(Hf)'.
40 MR. NIVEN, SPECIAL METHODS OF INTEEPOLATIXG.
The five numbers are taken from the tables for ogival-
headed shot. The law of retardation is formulated by
^ K ( ^ V
w 'Vioooy '
being the diameter of the shot and W its weight in lbs.,
and K^ is tabulated for every 10 feet of velocity between
certain limits.
Putting a = 1400 x 6"^ = 1146-1,
t =1400X6"!^ =1266-8,
c=UOO =1400,
eZ= 1400 xe^= 1547-2,
«=1400xe'^ =1710.
We then have, from the tables, as the values of K^^
ir.= 108-1,
jK;= 108-5,
i;=io4,
jr^ = 93-2,
ir=84-i.
Hence K, = 104-0 - 164 log j^ + 1416 (log j^j
+ 4400(logj^J+63000(log^J.
For values of v in the vicinity of 1400 the quantity
log-jTj: is small; for example, between the limits 1400^'^^
and 14006*, we may consider K^ as being approximately
represented by the first two terms. Now if K^ between
hose limits be of the form Av"^^ we have
= ^1400"»(l+nilog-^^ + &c.).
Hence ,n«-{J*.
PROF. LAMB, ON A THEOREM IN HYDRODYNAMICS. 41
This shows that the retardation of the shot, when it moves
with a velocity 1400 feet per second, varies approximately
as v'\
Nearly the same result is readily got at by the use of the
common interpolation formula, the values of K^ for intervals
of 50 feet difference of v being the given values.
NOTE ON A THEOREM IN HYDRODYNAMICS.
By Professor JET. Lamb, MA,
The well-known theorem of Helmfaoltz, that in a moving
liquid the vortex-lines always consist each of the same series
of particles, was deduced by him from certain equations
obtained by elimination of ^ (the pressure) from the ordinary
hydrodynamical eauations. Thomson afterwards gave a
proof, founded on tne theorem that the " circulation," that is,
with the usual notation, the value of
J{'udx + vdy + wdz) (1)^
taken round any circuit moving with the fluid, is constant
with regard to the time. The object of this note is to
exhibit the connection between these two modes of proof.
Let -4, -B, G be the areas of the projections on the coordi-
nate planes of any circuit moving with the fluid, and let —
St
denote a differentiation following the motion of the fluid.
We easily find
the integration being taken round the circuit. If the latter
be infinitely small, this becomes
- - = >4 (^ 4 ^^ ^B — -^r^^
Bt \di/ dz) dbc dx
-^O-^^-B^-C^ (2),
42 PBOF. LAMB, ON A THPfippw ,„
, u« A THEOEEM IN HTDEODYNAMICS.
where ^ = ^ j. ^ dw
-here f, ,, fare the instantaneous angular velocities
1 (^ dv\
w that his circulation-theorem ;„<.* e ,
the case in question, ' J""* "'^^"•ed to, becomes, for
that is, by (2), and similar equations,
f^-f^_„<^" ydu\
+ (^S+Bfj+coe «o
(3).
Since this must hoH fi,- • «. . ■'*
coefficients ofj^ 43 on thTlftIr A ^""^U '^^^K the
must separately vanish, whchJ.vl;*!^.''!" ''^ '^''' «q««t on
equations above referred to ^ ' ''"'' ^= <>» Helmholtz's
but^^^r^„jjr^^^^^ mereW to a hquid
Mr. Nanson has shown that this eJ^ '' * C""*^*'"" "fj'.
ongmal theorem follows from the !« I-'*"" ^^ Helmholtz's
pl«ed m (3) when € does nor^anSh ^ '''°' ''^•<^'» «" i™-
( 43 )
ON THE PRODUCT l\2\3\..n'.
By J. W, L. Glaisher,
§ 1. It is easy to show, by means of the formula
a),
that
(2),
where -4 is a constant independent of n.
The numerical value of A is best determined by giving a
particular value to n, and for this purpose the form in which
the equation is directly given by (1), viz.
1 logl + 2 log2 ...+ n logn = log -4
is rather preferable to (2).
Putting therefore n = 10, and multiplying throughout by
My =0.43429..., to 'convert the hyperbolic into £riggian
logarithms, we have
1 log,, 1 + 2 log,,2...+ 10 log,,10=log,,^ +50 + 5 + ^ij - 25M
■^ 172,000 " 50,400,000
10,080,000,000 950,400,000,000 7 '
The left-hand side =44-33401 00057 699..., and the series
—jr^-&c. = 0*00001 38691 457..., which when multiplied
by if becomes 0*00000 60232 9..., and it is thus found that
log,,^ = 0-10803 26967 2...,
whence A = 1-28242 7130,
in which value of A. as it is obtained by ten-figure logarithms,
the last figure may oe slightly in error.
44 MB. OLAiaHEH, ON THE PRUDUCT l'.2'.3'...w".
§ 2. The above method, viz. by giviog a particular value
to n in (2} or (3), would probably always afford the best
practical means of obtaining the numerical value of A to
any number of decimal places; but it will be Doticed, that
since the series in (3) ia semi -convergent, none of the equa-
tions written above contain a perfect analytical definition of
the constant A, as it is given in terms of series that ulti-
mately diverge. The object of what follows in §§ 3 and. 4 is
to obtain an expression for A in terms of a convergent series.
§3. When n is very large, we have from (2)
l'.2'.3'...n" = ^n*'''+^'^e-^ (4).
Multiply by 2'.2'.2»...2", =2*"*'^'',
and we have
2'. 4-. 6". . . (2«)- = Ai^*^ „!"«**>»* e-*"*,
which, when squared, becomes
2".4'.6'...(2«r = ^'2"^ »"■**** e-*"',
suppose n even, and write \n for n, and this is
2'. 4'. 6'. . .«• = ^'2"* n*"**^**^ e~*"\
Square again and divide by (4), which gives
2'.4\6'-w" _ .. -J i^
l'.3'.5'...(«-ir'-^^ " ^*'-
§4. Now let
/l+aiV /3+a:\' /"S + aiN' /n-l + a:\"''
whence
+-i ^ +i i -1 ^ +&«.■
+'-i T +J r. -i s>
+ -4r^+i
= ina!-J Qj? +i sjs* -\ »^*
J
MR. GLAISHER, ON THE PRODUCT 1^2^3^..«". 45
where ^=1 + J4 | + |...+ --1- ,
J , 1 1 1 «
and «r= 1 + oa + ^ 4 =; + &c.
Put aj= 1, and we have
or, as we may write it,
Multiply by 2.4.6...r?,= Trin*""^* e-***,
and this becomes
it which it only remains to determine the value of Q. This
is easily eflfected for, 7 denoting Euler's constant 0-57721...,
we know that
l-l-i+ J...+ -=7 + logw,
whence i + i + i •••+ o" ~ ^^^ "^ ^ ^^g^y
and l + 4+i...+ ^=7H-log(2n);
therefore, by subtraction,
l + i + i...+ 2^j=i7 + log2 + ilogn,
so that C = i7 + J log2 + i logw.
Substituting this value of Q in (6), that equation becomes
p-^!^;^4^, =2-M«*-i exp(- i7 + K- K + &CO.
Comparing this with (5), we have
^' = 2*7r* exp (- J7 + ^5, - 1^3 + &c.),
* exp (u) is written for e* when u is complicated.
46 MR. GLAISHER, ON THE PRODUCT l\2".3'...w".
and therefore
^ = 2^7r*expi(~i7+K-K + K-&c) (7),
which 18 the value of A expressed m terms of a convergent
numerical series.
§ 5. To obtain the numerical value of ^ from this series we
notice that the quantity in brackets subject to the exponential
sign is
which =-.j7 + (i-i + i-&c.)-hJ(^.-l)-i(^8-l) + &c-
whence
and, therefore, 3 log„^
= l!log,.2+iiog,.^-iif{i7+4-J(«,-i) + H».-i)-&c.}.
The series in brackets is found on calculation to be
= 0-5770109574 3...,
which, when multiplied by Jf,
= 0-25059 26748 1,
and if log2 + \ logTr = 057469 07649 8 \
therefore 3 log,^^ = 0-32409 80901 7,
and logio^ = 0-10803 26967 2,
agreeing with the value found in § 1 to the last figure.
§6. The series in (7) can readily be expressed as a
definite integral, for if
« , 1 1 1 1.
^^='^+2"' + F"^4''*"*^-'
then *r=^r(l-i),
80 that - i7 + i*» - \h + H ■" &®'
-j74(.-i)-f (1-^)^40 ,S).
Now logr (1 + 0?) = - 70? + \8jc* - JS'jX' + J/S.x* - &c.,
MB. GLAISHER, ON THE PRODUCT 1^2^3^..w'•. 47
whence
and X logr (1 + a?) die = - 7a;' + i S^o^ - ^S^x* + {S^x"" - &c.
80 that a; logr (1 4 ir) - r logF (1 + x) dx
•'0
= - lyx' + i S^x" - {Sy 4 &c.
Put a; = 1, and we have
-[ \ogr{l + x)dx = -iy + ^S^-\8^ + &c.',
•^
that is l-Jlog(27r) = -i7 + J5,-i)S,+ «S:c (9).
Put 0; = ^, and we have
i logr (f )-[* logr (1+x) da:
Jo
th^t is, log (^ V^r) - 2 I logF (1 + ar) cic
whence from (9) and (10)
1 + i logS-log-n-H-a I logr(l+a;jda?
-l74(.-|.)-f(l-i,)+&e.
from (8), thus
A"" = 2*7r"* exp {1 + 2 /,* logF (1 + a?) dir},
and therefore
A = 2^7r"* exp {^ + J/„i logF (1 + a:) dx].
( 48 )
ON THE TRIPLE THETA-FUNCTIONS.
By Professor Cayley,
As a specimen of mathematical notation, viz. of the
notation which appears to me the easiest to read and also
to print^ I give the definition and demonstration of the
fundamental properties of the triple theta-functions.
Definition. .9 ( CT, T, FT) = 2 exp 0,
where
= (^, J5, 0, F, G, H) (?, m, w)«-h2 (Z7, F, W) {I, m, w),
S denoting the sum in regard to all positive and negative
integer values from — oo to + oo (zero included) of Z, w, n
respectively.
5(?7, F, W) IS considered as a function of the argu-
ments (?7, F, TF), and it depends also on the parameters
[A, B, 0, F, G, H).
First property. 3 ( Z7, F, PT) = 0, for
tr=i{a:«+(^,S,G')(a,i8,7)},
W^}i[xni+[0,F, C)(a,/3,7)},
0?, y, «, a, )8, 7 being any positive or negative integer numbers,
such that ax -{■ j3y -\- fyz == odd number.
Demonstration, It is only necessary to show, that to each
term of S there corresponds a second term, such that the
indices of the two exponentials differ by an odd multiple
of TTt,
Taking 7, m, n as the integers which belong to the one
term, those belonging to the other term are
-(? + a), -(w + y3), -(n + 7),
(where observe that one at least of the numbers a, )8, 7 being
odd, this system of values is not in any case identical with
7, 7W, w). The two exponents then are
0, =(^, J5, 0, F, G, H) [I, m, w)« + 2 (ZJ, F, W) (Z, m, w),
apd 0', =±(^, j5, 0, F, G, H) (Z4 a, m + /3, n4 7)'
-2(^7, F, TF)(? + a,7n + /3,n + 7),
PROP. CAYLEY, ON THE TRIPLE THETA-FUNCTIONS. 49
viz. the value of 0' is
= (A, B, C, F, G, H) (I, m, «)• + {A, B, C, F, G, H) (a, /9, 7)"
+ 2 {A, B, C, F, G, H) {I, m, n) (a, /S, 7)
-2(Z7, F, W){l + (x, w + /3,M + 7),
and we then hare
0' - = 2 (^, B, G, F, G, H) {I, m, n) (a, y8, 7)
+ (A, B, C, F, G, H) (a, /3, 7)"
-2(Z7, V, TF)(2Z+a, 2»i + i8, 2n + 7).
Substituting herein for Z7, F, W their values, the last term is
= - {(2Z + a) a; + (2jn + iS) y + (2n + 7) z]
- 2 (^, B, G, F, G, H) {I, m, n) {a, /3, 7)
- {A, B, G, F, G, H) (a, /3, 7)',
and thence
e'-.0 = - {(2Z+a)a;+(27n + i8)y+(2ii + 7)«} th,
which proves the theorem.
As to the notation, remark that after (-4, B^ (7, F^ O^ H)
has been once written out in full, we may instead of
(-4, -B, (7, F^ Oy H)[ly 7n, w)'*, &c., write (-4, ...)(Z, tw, w)^, &c.,
and that we may use the like abbreviations
[A^ ...) (Z, Tw, n), to denote {A^ S, (?) (Z, m, w) respectively,
(fi,...)(?,7n,n), „ {H,B,F)[l,m,n)
((?,...) (Z, m, n), „ {O, F, G) [I, m, n) „
these are not only abbreviations, but they make the formulae
actually clearer, as bringing them into a smaller compass ; and
I accordingly use them in the demonstration which follows.
Second Property. If t/j, F„ W^ denote
U-\rxnri-^[A,H, G){a,fi,y),
V+yirt + {H,B,F)(a,^,y),
W+X7ri+{G,F, C)(a,/S,7),
respectively, where ar, y, 2, a, ^8, 7 are any positive or nega-
tive integers (zero values admissible), then
■9 {U„ F., TF.) = exp {- {A, B, G, F, G, H) (a, ^, 7)'}
xexp{-2(atr+^F4 7TF)}.^(Cr, F, TF),
or say = exp {- {A, ...) (a, ^, 7)'}
xexp(-2(a:7+^Ff7TF)j.3(C/, F, W).
VOL. VII. E
60 MR. MAETIN, ON THE CUBE ROOT Or 2.
Demonstration. Writing S ( U^, F„ TT,) = S.exp 0„ then
in the expression of 0, we may in place of Z, m, n write
Z— a, »n — /8, n — 7 ; we thus obtain
0. = (^, ...)(Z-a,«t-/3,«-7)'
+ {{I -a)\_U+ xiri+ {A, ...) (o, /9, 7)]
+ (m - /S) [ F + yTTt + [H, ...) (a, /3, 7)]
+ (n - 7) [Tr+ 2,r» + {G, ...) (a, /9, 7)]},
whicli 18
-2(^,...)(?,w,n)(a,^,7)
-2(aZ7+i9r+7TF)-2(ax+/3y+72r)7rt-2(^, ...)(«, i8,7)*
,. , . + (^,...)M,7r,
which IS
-(^, ...)(a,^, 7)»-2(atr+y3F+7?F)
■f 2 [(Z — a) a? + (tw — )8) y + (« — 7) z] iri.
Hence, rejecting the last line, which (as an even multiple of
TTt) leaves the exponential unaltered, we see that S{U^j V^ W^)
is = ^ ( Z7, F, W) into the factor
exp {- M, . . .) (a, ^, 7)'} . exp {- 2 (a 1^+ y3 F+ 7 TF)},
which is the theorem in question.^
In many cases a formula which belongs to an indefinite
number s of letters, is most easily intelligible when written
out for three letters, but it is sometimes convenient to speak
of the s letters ?, tw, ... w, or even the 8 letters Z, ... w, and
to write out the formulae accordingly.
COMPUTATION OF THE CUBE BOOT OF 2.
By Artemas Martin,
In vol. v., p. 172, Mr. Gray has given the cube root of
2 to 28 places, which he states was computed by Homer's
process, and in vol. Yi., p. 106, he has extended the root
to 32 places.
Having recently computed the value of the root to a
? greater number of places, by the method' of approximation
bund in Simpson's Algebra^ I submit it, with the work,
for publication.
MB. GRAT, ON THE CUBE BOOT OF 2. 51
Let -B = the true n^ root of a number N. and r = a near
Tit
approximate root, and put g= ^^ n > then (Simpson's
Algebra^ p. 169)
-^ = ^ + j(2g + 2n-l) + i(n-l)(2n-l)' ^^^7 ^^^^^
which he says (p. 165) " quintuples the number of figures at
every operation."
Takmg n = 3 we have for the cube root of JV,
P , r(2j4 3) ,
jB = r + /^ rx « 1 very nearly,
To compute the cube root of 2, take r = 1*25 = J, then
^^'^•^^ + 125^(255?.fr =* + ^*^»''
= flM3> =1-25992 10498 + ,
which is true to the last figure.
Now take r = feM, then ^ = fffm5ttlHlfii a^d,
after some reduction we get
= 1-25992 10498 94873 16476 72106 07278 22835 )
05702 51464 70150 +J'
which, if Simpson's remark Is true and I have made no
mistake, is correct to fifty places of decimals.
Erie, Pa., U.S.A.,
January 31, 1877.
VERIFICATION AND EXTENSION OF THE
VALUE OF THE CUBE ROOT OF 2.
By J^eter Gray, F.B.A.S,, &c.
I HAVE verified Mr. Martin's 50 decimal value of the
cube root of 2, by direct multiplication by means of the
arithmometer, and extended it to 56 decimals in the following
manner :
The cube root was divided into six groups, or periods of
eight figures, and in the work seven such periods, that is
56 decimals, were calculated. The work is given below, the
method of formation being as explained in the Messenger^
Vol. V. p. 172 (March, 1876).
£2
52 MR. GRAY, ON THE CUBE ROOT OF 2.
1.25992104 98948731 64767210 60727822 83505702 51464701 50
1.68740102 70346816
1 24667588 06820024
1 24667588 06820024
81601570 58109840
81601570 58109840
76512260 65117488
76512260 65117488
1 05210590 90977008
1 05210590 90977008
64841459 60720904
97908513 66510361 64841459 60720904
64086332 399105 10 62996052
64086332 399105 10 62996052
60089409 23293882
60089409 23293882
82627832 44164162
82627832 44164162
50923669
41947914 91184100 50923669
39331716 00316620
39331716 00316620
54084313
54084314
36878684
1.58740105 19681994 74751705 63927230 82603914 93327899 83290125
1.99999998 18130920
24797758 34975376
94181245 90537320
80543262 10591920
I 57071319 48556755 1 04074409 23495056
19475083 29849614 1 17585783 56909496
73965863 49836355 1 04938981
1 02811537 15957050 63255182 84845130
12747478 38616740 81735524 65933134
48414593 75593050 92346773
96399408 40701310 41403883 30128300
11952446 28237068 53500250
45395082 35436510
1 32557039 03578710 38821614
16435587 25729788
62421936
8169512040533605
10129279
79370053
1.99999999 99999999 99999999 99999999 99999999 99999999 96199719
Error 3800281
MR. ORAY, ON THE CUBE ROOT OP 2. 53
It Is only necessary to indicate very briefly the details of
the process.
In the first operation, that is in the formation of the
square, the first period of the root (In which the initial unit
is Included) is set on the machine, and multiplied by itself
and the succeeding periods in order, all the products after
the first being duplicated. The results are written down in
black ink.*
The second period is now set on and dealt with in the
same way; then the third, and finally the fourth; the colour
of the ink being changed for each new setting of the machine.
The initial product of each setting falls two periods to the
right of that of the preceding setting. The sum of the
results is of course 2*, which has to be multiplied by the
root.
Again setting on the first period of 2*, it is multiplied In
succession by all the periods of 2?. Setting on the second
period it is multiplied as before, beginning with the first
period of 2', and necessarily stopping at the penultimate.
And so with the rest, the colour of the ink being changed
for each set of results.
My machine is a ten-figure one, and in consequence I
am able to include the initial figure of the root In the first
period. This is the cause of sundry gaps which appear in
the work.
The result shows an error of 3800281 in the 51st and
following places : denoting it by e and the correction to be
applied to the root r by c, we have
c = /, = 798009.
or
giving for the last (the seventh) period of the corrected root
507 98009.
Availing ourselves of the previous work the verification of
the extended root is very easy, especially as the correction
extends only to one period.
The corrected seventh column is given below, and it will
be seen that only three products in all had to be revised.
These are marked by asterisks.
* The figures that are written in black ink are printed in modem type, those
that are written in red ink in antique type.
64 MR. ORAY, ON THE CUBE BOOT OF 2.
Conectod seTentli oolomn. Corrected sereiith oolumn.
(eorUinued),
60720904
60720904 66909496
64001480* 107472501*
64001480*
65933134
92346773
44164162 30128300
44 I 641 62 53600250
50923669 35436510
50923669 38821614
00316620 26729788
00316620 62421936
64084313 40533605
64084314 loi 29279
36878684 80636813*
85300981 99999999
Mr. Martinis fifty-place, value is thus completely verified.
It appears that the last figure, instead of 0, should conr
ventionally be 1.
The fifty-six value found above is
V2 = 1.25992 10498 94873 16476 72106 07278
22835 05702 51464 70150 79800 9
The value of the cube root of 4 also occurs as the result
of the first multiplication.
The value is
V4 = 1.58740 10519 68199 47475 17056 39272
30826 03914 93327 89985 30098 l.f
Londoni Uaroh 6, 1877.
t [In oonnezioii with this calculation of the cube root of 2 and of 4 to 56
decimal places I may mention two similar numerical calculations extending to
a great number of places, that are but little known. The first is the value of the
square root of 2, which was calculated by Mr. W. H. Colvill to 110 places, and
was yerified by actual multiplication by Mr. James Steel. The yalue is giyen
in De Morgan^s Budget of Paradoxet (1878) p. 293. The second is the yalue
of the real root (2.09465...) of the celebrated cubic equation a:*~2a;— 5 = 0, which,
at the instigation of De Morgan, was calculated by Mr. J. P. Hicks to 152 places,
as an example of the power of Homer's method. The result is giyen in yoL x.
p. 837 (part ii.) of the IVansactions of the Cambridge Philosophieal Society.
One hundred places had been preyiondy calculated by Mr. W. H. Johnston
{Mathematicianf t. iii. p. 289).— J. W. L. G.].
( 55 )
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, May 10th, 1877,— Lord Rayleigh, P.R.S., President^ m the chair.
Mr. Tucker communicated a short account of a paper by Dr. Hirst on the
correlation of two planes. In a former paper on the subject (Proceedings,
vol. v., p. 40) the nature and properties were described Jirst of an ordinary correla-
tion satisfying any eight given conditions; secondly of an exceptional correlation of
the first oraer, possessing either a singular point or a singular line in each plane,
and satisfying seven conditions ,■ and thirdly of an exceptional correlation of the
second order, having in each plane not only a singular point but also a singular
line passing through that point, and satisfying six conditions. Moreover, the two
following numerical relations were established between the (ir, \) exceptional
correlations of the first order, with singular points and singular lines respectivelv,
which satisfy any seven conditions, and the (/x, y) ordinary correlations which,
besides satisfying these same conditions, possess a given pair of conjugate points
or conjugate lines respectively (2v = /i + ir, 2/u = v + X). It was by means of
these relations that the numljer of ordinary correlations was determined which
satisfy any eight elementary conditions. Before they could be applied, however,
the exceptional correlations of the first order which satisfy any seven elementary
conditions had to be directly determined^ and this determination not un^requently
necessitated the consideration of the projective properties of curves of high order.
In the present paper the writer shows that the object just referred to can be
attained in a very much simpler manner by means of two general relations,
hitherto unobserved, connecting the number of exceptional correlations of the
second order which satisfy any six conditions with the numbers of exceptional
correlations of the first order, which, besides satisfying the six conditions in
question, possess a given pair either of conjugate points or conjugate lines. The
Secretary then read part of a paper by Prof. H. Lamb, of the University of
Adelaide, " On the free motion of a solid through an infinite mass of liquid."
Suppose that we have a solid body of any form immersed in an infinite mass of
perfect liquid, that motion is produced in this system from rest by the action of
any set of impulsive forces applied to the solid, and that the system is then left to
itself. The equations of motion of a body under these circumstances have been
investigated independently by Thomson and by Eirchhoff, and completely
integrated for certain special forms of the body. The object of the present
communication is, in the first place, to examine the various kinds of permanent or
steady motion of which the body is capable, without making any restrictions as to
its form or constitution; and, in the second, to show that when the initiating
impulses reduce to a couple only, the complete determination of the motion can be
made to depend upon equations identical in form with Euler's well-known
equations of motion of a perfectly free rigid body about its centre of inertia,
although the interpretation of the solution is naturally more complex. Free use
is made throughout the paper of the ideas acd the nomenclature of the theory of
screws, as developed and established by Dr. Ball.
Herr Weichold (Head Master of the Johanneum, Zittau, Saxony) sent a paper
(read in part by t^e Secretary) containing a solution of the irreducible case t.«., of
the problem to express the three roots of a complete equation of the third degree,
in the case of all these roots being realj directly in terms of its coefficients by
means of purely algebraical and really performable eperations, whose numb^
shall always be limited except in the case when all these roots are incom-
mensurable. Mr. H. Hart made three communications : (1) " On the Idnematic
paradox." Prof. Sylvester has described a system of Peaucellier's cells, the
poles of which all move in a straight line, but two of which^ not directly connected,
always ren^ained at a constant distance. Such a result is very easily obtained
hy means of the foUowing relations connecting six points, Af B, C, D, Ef F,
lying on a straight line (fig. 3). If :
AB,AC= a«
BC.BD = 4a^
FA, FE = 2a»
56 TRANSACTIONS OF SOCIETIES.
He then ipoke on the solotkm of the algdmkal equatuHi f{x) = bj Hnkwork,
oonsidenng three pomts, the prqpantioa of the eqnatiaa (pixt under the form
-( — T~T + M« = i^), the icpiCBC ii t it ion of the terms of this equation, and the
method of adding these terms. He shewed that for the aolotion of the cnbic
s^ + jM^ -f jo; + r = 0, treated imder the foam.
two ledprocatozB alone are reqmred. He then spoke on the prodnction of drcnlar
and rectilinear motion. The particular im)blem considered, he tiius enunciated
" to find if poesiUe the relation that must exist between the fourteai segments of
the ban puuxd as in (fig. 4) in order that the sy^sm may be capi^Te of free
motion. He showed that seven equations can be obtained connecting the fourteen
quantities only, so that any seven being given, the remaining seven can be
oetermined in terms of them. Mr. Hart &en proceeded to an i^lic^on to the
cases of five-bar motion laid before the Society at its April meeting. Mr. Kempe
stated that the cases submitted by Mr. Hart at the previous meeting had also
occupied some of his attention, and he proceeded to remark that he had
determined the positions that the lines GE, KM must have, and that the
determination of one involved the determination of the oth», as the {xmtion of
either turned upon tiie fact that the angles i^ A and H must be equal. Prof.
Cayl^ also made a few remarks on the sulqect.
Mr. J. W. L. Ghusher stated that he had had all the cases in which there
are more than 50 consecutive composite numbers looked out from Buickhardt's
and Dase^s tables, which extend from 1 to 3,000,000, and from 6,000,000 to
9,000,000. A list of the groups of composite numbers so found in the first three
millions was exhibited to the Society. The two most noteworthy stretches were
a group of 111 consecutive composite numbers between 370,261 and 370,373, and
of 113 between 492,113 and 492,227. These two yeiy long sets of numbers
without a prime occurring in the first half million were remar^ble, as he thought
they exceeded what one would a priori expect. The longest stretch noted in
the three millions occurred in the third million and consisted of 147 con-
secutive composite numbers, and the next largest occurred in the second million,
where there were two stretches, each of 131. The 147 consecutive composite
numbers were those between 2,010,733 and 2,010,881. and the stretches of 131
were from 1,367,201 to 1,357,333, and from 1,561,919 to 1,562,051. Questions were
put to tl e meeting for information by Profs. Cayley and Clifford.
Thursday, June, 14th, 1877. — Lord Rayleigh, F.R.S., President, in the chair.
Prof. Crofton, F.R.S., proved some geometrical ^eorems relating to mean values.
These theorems were chiefly interesting as examples of the employment of the
theory of probability to establish mathematical results ; they were of a kindred
nature with theorems given in the Phil. Trant., 1868, p. 185, and in Williamson's
'' Integral Calculus," second edition, p. 329. Mr. Merrifield made a few remarks
on the communication. Prof. ClifEord, F.R.S., read a paper on the canonical form
and dissection of a Riemann's surface. The object of the paper is to assist
students of the theory of complex functions by proving the chief propositions
about Riemann's surfaces in a concise and elementary manner. To this end
certain results of Puiseux's were assumed at the outset. Prof. Smith, in making
remarks on the paper, expressed his indebtedness to the author in having cleared up
a difficulty wluch presents itself in LUroth's paper on the subject. Prof. H. J. S.
Smith, F.R.S., gave a short account of a further communication upon Eisenstein's
theorem. Mr. Tucker communicated a paper by Mr. J. C. Malet entitled, " Proof
that every algebraic equation has a root." The Society's next meeting will be held
on the secpnd Thursday in Noyember*
B. TuoKBR, M.A., Hon, See,
( 57 )
MATHEMATICAL NOTES.
On a pair of Algebraical Equations.
Consider the equations
aj« + y+«» =1,
a^x 4 &,y + c^z — ajM = 0,
a^x -\-bj/ + c^z -zu=0^
whence we have
M =
a„ J„ c,
a„ J„ c.
«>» ^.. <'s
oj, y, z,
o„ 5„ c„ X
«,> ^,» c.» y
«s» ^> c„ «
M
y
u
x =
1. y.
«.
0, J.,
c„
a;
0, &.,
c..
y
0, J.,
c..
z
u
(1),
U '
A^, A^, A^ being the minors of a„ a,, a, in V- Thus
a;(^,-Cr)+y ^, +z A, =0, '
and similarly
SB A +y(A-f'") + «A =o»
a 0. +y C. +z{C,-U) = 0,
B^,G ... being the minors of J„ c, ... in V, whence we see
that uis a root of the equation
A-U, A, , A,
0.
0, , G.-U
= 0.
(2).
58 MATHEMATICAL NOTES.
Also » is a root of the equation
= 0,
(3).
The equations (2) and (3) are therefore such, that their roots
are connected by the curious relation (1), viz.
The direct verification is easy, for the equation (3) is
and, iiv virtue of the well-known values of the reciprocal
determinant and its minors, the equation (2) is
that is, on multiplying v and dividing by Z7",
(jy-(a.+j.+c.)(^y+(A+A+c',)^-i=o.
Paul Mansion.
X
On a Discontinuous Series.
We have identically
X = (aj- aj*-a:»)+(aj*-aj*-a^)...+ (x""*- a:'*-'- «;»-*)+ aj* ^^ ,
whence, for a? < 1,
aj=(a;-a;'-aj') + (a?-aj*-aJ*) + (aj'-aj*-aj') + &c.
Integrate the first equation between the limits and a;, x
being < 1, and
X* _ /a?* ^ a*\ /a?' a;' a;*\
2""U"3 4;"^U ""5 " 6>/-'
/a?* a^-' •aj'"\ r* «!-«* ,
\n 2/1 — 1 n/ j^ l—x '
When n = CO , the integral just written vanishes, and therefore
ix«= ( Jx'- K-ia;*) + (ix»- K- K) + (K- K- K) + &c.
Although both sides of this equation remain finite when a;= 1)
yet we may not give x this value ; for when aj=: 1, the series
changes its value abruptly from a value as near as we please
to \ to i — log2. This is evident, for we know that
i-[i-H3-i + *-i + f-J+&c.] = i-iog2.
MATHEMATICAL NOTES. 59
We may conclude from this, that
log2 = lim,,^ I a?" ^ dxj
which it is easy to verify directly ; and we also see that the
statement contained in most French treatises on the Integral
Calculus, that the integral of a convergent series, if itself
convergent, is a continuous function of the independent vari-
able with respect to which the integration is effected, even
when the series integrated ceases to be convergent for the
superior limit of integration, is not true. Paul Mansion.
On the determination of the Sign of any term of a Deter^
minant.
The following seems to me to be preferable to the usual
statement of the rule for determining the sign of any term of
a determinant. Denoting the disarranged subscripts by
a, i3 ..., suppose that the original place of the subscnpt a is
occupied by )8, that of ^ by 7, and so on until we come to \,
whose place is occupied by a. Gall a, )3, 7...\ a cycle, and
let all the subscripts be thus grouped into cycles ; let there
be m cycles, some of which of course may consist of single
subscripts. Then an interchange of two subscripts will in-
crease or decrease the number m by unity. For, first, let the
interchanged subscripts be e and p, belonging to the two
cycles
a...S8...\ and /L(...'n'p...o),
then the interchange converts these two cycles into the single
cycle 3. ^
(X. . . op. • .6)/L(. . .TTS. . . A.
Secondly, if the interchanged subscripts belong to the same
cycle, as e and p in
Ot. • • OS. • • TT/D . • . 0),
the interchange converts this into the two cycles
a...Sp,,,(o and 6...7r.
Thus, n denoting the order of the determinant, if we give to
each term the positive or negative sign according as n - m is
even or odd, an odd number of interchanges will always
change the sign of the term, and it is easy to see that n-^m
is the least number of interchanges that will convert the
given term into the principal term or diagonal.
W. WooLSEY Johnson.
Aaiiapolis Md., U.S.A.i
Dec. I, 1876.
60 MATHEMATICAL NOTES.
Summation of a Series.
If IP 4.2" + 3"+... n" be denoted by S^^p^ to find S^^^^^ In
terms of S^^p^ S^^ ^_j, &c., and so calculate generally the sums
of the powers of the natural numbers.
Let ABj BGy GD^ &c. represent 1", 2'', S**, &c. respectively,
draw BF^ CG^ DH^ &c. at right angles to AE and equal to
1, 2, 3, &c. respectively. Then the rectangles AF^ BG^
OH, &c. will represent l'^', 2*^^ ^'^\ &c.
Suppose DK is the w^^, produce EK to i, [KL = 1) and
complete the rectangles as in fig. 5.
Then AE=S^p^ jEL = n+l, and we have S^^^p^^-\- series
of rectangles gF^ hG^ kH^ &c. =[n-\-\) S^p.
In the series of rectangles gF^ hG^ &c./the r^ = S^ p.
Hence S^^p^, + S. ^,,, = (w + 1) ^„,,.
In the term S./S-^^, S^ p must be expressed in terms of r,
and then all values from' I to w given to r. The result will
give the relation between S^^p^^ and the sums of the lower
powers S^^p, S^^p^,, &c.
Ex. Letj> = l,
-»«.. + 2 (i»^ + ir) = (n+l)^...„
= Jn" 4 iw' + Jn.
K,s=(« + i)(K+K + i«)-4(i"'+l«),
which gives 8^^ , = i«* 4 i"' + i"' = {in (« +1)}*.
= (« + i) ih' + in' + in") - i (K + i«' + in),
which gives 8^^ ^ = ^n* + in* + in' - ^n.
;? = 4,
^„.. + S (i*-' + i*"* + i*-' - bV*-) = (« + !) -S,.*.
K,.={«+i)^„,*-i^„..+3V^.,.,
which gives S^^ ^ = Jn® + ^w' + ^i^n* - ^^w',
and so on for higher values of;?.
MATHEMATICAL NOTES. 61
„i)+i
To show that the two first terms In 8„ are + An",
Assume this to be true for (^), then
^«,..i + S (^ + K + &c.) = (n + 1) ^„,„
n^^'
If therefore the law be true for {j>) It is so for Q? + 1) .
J. M. Ceoker.
Solution of a Cubic Equation.
Substituting for a?, y -h w, we get
3/' + 3 (w +^) y' + 3 (n' + 2pn -^ q) y -]■ n"" + 3pn' + 3 jn + r = 0.
This may be put In the form
my' = [ay + 6)',
if {n + p) {n' + 3pn* + 3 jw + r) = (n" + 2;?n + q)%
which reduces to
{p^ - q)n^+ {pq - r) w + {q* —pr) = 0,
It will be found (most easily by making p = in the
usual way), that if the roots of the original equation are all
real, the roots of the quadratic for n are imaginary, and we
have to extract the cube root of a binomial of the form
a + ySi to determine h and m. Thus the method presents no
particular advantage over Cardan's. It is perhaps interesting
on account of the coefficients in the auxiliary equation for n.
R. W. Genese.
On the Theory of Envelopes.
The envelope of ^(a^7...\) = 0, a^Sy... being coordinates
and X a variable, is generally to be obtained by eliminating X
between this equation and ;^ = 0.
Let jB = be the result of the elimination. The logical
conclusion from the theory is that ^(\)=0 meets <^(\+c?\)=0
where it is met by jB = 0. If then ^ = passes through
62 MATHEMATICAL NOTES.
fixed points, 5 = will in general contain factors P=0,
Q = 0, &c. passing through the fixed points. These factors
therefore must not be regarded as forming part of the
envelope.
Ex, To find the envelope of the parabola circumscribing
a triangle ABO.
Taking CAy CB as axes, the equation to any one of the
parabolas is
aj (a? — a) + \*^ (y — &) + 2X.'cy = 0,
And the envelope seems to be
a:y = aj(a;-a)(y)(y-6),
i,e. aj = 0, y = 0, and - + f = l,
or -4B, -4(7, 5(7, which is obviously absurd.
The envelope is, we know, the line at infinity which may
perhaps be inferred from the fact that the envelope appears
m the form of the fourth degree, and reduces to three linear
factors only, the other being the line at infinity. This line
appears clearly if the problem be treated by the method of
Trilinear Coordinates.
E. W. Genese.
[The example, although simple, is an instructive one
Introducing «, /a for homogeneity, the equation is
X*y [y - hz) + 2\iiQGy + /i*a; (a; — cw?) = 0,
giving the envelope
cry [(aj- a2?) (y - fe) - ag^] = ;
that is, ayy {bx -Vay — abz) « = ;
viz. we have thus the four lines
aj = 0, y = 0, 5 + |-« = 0, z^O.
Writing these values successively in the equation of the curve,
we find respectively
/A*aj [x — az)=^ 0,
(Xy + /»«)'' = 0;
MATHEMATICAL NOTES. 63
VIZ. in each case the e<]^uation in \y /i has (as it should have)
two equal roots ; but in the first three cases the values are
constant; viz. we find \ = 0, a* = 0, JX — a/A = respectively ;
and the curves a; = 0, v = 0, - +f-» = Oarefor this reason
* ^ * a ft
.not proper envelopes.
It is to be remarked that writing in the equation of the
parabola these values X = 0, /* = 0, JX - a/A = successively,
we find respectively
x{z — az) = 0,
y(y-i«) = o,
{bx f ay) {]bx -{-ay^ abz) = ;
viz. in each case the parabola reduces itself to a pair of lines,
one of the given lines and a line parallel thereto through the
intersection of the other two lines ; the parabola thus becomes
a curve having a dp on the line at infinity.
In the fourth case j5=0, the equation in X, /* is (Xy+/ia:)*=0,
giving a variable value X^/A = -a;-7-y; hence « = 0, the line
at infinity is a proper envelope.
The true geometrical result is that the envelope consists
of the three points -4, J5, (7, and the line infinity ; a point qud
curve of the order and class 1 is not representaole by a
single equation in point-coordinates, and hence the peculiarity
in the form of the analytical result. A. Cayley.]
An Arithmetical Theorem.
In vol. XXVI. of the Mathematical Beprint of the
Educational Times Mr. Martin proposed tne following
problem (Question 5009, page 28. "Prove (I) that aU
powers of 12890625 termmate with 12890625, and (2) that
all powers of numbers terminating with 12890625 terminate
with 12890625." In the following note an attempt is made
to answer the question : How many numbers, of a specified
number of digits, exist in any given scale of notation, which
have the properties here predicated of 12890625 ?
The work is much simplified by observing that the pro-
perties in question belong to every number whose square
terminates in the number itself. Hence, if ^ be a number, in
the scale of r, consisting of n digits, we have to find the
number of solutions of
IP'-N^^Er'' (1).
64 MATHEMATICAL NOTES.
If -4, B be two co-factors of r" (1) may be replaced by
(2).
Here -4, B must be prime to each other, and N being less
than r" or AB^ we have
A<5, k<A^
and from (2) hA-kB=^l.
If we regard -4, B as known, and A', A' as particular solutions
of this last equation, the general solution (in integers) is
A = A' + 5f, & = k' + At^
t being any integer. It is clear that for one and only one
value of f^ we have
0<h<B^ 0<k<Aj
so that Ay B being determined there is one and only one
solution of (2) or (1).
It only remains to determine how many values A may
have. Let
r =a^.a^..,agy
where a^, a^...ag are powers of primes. Suppose or, = a*"
where a is a prime. Then Aj B being prime to each other,
A must be a multiple of a"* (or a,), or it cannot contain a
at all. There are thus 2' forms of -4, since either of the
q quantities a may be included in A or not. Of these 2'
values two are inadmissible, viz. when all the a's are included
in A (or -4 = ?•**), and when they are all excluded (or B=r')j
for these are inconsistent with the fact that N is less than r*.
Hence there are 2« — 2 values of N^ and no more, which
satisfy the conditions of Mr. Martin's theorem.
These numbers are given to ten digits each in the cases
r = 6, 10, 12 in my solution of question 5277 of the Educational
Times. They are
r = 6 , 3221350213, 3334205344,
r=10, 8212890625, 1787109376,
r=12, 21e6l63854, 9«)5«08369 , (« = 9 + l, e = 9 + 2).
The above note will apply to numbers such that the square
of each ends in its arithmetical complement, for here we
start with the equation
H. W. Lloyd Tanner.
If
MATHEMATICAL NOTES. 65
On a Glass of Definite Integrals*
u= I B coshxdx (1),
where iZ is a function of x and i, then, by integration by parts,
J*e«= bB sinJa3+ -j- cosbx — I -tt coahxdx.
Differentiating (1) with respect to tj
= / -7:3- cosoxax.
h
If, then, B be such that
d*B d^B ^ ,_»
^+i?^=« (2)'
then -^ — 6*w+ hU sinJaj+ -^ cosJa; = 0......(3).
Now from (2) we have
B = F[x + it) +/(a: - ««) j
and this value of B being substituted in (3), we have
of which the solution is
2Jw = u4c^ + Be-"" - c^/e-"<^ («) rf« + e'^'/e"^ [t) dt^
so that, except that the arbitrary constants have to be deter-
mined, we obtain the value of the definite integral
r {F[x -f it) +/ {x - it)] cos Jxdir,
J ^
expressed in terms of indefinite integrals with regard to t.
As an example, let
/(a: + i^)=~^,, F{x^it)^-^ ^.
''^ ' x + tt^ ^ ' x — it^
Of
then f{x 4 it) + F{x - it) = ^-^^ ,
and ^{/(a, + »V)4i^(.:-i^)} = -^,.
VOL. Vil. F
66 MATHEMATICAL NOTES.
Thus the quantity in brackets in (3) vanishes both when
x = and when a; = ao, so that, putting a = ao and ^8 = 0,
we have
and therefore the value of the definite integral
2t — — -5 dx
satisfies the differential equation, and is therefore of the form
Aef* + Be"^. If the limits be a and /8, so that the integral is
^^^ 7;« A f«c<>8Ja ^cosJS) - fsinJa sinJ/8)
Similarly, if
*^ ^ ' a? + tr ^ ' a; - tr
we find that the value of the integral
X cosbx
,
dx
satisfies the differential equation
d*U y^
~ 1 {f + <e)* {e+^Y I ^"{f + a' f + ^j'
The above process is a generalization of the method
employed by Mr. Glaisher for the evaluation of the integral
[Messenger y vol. I., p. 35, 1871), and was suggested by it.
Egbert kawson.
Hayant, 1876.
Theorem relating to the difference between the sums of the
even and uneven divisors of a Number,
If f{n) denote the sum of the uneven divisors of any
number n, and F{n) the sum of the even divisors of n (unity
and n being both included as divisors], then
/ W+/ (w- 1)+/ (n-3)+/ (n-6)+/ (w- 10) + &c.
= F{n) + F{n - 1) + F{n - 3) + i^(n - 6) + i^(n - 10) + &c.,
MATHEMATICAL NOTES. 67
where 1, 3, 6, 10... are the triangular numbers, and it Is
supposed that
/(n-w) = 0, F{n~'n)=n.
Examples.
I. ri=14,
8-f 14 + 12+ 1 + 1
= 16 + . + . +14 + 6.
IT. n = 15,
24+ 8+ 4 + 13 + 6+
= . + 16 + 24 + . + . + 15.
The theorem may also evidently be enunciated as follows :
If 0{n) denote the excess of the sum of the uneven
divisors over the sum of the even divisors of n, then
tf (n) + ^(n- 1) + ^ (n -^3) + ^ (n- 6) + ^ (n- 10) +&c. =0,
where it is supposed that
tf (w — w) = — w.
J. W. L. GliAlSHER.
On some Continued Fractions.
I. If in the two expansions
the series be converted into continued fractions, we find
log{a? + V(l+a?*)} _ X. 1.2a;^ 3.4ag' 5.63:*
V(l+aj*) "■ 1+ 3-2aj'+ 5-4ar"+ 7- 6a?* + &c.
1 1.2 3.4 5.6
" aj"^+ 3aj"*-2aj+ 5a;"^-4aj+ lx'^-Qo9+&o. ^
arc singg _ ob 1.2a;* 3>4a;* 5.6g*
V(l - X*) ■" r^ 2x* + 8- 4a;H5- 6jc*+ 7 - &c-
1_ 1.2 3.4 5.6
"■ x'^" 2ic+3a;"'- 4^+5j^ 6aJ+7a;''-&c. ^
f2
68 MATHEMATICAL NOTES.
and, putting a; = -^ , we have
2 1+V3 _ J_ _L J_ i^ 28
V3 ^^ V2 - 1+ 2+ 3+ 4+ 5 + &C. '
V2 . 1 + V3 _ 1 1.2 3.4 5.6 7.8
V3 ^^ "~V2 V2+ 2 V2 + 3 V2 + 4 V2 -f 5 V2 + &c. '
*'^" 1- 4- 7- 10- 13-&C. '
TT 1 1.2 3.4 5.6 7.8
2 V2 V2 - 4 V2 - 7 V2 - 10 V2 - 13 V2 - &c. *
where 1, 6, 15, 28, ... are the alternate triangular numbers.
Putting a; = i, we have
2 , 1 + V5 1 1.2 3.4 5.6 7.8
log
V5 ^ 2 2+ 5+ 8+ 11+ 14 + &c.'
TT JL^ 1^ 3.4 5.6 7.8 ^
3 V3 " 2 - 7 - 12 - 17 - 22 - &c. •
If we take a; = 1 in the first continued fraction, there
results the equation
log(l + V2) _ J_ Iji? 2L4 5^ 7.8
V2 "1+1+1+1+ 1 + &C.'
which may be compared with the continued fraction in the
formula for ^tt,
X - 1 - -i- ill ^ i:^ 4.5
^'^ 1+1+ 1+ 1+ 1+&C/
^j l+a? + a;' + a;'^ + a?'^ + a;'^ + a;" + & c.
l-aj-(c» + a« + a;'^-a;"-aj''' + &c:
■"1- 1 +a;- l+o;'- 1 + aj*- 1 + a;^ - &c. '
the n + 1^^ partial fraction being
a:--a;^-"
l+a;«*-''
this continued fraction is derived from the series in VI.,
p. 111., vol. V. (November, 1875).
J. W.' L. Glaisher.
1
1
♦ See p. 79.
MR. BAWSON, ON COGNATE EICCATIAN EQUATIONS. 69
A simple proof of a Theorem relating to the Potential.
The value of a function, which satisfies Laplace's equation
within a closed, surface, is determined by the values at the
surface. If two surface distributions be superposed, the value
at any internal point is the sum of those due to the two
surface distributions considered separately. By means of this
principle a very simple proof may be given of the known
theorem that the value of the potential at the centre of a
sphere is the mean of those distributed over the surface.
On account of the symmetry it is clear that the central
value would not be affected by any rotation of the sphere, to
which the surface values are supposed to be rigidly attached.
Thus, if we conceive the sphere to be turned into n
different positions taken at random and the resulting surface
distributions to be superposed, we obtain a new surface
distribution, whose mean value is n times greater than before,
which determines a central value which is also n times
greater than that due to the original distribution. When n
is made infinite, the surface distribution becomes constant, in
which case the central value is the same as the surface value,
from which it follows that in the original state of things the
central value was the mean of the surface values.
Eayleigh.
An Identity.
The following remarkable identity is given under a slightly
different form by Gauss, Werke^ t. iii., p. 424,
A. Cayley.
ON COGNATE EICCATIAN EQUATIONS.
By Robert Ratvson*
1. It is readily seen that the equation
^ + ^+5a;V-cx» = (1)
is transformed into
f^:^y + hary-Cx'-' = (2),
if buy =^ ex' (3).
70 MR. RAWSON, ON COGNATE RICCATIAN EQUATIONS.
After considerable hesitation the system (1), (2), and (3)
has been designated a cognate Riccatian, a term suggested
by Sir J. Cockle's papers " On Linear Differential Equations
of the Second Order " (Messenger^ vols. i. and ii.).
2. We also have
dy . a-a--2n-2 ^;. ^^^^^, ,^ .
^ + y-^f>^ y -ex =0 (4),
"^ "=% + — 5— *" (^)'
and both of these transformations are particular cases of the
following :
dy^ f g + g + 2&^a;^''"'M
caj*
where w = *= — h-^ay' (7).
hy ^
3. In (2) take a = 0, then it becomes
|-f + J^V-«'" = (8),
where huy^c (9).
Again, in (4) take a = 2a — 2n — 2, and it becomes
^-^-Ja;-*«»+«-«-y««caj'^-« = (10),
. "" cx^"^ 7i + l-a ^., ,,,,
where m= — f + j x ^ (11).
by h
It is easily seen from (8) and (10) that if ^ (a, w, m) be
the condition of solubility of (1), then ^ (— a, tw, w) and
^ (- o, wi + 2w + 2 — 2a, 2a — w — 2) are also conditions of
solubility of the same differential equation.
4. The following cognate Riccatlans are intereetlng, as
they give other conditions of solubility of equation (1) besides
those in the last article :
^ -. £2^ + 6x-"-»y*-. ca;"'*-"^* = (12),
cue 'X
where x^H=y+"-^^x^' (13);
^y _ ^y + hx'^-^y' - caj"^*"" = (14),
CIX X
»
MR, RAWSON, ON COGNATE RICCATIAN EQUATIONS. 71
^^'"•^ "" iy + Ca + mH-ljar"-' ^^^^^
From these it follows that ^ (- a, — w — 2, tw + 2n 4 2)
and ^ (— «, — TW — 2 — 2a, 2a + 2w + n 4- 2) are also conditions
of solubility of (1).
In particular cases it is possible that some of the above
conditions of solubility may be identical. The following
equations are a little more general than the above :
g-f "-"t^"-^% -i-5.-y-c.-=o...(i6),
, . »+l — a ^_, ,^^.
where u = Qcfy-] ^- — x \ (17);
^+ ^ y + Ja;"-*y''-caj"^- = 0....(18),
ex*
where w = t — —, — ; . ,. ,^ ,i (19).
*y + (« + w + 1) « ^
5. Particular Cases of Solvhility of the Equation (1).
In (2) put a 4 a = and m — a = n + a, then it becomes
^ +(V-c) »!'"*"' = (20),
which is easily integrated by the usual methods. It is seen,
therefore, that (1) is soluble when 2a + m - w = 0.
In (4) put a — a — 2w — 2 = and w — a = ri + a, then it
assumes the form (20). Equation (1) is, therefore, soluble
when 2a = w + 3w + 4.
Equation (1) is homogeneous, and, therefore, soluble when
7w + n = 2.
In (18) put a — a + 2m + 2 = and ?n — a = ri4a, then
it assumes the form (20); therefore (1) is soluble when
2a+3m-fn + 4 = 0.
6. Having regard to a series of transformations similar
to that employed in (4), it will conduce to simplicity to make
a = m — n, then (4) becomes
dj _^ a-{m^n^2) y^ J.y- c."= (21),
dx
X
- ex ~ n + 1 — a _- t ,^.
where w= -y — + r a? (22).
If i of these cognate Riccatians be taken in accordance
with (21) and (22), which are forms well adapted for such
I
72 MB. RAWdON, ON COGNATE RICX^ATIAN EQUATIONS.
a aeries of transformations, then a,. = a — t (wi + w + 2). ^ Apply
this equation to the conditions of solubility of (1), given in
Art. 5, and we obtain the following values of i which will
render (1) soluble :
._ 2a 4-711 -n _ 2a-7ii-3n— 4 _ 2a+3fii+w+4 _ wi+3n44--2a
* " 2(w+n+2) " 2 (wiH- n + 2) ~ 2(wi + n-|- 2) " 2 (wH-ri + 2) '
In these equations t must be a positive integer. Boole's
resplts are included in the above by suitable substitutions.
Had Boole assumed y = , h t instead of y = — I- t (see
Boole, Differential Equations^ p. 93), the resulting equation
would have been more simple for further transformations.
7. In (18) put a = m — w, then it becomes suitable for a
series of transformations
dy a-Hwi + n + 2 ,«, m r^ /^**\
^+ y-VbxY-cx =^ (23),
where u = , -. -r — ^i (24).
% + (« + ^ + 1) » ^ ^
If t of these cognate Biccatians be considered in accordance
with equations (23) and (24), we obtain o, = aH-i(wi + n + 2).
Apply this equation to the conditions of solubility given in
Art. 5, and we obtain the following values of % which will
make (1) soluble :
._ n — ^n — 2a __ 7?i + 3n + 4 — 2a
*"" 2 (w -f n+ 2) ■" 2 (w + 71 + 2)
_ ^ 3m_+n+4 + 2a _ tw -f 37i + 4 + 2a
" 2 (tw -I- 71+2) 2(7n + w+2y *
where i is a positive Integer.
8. Equation (1) becomes the ordinary Biccatrs equation
when a = and ti = 0. Substitute these values in the above
conditions, and there results
4t
7n = - -T--. — - .
2t±l
This condition renders soluble the equation
^+i«" = «»" (25),
as is well known.
Havant, 1876.
( 73 )
ON THE RESOLUTION OF THE PEODUCT OF
TWO SUMS OF EIGHT SQUARES INTO THE
SUM OF EIGHT SQUARES.
■
By J, J, ThomBon, Trinity College, Cambridge.
The product of two sums of eight squares seems to have
heen first resolved into the sum of eight squares by Mr. J, T.
Graves. Prof. Young, in a long paper on the subject in the
Irish Transactiona for 1848, works out the resolution, and
also gives a proof that the corresponding proposition does
not hold for sixteen squares, though Le JBesgue, in his
Theorie des NonibreSj says Genocchi has. proved that the
product of two sums, each of 2" squares, can be expressed
as the sum of 2" squares. Brioschi, in CrelU^ t. 52, deduces
the resolution for eight squares from some propositions about
determinants. The following proof, however, is shorter than
any of those mentioned above :
To prove
(\ «i«.
(I a a
la.".
i\ ''*"'
(I a. a
OL OL
+
+
a.a,
6 t
+.
+
«6«.
a7«B
+
» •
8 •
«.«7
«4«.
)'
(\ «i«.
+
+
a a
*
+
+
«.«.
«.«.
«.«.
«.«.
+
+
0>J^n
OA
+
% 7
4
6 8
4
«A lY
«.«4
«.«« lY
74
MB. THOMSON, ON SUMS OF EIQHT SQUARES.
We can see that this woald be true if, after expanding
the squares, the terms involving the products of the deter-
minants disappeared ; for, to take an example,
«.«.
«.«»
= « + «,V - 2a,a,o,a„
the first two terms occur on the left-hand side of the equation,
and the last term is cancelled by a term + ^ajx^ajx^ from the
first square on the right-hand side of the equation ; we thus
get all the terms we want from the squares of the deter-
minants, and if the sum of the products of the determinants,
two and two, vanishes, the equation will be true.
To prove that the sum does vanish we notice
+
«4«l
OL OL
= 0,
if we denote
a.a„
a.a.
by (1.2).
We may express this
(1.2) (3.4) + (1.3) (4.2) + (4.1) (3.2) =0 (I.).
Write, for convenience, the last seven terms on the
right-hand side again, usin^ this notation,
{(1.2) + (3.4) + (5.6) + (7.8)}«
+ {(1.3) + (4.2) + (5.7) + (8.6)}«
+ {(4.1) + (3.2) + (5.8) + (6.7)}«
+ {(1.5) + (6.2) + (7.3) + (4.8)}*
+ {(1.6) + (2.5) + (3.8) + (4.7)}*
+ {(1.7) + (8.2) +(3.5) + (6.4)}*
+ {(1.8) + (2.7) + (6.3) + (5.4)}*.
We can easily prove that the sum of the products vanish,
for suppose we have product (6.2) (7.3) from the fourth row,
we have (3.2) (6.7) from the third row, and (2.7) (6.3) from
the seventh row, and the sum of these three products vanishes
by equation (I.). There will be 42 products, two and two, but
in applying this process we take Into account 3 each time, so
that we have only to go through this process 14 times, and
we shall thus find that the sum of the products vanishes.
3
76 MR. GLAISHER, ON SERIES AND
7r» _ 1 , 1.3 / ^ 1 \ ^ 1.3.5 /I 1 \
i8""*2'^^2ll^ + 3"V + ^2X6 l^'^S"-^ W
. , 1.3.5.7 /, . 1 1 1\ p , ...
+ ^2X678 l^+3-» + 5^+7-»j-^*" (^")-
V3 2.3 ^ 3.4.5 V 27 "*" 4.5.6.7 V "^ 2^* "** 37
. 1.2.3.4 /, I 1 n « , ....
+ 5:6:7:8:9 V "^ 2^ -^ F- -^ rv + *' ^^^")-
TT* 2 , 2.4 /, 1\ , 2.4.6 /, 1 n ■ , . ,
48 =^ 3 +* 3:5 i^ ^ 2-^j -^ i 3X7 I' + 2- -^ 3-,J+&c...(xiv).
1.2 /, 1\ . 1.2.3 /I 1\ n
1X6 l^+ 2-0 ■*- 4X6:778 i^+?-^FO + *^-
(^v).
1944 2.3.4 "^ 3
§2. Of these (i) is well known, and is obtained at once
from the series for arc sina* by putting x=l] it is included
for the sake of completeness, as it somewhat resembles in
form several of the others, (ii) is obtained by putting x = 2'*
in tl^ie formula
arcsina5 2 ^ 2.4 . 2.4.6 , o , -x
V(r:^='^+ i'" -^3:5^ + 3X7 '^ +'^"-(^^')-
Differentiate thb equation, and we have
a; arc sina; , 1 2 , 2.4 ^ , 2.4.6 - p / .-x
in^r "*'l=^=^"^i*'^i:3^-^TX5^ +&c....(xyu),
which gives (iii) on taking x = 2"*. If in (xvi) we replace x
by ^a;, we find, after transforming the coeflScients as explained
in § 3, that
^ arcsin^aj 1 s. 1-2 5, 1-2.3 - « , ....
* V(4^^ ='"' 273 ^ + 3X5 '^ -^ 4X6:7 ^ +**'• -(^^"0'
which gives (iv) when aj= 1. Differentiating (xviii),
(x arcsiniaj 1) ^ 1, 1.2 4 1.2.3 - «
( (4 — a;)5 4 — a;j 2 3.4 4.5.6
(xix),
PRODUCTS FOR TT AND POWERS OF TT. 77
from which (v) is derived by putting a; = l. (vi) Is obtained
by subtracting (iv) from (v). (vii) and (viii) follow from the
equation
(arcsina;)* = a;' + J - a;*+ J^a;* + i Yf ^'' + ^^- ••• (^^)j
by putting x = 2"* and a? = 1. (ix) is obtained from
2(arcsin^aj)*==-x» + ^;^aj*+-^'^ga^^ (xxi),
and (x) from
2 [x (arc sin Ja;)' + 2 arc sin Ja: V(4 — a?*) — 2a?} — Ja;'
— a? + x-\ a: + o^c.fxxii).
;; ^ Q>| KCT^yl /;fi7Qa^ v»*v...yA-«.Ety,
2.3.4.5 3.4.5.6.7 4.5.6.7.8.9
which results from the integration of (xxi). The integration
of (xx) gives
2
x (arc sinaj)* + 2 arc sina; V(l — a?*) — 2aj — Ja?" = ^ q~R ^" "^ ^^'
(xxiii),
from which (xi) Is derived.
(xii) and (xiii) are obtained by equating the coefficients
of X in
. , x[x^^V) . a:(aj*-l«)(a:'-3') . , . ,
8in^7raj= x ^-^-i — - + -^ rr ^— &c. ...(xxiv),
• 1 1 /of a:(a:«-l«) ^ aj(aj»-l')(a:»-2') . 1 , ,
sinj7raj=Jv3jaj . -- + -^ '— &c.k..(xxv),
and (xiv) and (xv) by equating the coefficients of a?* in
, , a;" a;*(aj*-2») a;Va;' - 2«) (a:« - 4«) ^
COSiTTX^l- -J + ^^, ^ - -^ ^ ^ +&C.
(xxvi),
x^ a;«(aj'-l*) a* faj' - 1*) (aj' - 2') „
cosj7raj=l- -J + -^^^-j ^' - -^^ -^ i + &c.
(xxvii).
§3. The equations (iv) and (v) were given in vol. ii.,
p. 143, of the Messenger (January, 1873) ; they were there
derived from (xxv) and (xxvii) by equating the coefficients
78 MR. GL^ISHERy ON SERIES AND
of X and x*. There is an error in (iv) as it appears in vol. II.^
p. 141, which was corrected in vol. ii., p. 153.
The series for tt, tt*^ tt', &c., alluded to in vol II., p. 142,
are included in those given in § 1. They were obtained in
the first instance by equating coefficients in (xxiv)...(xxvii),
and in the other similar formulae which occur in vol. II.,
pp. 138-142 and 153-157; but the method of §2 is rather
preferable, as the formulae for arc sin a; and (arcsina;)'* are
better known than (xxiv)...(xxvii). The peculiar form of the
terms in (iv) and (ix) suggested the determination of the
values of the series in (v) and (x).
The coefficients in (xvi) and (xviii) appear to be very dis-
similar in form, although they only differ by powers of 2 ;
the transformation employed is
(2') 3.5 =3.4.5 , (2*) 3.5.7.9 =5.6.7.8.9,
(2") 3.5.7 = 4.5.6.7, (2*) 3.5.7.9.11 = 6.7.8.9.10.11, &c.
The proof is very simple, viz. 3.5.7. ..(2n+ 1)
2.3.4... (2n+l) (2n + l)I 1, ,, , ^x ,^ ,.
- 2.4.6:..2n ^ = ^-2^^-^(«+l)(" + 2)...(2n^■l).
We also have, on multiplying by 2,
(2') 3.5=4.5.6 ; 2* (3.5.7) = 5.6.7.8 ; (2*) 3.5.7.9 = 6.7.8.9.10, &c.
§ 4. The most curious of the series is perhaps (v) ; it may
be written
27r , 1 1 1 p
where G^^^ denotes the number of combinations of n things
r together, and also in the form
4 V^ , 1.1-2. 1.2.3 . 1.2.3.4 . « , ....
9-73 "* = 3 + 475 + 5-:6:7 ■*• 6^:8:9 ■*• *^- (^^^^^)-
In Connexion with (xix) it may be mentioned that the
series with reciprocal coefficients, viz.
1 ^1.2 ^ 1.2.3 ^ .*
is the development of (1 — 4aj')'*, and is convergent if a? < J.
524
PRODUCTS FOR TT AND POWERS OF TT. 79
§ 5. If we eqaate the coefficients of x^ in (xxvii), we have
TT^ _ 1.2 / 1 \ 1.2.3 / 1 1 1 \
24;880 " 3.4.5.6 U'2V "^ 4.5.6.7.8 U'^* ■*" 1'3» '^ 2'SV
1.2.3.4 /^ 1 1 1 1 1 1 \ g
■*■ 5.6.7.8.9.10 U''2' ■*■ ^3" "*" 1''4'* ^ 2'3' "*" 2V "^ 3VJ "^^^^
(xxix),
and similar results follow from equating coefficients in (xxiv),
&c. If we equate the coefficients of a?', we have series in
which the sums of the combinations of 1*', 2"', 3"*, ... three
together are involved, and so on.
§6. By converting (ii.), (iii.), (v.), and (Iv.) into continued
fractions, we find
2 _ 1.1 2.3 3.5 4.7 5.9
TT 4-7- 10- 13- 16- &c.'
w 2.1 3.3 4.5 5.7 6.9
7r + 2 5- 8- 11- 14- 17-&a'
47r-3 V3 _ L4 ^ 5^ 7.10
27r + 3 V3 ""8- 13- 18- 23-&C.'
TT 1 1.2 3.4 5.6 7.8
3 V3 2-7-12-17- 22 -&c.'
and with this last result may be compared the two equations*
1 2.3 4.5 6.7 8.9
1 =
1^
3- 7- 11 - 15- l9-&c.^
1.2 3.4 5.6 7.8
5 - 9 - 13 -. 17 - &c. •
§7. II. Products. If
"'-('-^)('-?)('-.M('-Y->v
^'•('-?)('-?)('-fO('-.T>-
♦ Proceedings of the London Mathematical Societi/f t, v. p. 83 (1874).
80 MR. GLAISHER, ON SERIES AND PRODUCTS FOR ir.
in which respectively all prime numbers, all composite
numbers, all uneven prime numbers, all uneven composite
numbers are involved, so that IT, = f jFJ,, then
n = ^ p=^
I I
SO that
n, TT* P, 7r»
n^ 72 ' F^ 256 •
§ 8. These results are readily proved, for
TT* - J. 1 I
2« + 3:e + ^.
~=l + :^ + ^jj + 7^+ &c.
as is evident from Wallis's formula, or by putting a; = 1 4 e
(e infinitesimal) in
C03i,nr=(l-a,')(l-J)(l-i;)...,
as is evident by putting a; = 1 + 6 in
SmTTOJ
=(i-a^(i-f;)(i-^:)....
Thus P P, = Jtt, and n^n, = J,
and these, in conjunction with
give n„ P„ n,, P, as^above.
( 81 )
ON AN ARITHMETICAL THEOREM OF
PROFESSOR SMITH'S.
By Professor Paul Mansion.
I. We may write the theorem* thus :
1, 1, 1, 1, 1, l,...l
1, 2, 1, 2, 1, 2, ...
1, 1, 3, 1, 1, 3, ...
1, 2, 1, 4, 1, 2, ...
1, 1, 1, 1, 5, 1, ...
X> JUy Oj i0J Lj \}j •«•
= ^(l)^(2)0(3)^(4)...^(i),
The determinant is formed according to the three following
laws: 1\ The principal diagonal consists of the series of
natural numbers. 2". The determinant is symmetrical.
3\ The m ~ 1 first elements of the rn^ line (or column) are
the same as the elements situated in the same columns (or
lines) on the parallel to the second diagonal which joins the
(m — 1)"* element of the first line and of the first column.
If i = a*i''...Z\ we suppose that
^(i) = (a--a-^)(J/'-J^>)...(P^-Z>^-^) = 7,-?, + Z3-Z, + &c.
II. Suppose that 1, 2, 3... represent, not the natural
numbers, but any algebraical symbols a;,, a;,, a;,... and that
<f} {L) = ?, - ?, 4 ?, - ?4 + &c. = xi^ - a:^ + a?/, - xi^ + &c.,
then Professor Smith's theorem Is still true.
III. In the determinant, when 1, 2, 3.,,L have an
arithmetical signification, we may write instead of them
S*(l), 2^(2), 2<^(3),..., 2^ (Z) = ^ (L.) + ^ (J.J + &c.,
or, putting for shortness (L) = <(> (Z),
2(1), 2(2), 2(3), ..., 2 (i) = (A) + (i,) + &c.
* Professor H. J. S. Smith's theorem was published in the Proceedings of the
London Mathematical Society, vol. vii., pp. 208—212 (On the Value of a Certain
Arithmetical Determinant) ; the theorem is also stated in vol. vi. p. 81 of the
Messenger (June, 1876).
VOL. VII. G
82
M. LUCAS, ON THE DEVELOPMENT
Thus
(I), (1) , (1) ,
(l),(l) + (2), (1) ,
(1), (1) ,(l) + (3),
(1)
=(1) (2) (3)...(i).
(1) ,(i,) + (A)+&c.
IV. Finally, instead of (1), (2), (3)... , we may write any
algebraical symbols ; which gives the solution of this question :
Express any product (1) (2) (B)...{L) as a determinant formed
according to the above laws, except that the terms of the
diagonal are not the series of natural numbers.
Example*
3/, i0*4/, iC • SC , SD
•Z/, iV , ^Xj X , X
•Z/, ^Xm X , oCt/, X
a;' =
X • X
Xm X . Su
X • X
We maj call sucb determinants Smithian determinants.
ON THE DEVELOPMENT OF (frTs)"
IN A SERIES.
By M. Jadouard Lucas.
MM. Laurent and Le Paige* have recently given the
development of t«^= f zA in a series of ascending powers
of z. This can also be performed in the following simple
manner :
Let B^ p be the coefficient of — r^ in the development
of M* ; we shall call B^ ^ the p^ BemouUian number of order a.
We have
wherein we are to replace powers of 5, and B^ by second
suffixes and B^ ^ by the n^^ BemouUian number, so that
* Laurent, Nouvelles Annales de MathematiqueSf 1876, t. xiv. p. 355.
Le Paige, Annaks de la Societe scientifique de BruxelleSf 1876.
ir^J
OF ( : -i 1 IN A SERIES. 83
"V^e have then, by definition,
5„,,=[5/+5."+...+ 5/-T,
on replacing [5,^*^]" by B^ ^, and, more generally,
When a is a negative integer, we obtain, on changing
the sign,
'-m--
whence, denoting by (7„ ,, the number of combinations of
a things taken p together,*
5 _. =
(-ir
X [a--- a,, (a- ir-+ a,,(a-2p...± (7.,^ J. '
But
so that ^ a « = • ,;, \ ^, — 7 r (1).
As a particular case,
(-1)'
By diflTerentiation,
-^'' J) -fl
2?-—^ = a (s + 1) M«- aw**^* ,
and, equating the coeflScients of —jr^ — p -^ ,
1.2.o.»»^JI^ — Ij
a^<.«.. = (*-i>) -B-,.-^ «P^a,,-. (2).
Thus, for a = 1,
Put ^ = -B>,
and we bare the symbolical formnla
1.2.3...5„^,,,=i>(i>-l)...(p-a)£^'(l-^(2-£)...(a-2J)
(3),
in which the powers of B are to be replaced by suflkes^
G2
1
84 M. LUCAS, ON THE SUCCESSIVE SUMMATIONS
Beplace p by p — 1, and we obtain
1.2.3. ..aJ?^^,^^
= (^-l)(p-2)...(;>-a-l)-B'^'(l-5)...(a-5)....(4).
Multiply (3) by a-p+1, (4) by p(a-f 1), and add,
taking account of the formula (2) ; we thus find
1.2.3...a(a-f 1)5,^,,
= (^«l)(p-.2)...(p-a-l)5^*-'(l-5)(2-5)...(a-5)
x[p(a-2> + l)5+p(a + l)(p-a-l)].
Simplifying this, we reproduce the formula (3), a being
changed into a+ 1. This formula, true for a = 1, 2... is thus
generally true : it expresses the BernouUian numbers of order
a as a linear function of a consecutive BernouUian numbers
of the first order.
Paris, September, 1877.
ON THE SUCCESSIVE SUMMATIONS OF
l"+2" + 3"...+»'".
By M, Mdouard Lueat.
Let ;S',^„(a;) = l" + 2" + 3"-t...+ a;'",
-»,.- W = S^ul (1) + ^p-..- (2) + ^,-..« (3) +•••+ S^,„ (x),
and suppose that
^ o _ a?(a?+l)(a?4 2)...(ar-h;?~l)
^0,0= A, /5„,- 1.2.3...P
We have the symbolical formula
in the development of which the m + 1 powers of B^ are to
be replaced by second suffixes and B^^^ by the rfi^ BernouUian
number, with its proper sign. DiflFerentiating the two sides
of the equation, we have, as a formula to calculate the
BernouUian numbers, the identity
or, more generally,
/ (a; + 1) =/(« + 1 + 5.) -f{x + 5.).
OP ^ + 2''* + 3^..+ a^ 85
To calculate 8^ ^, we form the table
1"* + 2**,
1*^ + 2"* + S"*,
I <n I ckM I o(M . I ^^m
and add the columns; the sum of the j?^^ column is
{x-p + i)p'' or {x + 1) p" - p'^\
Thus S^„={x + l)8,^„-8,,^, (2) ;
or, symbolically expressed,
For example S^, = ^ , 1\ -,
^ *'* 1.2.3 *
^ _ a; {a; +1)^0? + 2) _ a?(a;+ 1) (a; + 2) (a;'+ 6a? + 3)
«»8 12 * «»» 60
In general
5,"(a; + l^;8,)(a:4 2-5.)...(a:+y~;g,) ' , .
^^^'-"~ 1.2.3...;> ^"^^^
In fact, changing x into a; + 1 the first side of the
formula (3) is increased by
S,^^{x+l)j
and the second by
^(aj + 2-i8;)(aj+3-/S,)...(aj + p-5J,
that is, by the second side of (3), when x is replaced by a? + 1
and » by p- 1.
The formula (1) gives by summation
and generally, after ^ summations.
»»H-1
'*'•»" m + 1 .
Changing^ into p + i, we deduce
_ (8, + B, + BJ^ - S," (B, + Br* iSr,,,Br^
(«»+l)(«i + 2) "" tn + 1
*-l,m
86 MB. DRACH, CUBE ROOTS OF PRIMES TO 31 PLACES.
and putting
we have
'^'"* (m+l)(w + ?) m+1 *
In general we should find in the same manner
'^'»" {m + l){m + 2)...(m + q) (m+ l)p...(m + 2- 1)
(m + l)...(w + j'-2) '" w + 1 ^ '*
Putting p = 0, we obtain the development of jSg^^ as a
function 8^ or of x.
Paris, September^ IS77.
CUBE BOOTS OF PBTMES TO 31 PLACES.
By 8. M. Drach, F.R.A,8.
The 56-place values of the cube roots of 2 and 4 given
on p. 54, have reminded me that twelve years ago I cal-
culated the values of the cube roots of the primes from 2 to
127 to 33 places. The extraction of cube roots to a number
of decimal places is so troublesome that it seems desirable
to publish these values, which are given in Table I. They
were obtained by the usual process of extracting cube roots,
and were verified by actual multiplication, the multiplication
being contracted throughout to 33 places.
The quantity 8, when preceded by 31 ciphers, is the
amount by which the cube of the quantity in the second
column differed from the first column, that is to say, for
example, by cubing the quantity
1-25992 10498 94873 16476 72106 07278 399,
retaining 83 places throughout the process, I obtained as the
cube
2 - -00000 00000 00000 00000 00000 00000 003,
and in general, cube of number in second column = first
column, + e preceded by 31 ciphers.
The product of the root oy itself gave me the root of
the square of the number, and Ta,ble II. contains the cube
roots of the squares of the primes from 2* to 127* found in
this manner.
L_
MR. DRACH, CUBE ROOTS OF PRIMES TO 31 PLACES. 87
All the cube roots ought to be accurate to 31 places
at least.
I may mention that they occur in my lai^e MS. table of
Binary Squares, a copy of* which is in the library of the
Royal Society, and another in that of the French Institute.
Table I.
N VN 8
2 1-259921049894873164767210607278 399 -03
3 1-442249570307408382321638310780108 +00
5 1-7099759466 766969893531088 72543 830 -02
7 1-9129311827 72389101199116839548 756 +03
11 2-22398 00905 69315 52116 53633 76722 157 - 05
13 2-35133 46877 20757 48950 00163 39956 921 - 00
17 2-5712815906582353554531872 08741445 -09
19 2-6684016487 219448673396273 71970 829 -00
23 2-84386 69798 51565 47769 54394 00958 652 - 06
29 3072316825685847 293312638036360183 -08
31 3-14138 06523 91393 00449 30758 96462 750 4 17
37 3-33222 18516 45953 26009 54505 05135 051 - 04
41 3-44821 72403 82780 38410 86376 34932 233 - 14
43 3-50339 80603 86724 17061 43337 58189 130 - 19
47 3-60882 60801 38694 68925 251 72 93399 702 + 00
53 3-75628 57542 21072 00661 21096 32059 320 + 09
59 3-89299 64158 73260 54646 14847 57149 833 - 01
61 3-93649 71831 02173 19582 88513 73032 163 - 16
67 4-061548100445679789082061585799 224 -02
71 4-14081 77494 22853 25000 45188 09325 572 + 05
73 4-17933 91963 81231 89205 63766 71392 658 + 09
79 4-29084 04270 2620711162 8314233454271 4 09
83 4-36207 06714 54837 56471 39794 76679 005 4 06
89 4-46474 50955 84537 63343 39684 80965 127 - 09
97 4-59470 08922 07039 80609 42964 64422 309 - 02
101 4-65700950780383563042 90105 64845024 4 28
103 4-68754 81476 53597 85820 73434 67619 767 4 20
107 4-74745 93985 23400 36029 37741 28935 120 - 18
109 4-77685 61810 35016 96494 37334 65132 732 4 22
113 4 83458 81271 11639 19899 42141 29566 902 - 05
127 5-026525695313479181137406871623742 -31
88 MR. DRACH, CUBE ROOTS OF PRIMES TO 31 PLACES.
Table II.
N IP ^2r
2 4 1-5874010519 68199
3 9 2 080083823051904
6 25 2-92401 77382 12866
7 49 3-659305710022971
11 121 4-94608 74432 48700
13 169 5-52877 48136 78872
17 289 6-611489018457945
19 361 7-12036 73589 01993
23 529 8-08757 93990 90064
29 841 9-43913 06773 92360
31 961 9-86827 24032 18973
37 1369 11-103702468586785
41 1681 11-89020 21368 72692
43 1849 12-27379 79695 21461
47 2209 1302362 56766 89216
53 , 2809 1410968 26673 64167
59 3481 15-15542 10940 02052
61 3721 15-496010072571344
67 4489 16-49617 29722 33909
71 5041 17-146371633935343
73 5329 17-46687 6118408521
79 6241 18-411311570202443
83 6889 19-02766 05427 66457
89 7921 19-933948768546182
97 9409 21-11127 6288848167
101 10201 21-68773 75557 75323
103 10609 21-97310 76365 70676
107 11449 22-538370740628166
109 11881 22-8183549742 92446
113 12769 23-373242358808827
127 16129 2526596 0565646655
47475 17056 39272 087
11453 00568 24357 885
0655067873 60137 984
517238073310119 419
86832 36036 57530 282
14147 23447 73085 347
00316 10465 36489 311
65206 96105 01623 781
32667 40060 25366 363
98272 06416 58182 980
92743 85838 62197 395
337446748652528462
61765 69351 98292 247
01832 02873 77401 952
423483196444970 526
77413 93952 61688 715
57923 35861 72329 105
48412 72009 22079 464
80130 79264 30167 879
48686 08965 02758 734
194618245841194 894
397100031834230104
44342 27059 68069 770
08877 61218 08447 890
62752 48074 83694 822
39559 26330 98920 636
4676931933 87120413
32306 04759 95760 252
77186 38932 28105 547
22094 28992 55751 637
34276 77391 28779 193
23, Upper Bamsbury Street, London,
September 3, 1877.
( 89 )
ON CERTAIN PARTIAL DIFFERENTIAL
EQUATIONS OF THE SECOND ORDER
WHICH HAVE A GENERAL FIRST
INTEGRAL.
By H. W. Lloyd Tanner, Jf.-4.
1. Adopting the usual notation for partial differential
coefficients, and representing by F, T, functions of a;, y, z^p^ j,
it is proposed to find under what conditions the equation
5+r^+F=0 (1),
has a first integral of the form
/(i>j ?> «) a?, y) = <^ {x) (2),
where is arbitrary.
This problem was discussed in a former paper,* but the
results there obtained were incomplete ; and, as the question
is not without interest, I venture to submit the present paper,
in which it is treated somewhat differently.
2. If we differentiate (2) with respect to y, ^(o?) is
eliminated, and we obtain the equation
dp dq dz- dy *
which should be the same as (1). On comparing this with (1),
we get
I"'*! («»«).
^=A*| (3'*).
If from this system we eliminate /a, /, we shall get the
relations between T, V which we seek. The only difficulty
in the way of performing this elimination arises from (3, c),
* Messenger^ toI v. p. 133.
90 MB. TANNER, ON CERTAIN PARTIAL DIFFERENTIAL
and this difficulty would be overcome if we could separate
(3, c) into two equations
-"I •
2 being some fiinction of F, T.
Now supposing this separation ta be accomplished, the
expression
dp-\- 2i?2+ScZi&+(F-jS)rfy, =^fidf (4),
becomes an exact differential on being divided by /a. To
ensure this three conditions are necessary and sufficient
(Boole, Differential Equations^ Chap. Xll. Art. 9), viz. these
conditions are
|(r-jS)-f-r|(F-22)+(F-j2)f.O
|_|(F-5S)-(F-j2)| + 2|(F-j2) = 0)...(5).
dz dq dp dp
The equations (5) constitute a complete solution of our
problem. For some applications, however, it is convenient
to transform them slightly. From the first of (5) subtract
the third multiplied by q. Some easy reductions then give
dV fdT\ ^^dT ^dV
2=^-U)+^^-^^ («)'
which fixes the form of 2. It is not difficult to verify by
means of (3) that the equation
2-M^
is identically satisfied when 2 is given by (6).
The last two equations of (5) may be written
/rf2\ dV dV „d^
dz 1^^^^^-^-^ ^^^'
EQUATIONS OP THE SECOND ORDER, &C. 91
The system (6), (7), (8) is obviously equivalent to the
system (5).
3. It may be well to illustrate the above results by
applying them to particular cases of a more or less simple
cnaracter.
Case L 2 = 0. Here (6), (7), (8) become
dg dy dp dp ^
dV ^ dT ^
^ = «' ^ = ^'
and^the first integral is
4>[x)^^-{dp-\-Tdq-^Vdy).
Case IL V— jS = 0. The system (5) reduce to
dy" ^ dy" '*
dT d2 d2 ^dT^^
and (4) becomes
dp+Tdq+ Idz^^fidf,
y
or djp +Tdq + — c& = fidf.
Case IIL F= 0. In this case (6) becomes
-(f)-
Then (7), (8) take the forms
<rr , „ <?'r . , d^r ^
dz dq \dy) dp \dy ) \dy ) dp'' '
while (4) becomes
dp+Tdq-{^) i^dz-qdy)=,^f.
92 PROF. CAYLEY, INTEGRATOR FOR THE CALCULATION
Case IV. T^ 0« We have In this case to deal with a
transformation of the yerj important equation
discussed by Ampfere. ^^
Here (6) becomes 2 = -3- ,
Hence V is linear in qj and (1) may be Written
8 + Qq + Z=0 (9),
Q^ Zj being functions of a?, y, z^ p. Taking this form, we
have
The equations (5) now reduced to one only, viz.
dO dZ „dQ ^dZ ^
ay dz dp dp '
and the first integral is found from the equation
dp + Qdz + Zdy = y^df.
These results agree with those given by Ampere.
Sepiemb4r, 1877.
SUGGESTION OF A MECHANICAL INTEGRATOR
FOR THE CALCULATION OF nXdx-^Ydy)
ALONG AN ARBITRARY PATH.*
By Professor Cayhy,
I CONSIDER an integral J[Xdx + Ydy\ where X^ Y are
each of them a given function of the variables {xj v) ;
Xc2b + lWy is thus not in general an exact differential; but
* Read at the Biitish Associatioii Heetixig at Plymouth, August 20, 1S77.
OP j{Xdx -VYdy) ALONG AN ARBITRARY PATH. 93
assuming a relation between (aj, y), that is a path of the
integral, there is in eflFect one variable only, and the integral
becomes calculable. I wish to show how for any given
values of the functions X, Y, but for an arbitrary path, it is
possible to construct a mechanism for the calculation of the
integral : viz. a mechanism, such that a point D thereof
being moved in a plane along a path chosen at pleasure,
the corresponding value of the integral shall be exhibited
on a dial.
The mechanism (for convenience I speak of it as actually
existing) consists of a square block or inverted box, the
upper horizontal face whereof is taken as the plane of xy^ the
equations of its edges being y = 0, y = l, a; = 0, x=\ respec-
tively. In the wall faces represented by these equations,
we have the endless bands -4, A\ By B respectively ; and in
the plane of xy^ a driving point i), the coordinates of which
are (a?, y), and a regulating point ^, mechanically connected
with 2>, in suchwise that the coordinates of R are always the
given functions X, Y of the coordinates of D ;* the nature
of the mechanical connexion will of course depend upon the
particular functions X, Y,
This being so, D drives the bands A and B in such
manner that to the given motions dx^ dy of D corresponds
a motion dx of the band A^ and a motion dy of the band B ;
A drives A' with a velocity-ratio depending on the position
of the regulator R in suchwise that the coordinates of R
being X, Y^ then to the motion dx of A corresponds a motion
Xdx of A' ; and, similarly, B drives B' with a velocity-ratio
depending on the position of if, in suchwise that to the
motion dy of B corresponds a motion Ydy of B\ Hence, to
the motions dx^ dy of the driver D^ there correspond the
motions Xdx and Ydy of the bands A' and B' respectively ;
the band A' drives a hand or index, and the band B' drives
in the contrary sense a graduated dial, the hand and dial
rotating independently of each other about a common centre ;
the increased reading of the hand on the dial is thus
= Xdx + Ydy ; and supposing the original reading to be zero,
and the driver D to be moved from its original position along-
an arbitrary path to any other position whatever, the reading
on the dial will be the corresponding value of the integral
j[Xdx-^Ydy).
It is obvious that we might, by means of a combination
* It might be convenient to have as the coordinates of R^ not X, 7 but £, i|,
determinate functions of X, Y respectively.
94 PROF. CAYLEY, INTEGRATOR FOR THE CALCULATION
of two such mechanisms, calculate the yalue of an integral
Sf{u) du along an arbitrary path of the complex yaiiable u,
= a?-|-ty; in fact, writing /(j? + ty) = PH-tQ, the diflFerential
is {P'^tQ)(dx + idi/\ ^Fdx-Qdi/ + i{Qdx-\'Pdy); and we
thus require the calculation of the two integrals J{Pdx — Qdy)
siiif[Qdx-\'Pdi/)j each of which is an integral of the above
form. Taking for the path a closed curve, it would be very
curious to see the machine giving a value zero or a value
different from zero, according as the path included or did not
include within it a critical point ; it seems to me that this
discontinuity would really exhibit itself without the necessity
of any change in the setting of the machine.
The ordinary modes of establishing a continuously-variable
velocity-ratio between two parts of a machine depend upon
friction ; and in particular this is the case in Prof. James
Thomson's mechanical integrator — there is thus of course
a limitation of the driving power. It seems to me that a
variable velocity-4*atio, the variation of which is practically
although not strictly continuous, might be established by
means of toothed wheels (and so with unlimited driving
power) in the following manner.
Consider a revolving wheel A^ which by means of a link.
JBCj pivoted to a point B of the wheel A and a point C of
a toothed wheel or arc J9, communicates a reciprocating
motion to Z> ; the extent of this reciprocating motion depending
on the distance of B from the centre of Aj which distance, or
say the half-throw, is assumed to be variable. Here during
a halfnievolution of Aj D moves in one direction, say up-
wards ; and during the other half revolution of A^ D moves
in the other direction, say downwards ; the extent of these
equal and opposite motions varying with the throw. Suppose
then that D works a pinion E^ the centre of which is not
abs<Jutely fixed but is so connected with A that during the
first half revolution of A (or while D is moving upwards),
^ is in gear with i>, and during the second half revolution of
^, or while D is moving downwards, E is out of gear with
D; the continuous rotation of A will communicate an in-
termittent rotation to E^ in such manner nevertheless, that to
each entire revolution of A or rotation through the angle 2ir
there will (the throw remaining cimstant) correspond a
rotation of E through the angle n .2x, where the coefficient n
depends upon the tlurow.* And evidently if ^ be driven by
* If inateadef ihewhedorMc2>wit^aredpiocatixi|rci>ci>^
a dooUe rack D with a reciprocating rectilinear motion, such that the wheel £ ia
OF JlXdx + Yd;/) ALONG AN ARBITRARY PATH. 95
a wheel A\ the angular velocity of which is - times that of
A*
-4, then to a rotation of A' through each angle — , there
will correspond an entire revolution of Ay and therefore, as
before, a rotation of J5? through the determinate angle w.27r;
hence, X being suflSciently large to each increment of rotation
of A'y there corresponds in E an increment of rotation which
is n\ times the first-mentioned increment; viz. £ moves
(intermittently and possibly also with some "loss of time"
on E coming successively in gear and out of gear with D,
or in beats as explained) with an angular velocity which is
= n\ times the angular velocity of A'. And thus the throw,
and therefore n being variable, the velocity-ratio n\ is also
variable.
We may imagine the wheel A as carrying upon it a
piece L sliding between guides, which piece L carries the
pivot Bj of the link -B(7, and works by a rack on a toothed
wheel a concentric with Aj but capable of rotating inde-
pendently thereof. Then if a rotates along with Ay as if
forming one piece therewith, it will act as a clamp upon 2/,
keeping the distance of B from the centre of Ay that is
the half-throw, constant; whereas, if a has given to it an
angular velocity diflFerent from that of Ay the effect will be
to vary the distance in question; that is to vary the half-
throw, and consequently the velocity-ratio of A and E. And,
in some such manner, substituting for A and E the bands
A and A' of the foregoing description, it might be possible
to establish between these bands the required variable
velocity-ratio.
placed between the two racks, and is in gear on the one side with one of them
when the rack is moving upwards, and on the other side with the other of them
when the rack is moving downwards ; then the continuous circular motion of A
will communicate to j^ a continuous circular motion, not of course uniform, but
such that to each entire revolution of A or rotation tlnrou^h the angle 2'7r, there
will correspond a rotation of JE through an angle n . 2'7r as before. This is in fact a
mechanic^ arrangement made use of in a mangle, the double rack being there t^ft
follower instead of the driver.
( 96 )
TRANSACTIONS OF SOCIETIES.
The Meeting of the British Association at Plymouth,
The forty-seventh meeting of the British Association for the Advancement o€
Science was opened at Plymouth on Wednraday, August 15, 1877, under the
Presidency of Professor Allen Thomson. The Sections assembled on Thursday
morning, August 16, the following being the list of officers of Section A (Mathe-
matical and Physical Science) :
President.— ?roieaiaor G. C. Foster, F.R.S.
Vice-Presidents.— ^roieefsor J. 0. Adams, P.R.S. ; Professor W. G. Adams^
F.R.S. ; Professor Cayley, F.R.S.; Professor S. flaughton, F.R.Sv; Professor
Bartholomew Price, F.R.S. ; Lord Rayleigh, F.R.S. ; Sir W. Thomson, F.R.S.
Secretaries. — Professor Barrett ; «f. T. Bottomley ; J. W. L. Glaisher, F Jl.S. ;
F. G. Landon.
Among the mathematicians present at the meeting, besides those mentioned
among the officers^ were Professor R. S. Ball, F.R.S. ; Professor D. Bierens de
Haan : Professor J. D. Everett ; Mr. R. B. Hay ward, F.R.S. ; Mr. W. M, Hicks ;
Mr. H. M. Jeffery; Mr. T. P. Kirkman, F.R.8w; Professor C. J. lAmbert; Mr.
C. W. Merrifield, P.R.S. ; and Professor R. K. Miller.
In his inaugural address ProfiBssor Carey Foster chiefly treated of the relations
between Mathematics and Physics, consiaered from a physical point of view.
The Section did not meet on the Saturday, but on Monday, August 20, a depart-
ment of mathematics sat under the Presidency of Professor Ca^ey, when the
following papers were read :
J. W. L. Glaisher. — Report of the Committee on Mathematical Tables.
Prof. D. B. de Haan. —On the Variation of the Modulus in Elliptic Integrals.
Prof. J. 0. Adams. — On the Calculation of BemonUi's Numb^ «p to B„ by
means of Staudt's Theorem.
Prof. J. C. Adams. — On the Calculation of the Sum of the Reciprocals of the
First Thousand Int^ers, and on the Value of Filler's Constant to 260 Places of
Decimals. /
Prof. J. C. Adams. — On a Simple Proof of Lambert's Theorem.
H. M. Jeffery.— On Cubics of the Third Class with Three Single Foci.
H. M. Jeffery.— On a Cubic Curve Referred to a Tetrad of Corresponding
Points or Lines.
Sir William Thomson.— Sblutions of Laplace's Tidal Equation for ceitaio
Special Types of Oscillation.
Prof. Cavley. — Suggestion of a Mechanical Integrator for the Calculation of
an integral / P^ + ^y) along an Arbitrary Path.
J. W. L. Glaisher. — On the Values of a Class of Determinants.
J. W. L. Glaisher. — On th£ Enumeration of the Primes in Burckhardt's and
Dase's Tables.
F. G. Landon.— On a method of deducing the sum of the reciprocals of the
first 2Pn numbers from the sum of the reciprocals of the first n numbers.
Besides these, the following papers reladng to mathematical subjects were alst^
read during the meeting : —
Prof. J. C. Adams. — On some recent advances in the Lunar Theory.
Prof. 8. Haughton. — The Solar Eclipse of Agathocles, considered in reply to
Professor Newcomb's criticism on the coefficient of the acceleration of the moon's
mean motion.
Prof. S. Haughton. — Summary of the first reduction of the Tidal Observations
made by the recent Arctic Expedition.
Prof. Osborne Reynolds. — On the rate of progression of groups of waves and
the rate at which wiergy is transmitted by waves.
The sum of £100 was granted to the Committee on Mathematical Tables for
commencing the calculation of a factor table for the fourth million, in continuation
of Buiekha^fis tables of the first three millions. Dase's tables extend from
6,000,000 to 9,000,000 so that there is a gap of three millions which it is now
proposed to fill up. The whple sum granted on the leconuneBdation of Section A
was £240. A committee was appointed to invite reports oq diffisrent special
branches of science.
The Association adjourned on Wednesday, August 22, till Wednesday, Aqgist
14, 1878, at Dublin. Mr. W. Spottiswoode is the President elect. In 1879 the
meeting of the Association will be held at Nottingham ; and it is probable that the
meeting will be held in 1880 at Swansea, and in 1881 at York, the city where the
fi.-st meeting of the Association took place fifty years before. J. W. L. G.
( 97 )
ON A SIMPLE PROOF OF LAMBERTS THEOREM.
By Professor /. C. Adams, M,A., F.B.8.
The following proof of Lambert's theorem, which I find
among my old papers, appears to be as simple and direct
as can be desirea.
Let a denote the semi-axis major and e the eccentricity of
an elliptic orbit, n the mean motion and fi the absolute force.
Also let r, r' denote the radii vectores, and Uj u' the
eccentric anomalies at the extremities of any arc, k the chord,
and t the time of describing the arc.
Then r = a(l — e cosu), r' = a (1 — 6 cost*'),
A? = a* (cosu - cosu')* + a" (1 — e") (sinu - sinu')*,
and . n^sa (^U=w — w' — e(8inu — sinw');
_ »Bm*-^- sm'^- + (!-««) cos" -^ 8in« -^-
. ,u—u' f, . t" + u'\
/ u4*u\.tf— u'
and n<aM-M-2f«cos— ^J sm—^*
H^nce we see that if a, and therefore also it, be g^ven
then r-i-r'jk and t are functions of the two quantities u — u'
, u+u!
and ecos-^ — .
Let tt — u's=2a and 6C0S— ^ — =scos)8, then
9*4 9*'
= 1 — cosa cosiS,
2a
/C • • A
-:- ssma sinp;
2a '
therefore ^jtili^^ i-cos(i8 + a),
TOL. VII. H
98 , PROF. ADAMS, ON A SIMPLE PEOOP
and — = 1 - cob()8 — a) ;
also nt=s2a^2 sina co8)3
= {y34a-8in(^ + a)}-{y3-a-Bin(y3-a)}.
The first two of these equations give ^8 + a and ^ - a in
terms of r + r' + k and r'\'r' — k^ and the third is the expres-
sion of Lambert's theorem.
An exactly similar proof may be given in the case of an
hyperbolic orbit.
Let i (a** 4 6"**) be denoted by csfi (w),
and i (e** — c"**) by snh (w),
which quantities may be called the hyperbolic cosine and
hyperbolic sine of m. Then we have
csh'(u)-Snh'(u)=«l,
csh (u) 4 csh (t* ) « 2 CMi — - — csh — ^— ,
2 2
csh (m) — csh(u') = 2 snh snh — -— ,
snh(u) — snh(w') = 2cnh— — - snh — j— .
The coordinates of any point in the hyperbola referred to
its axes may be represented by
or = a cnh (w),
y = a sl{^ — 1) snh (w).
If K, u' denote the values of u corresponding to the two
extremities of the arc, we have
r = a {e csh (tt) — 1}, r'^a{e csh {u') - 1}^
k^ = a'* {csh (u) - csh {u')Y 4 a* (e* - 1) {snh (u) - snh (w')}« ;
r4r' / .u + u\ .u^u' ,
— r — = e csh — t; — 1 csh — ^r 1,
2a V 2 y 2 '
or
Bnh»!i^(e'c8h"«-±ii^-l).
OF lambebt's theobek. 99
Also, twice the area of the sector limited hy r and /
= a' \/(«* — 1) {^ (soh u - snh vi) — (w — v!)\
=oV(e'-l){2(ec8h^')snh!^-(tt-«')};
and twice the area described In a unit of time Is
Hence t— \--\ -12 \e csh — — j snh -^ (w - w'jv ,
and therefore if a be given, t-\-t\U and t are functions of
the two quantities e csh — - — and u — w'.
Let w — m' = 2a and 6 csh — - — = csh (/8), which is always
possible since e is greater than 1, then
2a
= csh(^)csh(a)-l,
k
— =snh(^)snh(a);
therefore — ^ = csh (^8 + a) - 1,
2a ^ * ' ,
and — = csh ()8 - a) — 1.
ad t
-j {2 csh (^) snh (a) - 2a}
= (-y{snh(^ + a)-{^4a)-snh(/S-a) + (y3-a)l.
As before, the first two of these equations give iS + a and
^ - a in terms of r + / + A and r + r' - A;, and the last Is the
expression of Lambert's theorem in the case of the hyperbola.
When the orbit is parabolic a becomes infinite, and since
r + t\ and h are finite, the quantities a and j8 become inde-
finitely small.
Hence I±rl±^ = 1 - cos(/3 + a) = i (/S + a)' ultimately,
2a
'"''/" ^ = 1 - COS (/3 - o) = i (/3 - «)" ultimately.
r2
100 PROF. MANNHEIM, ON THE WAVE SURFACE.
-J {^ + a-B\n(fi + a)-{l3-a) + Bm{fi-a)]
- [ff {* 03 + «)• - i (y9 - «)•} ultimately
which is Lambert's theorem in the case of the parabola.
ON THE WAVE SURFACE.
By A» Mannheimf Professor at the Ecole Polytechnique of Paris.
The wave surface cuts each of its principal planes and the
plane at infinity in a conic and a circle. In each of these
planes the circle and the conic have four points of intersection,
which are the conical points of the wave surface. We shall
show that it is easy to deduce from these well-known results
that there are certain planes touching the wave surface along
lines which are circles, and that planes parallel to these cut it
in curves which are anallagmatics of the fourth order.
Fig. (6) shows the traces of the wave surface in the
principal planes, the circles being represented by dotted lines,
and tne extremities of the axes being denoted by letters
which also denote their lengths. In the plane of yz the
conical points are imaginaiy^ but they lie upon the common
real chords of the circle and the conic in this plane.
One of these chords el is parallel to the axis of y and is at
-^ — 75] . Also in the
plane of xy we have the chord dm parallel to the axis of jf
J? «) •
Every plane passing through one or other of these straight
lines cuts the wave surface in a curve having two double
points. The wave surface is therefore cut by the plane con-
taining the two straight lines e7, dm in a curve of the fourth
order which has four double points, viz., in a curve which is
decomposable into two conies.
But from the values of oe and od it follows that
1
i + -
oe
-' "^ 0(? "■ P '
PROF. MANNHEIM, ON THE WAVE SURFACE. 101
and consequently that edissL tangent to the circle of radius b.
The plane (elj dm) is therefore abo tangent at g to the wave
surface, and o^ is thus a double point of the section made by
the plane. But this section consists of two conies : the point
ff must therefore be common to these conies. They thus nave
five common points and must be coincident. The plane
{elj dm) therefore touches the wave surface along one of these
conies. We shall show that these conies are circles.
Cut the wave surface by a plane perpendicular to the plane
of xz. This plane cuts the plane of xz in a straight line
which is an axis of the section determined by it in the wave
surface, and it cuts the plane [el^ dm) in a perpendicular to
this axis. If we move this cutting plane parallel to itself
to infinity, the curve of intersection is composed of a circle
and a concentric eonic. and the intersection of the plane {el^ dm)
with the plane at infinity, which is, as we have said, perpen*-
dicular to an axis of these curves, cannot be one of their
common tangents but must be one of their common chords.
Thus the singular tangent planes of the wave surface cut the
plane at infinity along the common chords of the circle and
the conic situated in that plane. Consequently the conies
along which they touch the wave surface pass through points
situated on the imaginary circle at infinity, and are therefore
circles. As the wave surface has a centre, its singular
tangent planes are symmetrical two and two. There are
then two singular tangent planes for each of the chords
common to the circle and the conic, situated in the plane at
infinity, viz., the wave surface has twelve singular tangent
planes. Of these twelve tangent planes, only four are real,
since on the plane at infinity there are only two real common
chords.
Planes parallel to the singular tangent planes pass through
the chords common to the circle and the conic situated on the
Elane at infinity, they therefore cut the wave surface in curves,
aving for double points the conical points situated on these
chords. And as these double points are upon the circle at
infinity we see that planes parallel to the singular tangent
planes cut the wave surface along anallagmatics of the fourth
order*
♦ I have already proved, in a very different manner, a particular case of this
tiieorem in a communication made to the Fiench Association for the Advancement
of Science at the meeting at Nantes (1875).
( 102 )
ON LONG SUCCESSIONS OF COMPOSITE
NUMBEKS.
By J. W. L. Glauher.
§1. Burckhardt's tables (1814-1817) give the least
factor of every number up to 3,036,000, and Base's tables
(1862-1865), the least factor of every number from 6,000,000
to 9,000,000; the primes in these tables are marked by a
dash, or short line, facing them. The number of primes
between any limits can therefore be obtained by counting
the number of dashes; and in this manner I eflTected the
enumeration of the primes in the six millions over which
the tables extend * .
v^Uinff, for convenience of expression, the hundred
numbers between lOOn and 100 (n + 1) the (n + 1)*^ century^
then the enumeration was made by centuries, that is to say,
the number of primes in each century was obtained by
counting and entered in its proper place upon a printed
form.
The following specimen exhibits five colunms of one of
the tables so formed :
No. 270.
20
9
* ■
30
6
40
6
50
10
60
9
21
7
31
:7
-41
9
51
6
61
8
22
6
32
ii3
42
6
52
7
62
5
23
6
33
:5
43
6
53
6
63
6
24
4
34
9
44
6
54
5
64
6
25
2
35
5
45
8
55
7
65
6
26
7
36
10
46
4
66
8
66
8
27
9
37
7
47
9
57
3
67
7
28
8
38
5
48
7
58
8
68
3
29
4
39
7
49
17
59
9
69
8
62
64
77
69
66
This portion of the table shows that the number of the
primes between 2,702,000 and 2,702,100 is 9, between 2,702,100
and 2,702,200 is 7, and so on ; or, in other words, that the
27,021**^ century contains 9 primes, the 27,022*^ contains 7
primes, and so on. The results were then classified in tables •
* A preliminary account of the results is printed in the Proceedings q/ th€
Cambridge Philosophical Society j vol. III., pp. 17-23, 47-5G, (1876-1877).
MR. OLAISHEB, ON COMPOiUlTE NITMBESS. 103
and those publislied in the Proeeedifngs of the Cambridge
Fhilosopkical Society give the number of centuries, in each
group of 100,000 in the six millions, which contain n primes
ior n as argument. Thus, for example, the eighth column of
the table for the third million show& that in the thousand
centuries between 2,700,000 and 2,800,000 Uiere is no century
that contains no prime, 2 centuries each of which contains one
prime, 7 centuries each of which contains two primes, and so
on, there being 195 centuries containing six primes and one
century containing as many as seventeen primes.
The following is a similar table, in which each column has
reference to a million :
Number of centuries^ each of which contains n primes,
n
1
2
3
4
6
6
7
8
9
10
11
12
13
14
15
16
17
21
26
No. of
primes.
§2. Whenever there is no prime in a century there must
be a succession of at least 100 consecutive composite numbers,
and whenever there is only one prime there must be a suc-
cession of at least 50, but, of course, a 2-priine century or a
first
second
third
seventh
eighth
million.
ninth
xuiUlon.
million.
milUon.
million.
million.
1
1
6
4
4
3
16
25
28
30
34
29
72
97
173
171
178
140
257
338
482
541
570
372
667
775
1,049
1,066
1,078
801
1,253
1,408
1,603
1,691
1,742
1,362
1,743
1,878
1,948
1,993
1,966
1,765
2,032
1,997
1,916
1,754
1,788
1,821
1,612
1,526
1,366
1,394
1,278
1,554
1,182
1,036
840
787
778
1,058
691
558
374
360
390
592
311
227
156
155
143
316
113
98
46
40
38
122
39
28
10
10
11
32
7
6
3
2
2
20
3
1
2
8
1
3
It
1
1*
'8,499
70,433
67,885
63,799
63,158
62,760
* 1 and 2 are hoth counted as primes.
t The specimen on the preceding page (which is half of one of the small tables,
and therefore -j^\j^ of the whole table) was chosen so as to include this (the
270,60th) century. It will be noticed that there is no other century in the second
or third million that contains so many as 17 primes, and that in the third million
there is no century contaimng 16 primes, and only one in the second million.
104 MR. OLAISHEB, ON IjOHO 8U0CEMI0K8
3-priiiie century need not indicate a long sequence. In order
to find long saccesBions of composite namberS| I had all the
instances in which there were 0, 1, 2 or 3 primes in a century
looked oat in the tables^ and the cases noted down in which
the sequence was 50 or upwards. The number of sequences
exceeding^ 50 was very great,* and therefore in the fol-
lowing hst I only give the sequences of 79 and upwards in
the first millioni and of 99 and upwards in the second and
third millions, that were thus founct :
FlBST MILLION. — Sequences of 79 and upwards.
Lower limit. Upper limit. Seqaenoe.
155,921
156,007
85
338,033
338,119
85
370,261
370,373
111
461,717
461,801
83
492,113
492,227
lis
544,279
544,367
87
604,073
604,171
97
682,819
682,901
81
816,729
815,809
79
818,723
818,813
89
822,433
822,517
83
838,249
838,349
99
843,911
844,001
89
863,393
863,479
85
927,869
927,961
91
979,567
979,651
83
982,981
983,063
81
SECOND MILLION.-
—Sequences of 99 and upwarda.
Lower limit.
Upper Limit.
Seqnenoe.
1,349,533
1,349,651
117
1,357,201
1,357,333
131
1,388,483
1,388,587
103
1,444,309
1,444,411
101
1,468,277
1,468,387
109
1,561,919
1,562,051
131
1,648,081
1,648,181
99
1,655,707
1,655,807
99
1,664,123
1,664,227
103
1,671,781
1,671,907
125
1,761,187
1,761,289
101
1,775,069
1,775,171
101
1,895,359
1,895,479
119
* Tablet containing these sequenoes were exhibited to the Meeting of the
London Mathematical Society, May 10, 1877 (see Mtstengtry ante p. 66).
upper limit.
2,010,881
Sequence.
147
2,238,931
107
2,244,091
103
2,305,271
101
2,314,547
2,346,089
2,598,091
2,614,987
2,637,911
107
99
109
103
111
2,784,473
2,867,213
2,898,359
99
105
119
OF COMPOSITE NUMBEBS. 105
THIRD MILLION. — Seqiiences of 99 and uptoards.
Lower Limit.
2,010,733
2,238,823
2,243,987
2,3'05,169
2,314,439
2,345,989
2,597,981
2,614,883
2,637,799
2,784,373
2,867,107
2,898,239
Thus the 85 numbers between the primes 155,921 and
156,007 are composite, and so on, viz. the numbers in the
first two columns are primes, and the numbers intermediate
to the lower limit and the upper limit are all composite.
It is to be specially noted that the above list is not
complete, as, of course, long sequences may occur when the
number of primes in adjacent centuries is 4 or more. Biit
the parts of the table where there were but few primes in
a century were naturally most likely to be fruitful in long
sequences of composite numbers. I ought also to state that
I am not certain even that all the sequences obtained from
the O's, I's, 2's, S's are Included, as the work was merely
performed once, and in view of the necessary incompleteness
of the list, it did not seem worth while to undertake a
thorough examination or a duplicate calculation, in order to
prove that no sequence had been omitted. The list is,
therefore, merely a collection of long successions of composite
numbers, without regard to completeness. I have, of course,
verified that all the sequences given are correct.
It may be noted that the longest sequence (147) was
obtainisd firom a 3-prime century, the two sequences of 131
in the second million were each obtained from 1-prime cen-
turies, and in the first million the sequence of 113 resulted
firom a 2-prime century, and that of 111 from a 3-prime
century. The 0-prime centuries produced sequences of 125
and 111.
Although a 3-prirae century is less likely to produce a
long sequence than a 2-prime century, still the number of
the former is so much greater that the sequences obtained
from them are almost equal in importance to those derived
106 MR. GLAI8HEB, ON C0MP06ITE NUMBEBS.
from the 2-prime centuries; and a similar remark will applji
tboagb probably in a less degreej to tbe 4-prime and other
centuries, which were not examined.
§3. The great length of some of the seanences and tbe
number of long sequences are, I think, remarkable ; and tbe
most noticeable of all is the sequence of 111, which occurs
so early as 370,261.
It IS known that any sequence of composite numbers,
however long, must occur at a certain definite place in the
series of natural numbers, for if p be any prime, q the next
prime to^, and 2, 3, 5...i; all the primes up to ^, then the
J — 2 numbers immediately following (2.3.5.7...^] + ! must be
composite, for (2.3.5... ^)i-2 is divisible by 2, the next number
by 3, the next by 2, and so on, every number obviously
having a divisor, up to and including (2.3.5...^) 4 (j- 1).
The J- 2 numbers between (2.3.5...^) + 1 and (2.3.5...^) + j
must therefore be composite, and the limiting numbers may
be either prime or composite, so that we have a sequence of
at least y — 2 composite numbers.
The prime next below 111 is 109, and the next prime
is 113, so that of necessity the 111 numbers following
(2.3.5.. .109) + 1 (i)
are composite; we therefore have a sequence of 111 immedi-
ately following
279,734,996,817,854,936,178,276,161,872,067,809,674,997,231
(ii),
this number being the value of (i).
The number (li) consists of 45 figures, and is enormously
greater than 370,261, where a sequence of 1 11 actually occurs,
so that long sequences are met with far earlier than the
theorem requires them to happen. In fact,
2.3.5. ..17= 510,510
2.3.5...19 = 9,699,690,
so that we cannot predict from the theorem a sequence of
more than 21 in the first ten millions. The sequences for the
seventh, eighth, and ninth millions, as also some remarks
upon the sequences that occur between 1 and 30,000, will
be given in a continuation of this paper.
( 107 )
NOTE ON A DIFFERENTIAL EQUATION.
By H. W. Uoyd Tanner, M.A.
The eqaation is
— •- ^ +...M"-'^=0
dx. dx, '"^ ' dx.
(1).
The following form of solution may be worth record.
Let A^ represent the determinant formed by omitting the
m*^ column of the system
d(f>^ d^^ d(f>^
dx^ * dx^ '"' dx^ *
dx,>
#.
<^.'
dx, '
dx. '
where ^,...^^, are arbitrary functions of a;,. ..a;,. Then the
general solution of (1) is
»«=^«('« = l,2,-«) (2).
That (2) is a solution of (I) follows if we can prove
dx^ dx^
or, as it may be written,
d d
dx. dx^'^'-^'^ ir~^
(3),
dx^ * dx^ *'*' dx^
d^ d4>^ d(f>,
dx^ ' dx^ ' "
dx.
= 0,
.(4).
#1,-1 #n-1
dx, * dx^ '*"
dx_
Now if (3) or (4) be expanded there will be In each terra
one, and only one, factor of the form , ,* . This factor
will occur in two terms, multiplied by the same coefficient, but
'
108 PROF. TANNEB, ON A DIFFERENTIAL EQUATION.
with opposite signs. For instance, take , ,* . The
coefficient of this in each of the two terms in which it occurs
is the Jacobian j[' \ } but one term is positive and the
other negative. HencCi as there are no terms in the
expansion of (4) without a factor of the second order, and
terms containing such factors annul each other, (3) and (4)
are identically true; or (2) is a solution of (1).
It is ako the general solution of (1). To prove this we
show that the equations
«m = ^^«(^=^2,...n) (5)
impose no restriction upon 2^, and then making these substi-
tutions in (1) we determine the general form of \. It will
be found that \ is such that (5) is equivalent to (2).
Fii-st then, (5) do not imply any relation between «i,...«?^,
in other words ^„...0„.„ \ can be determined so as to
satisfy (5) whatever z^...z^ may be. In fact, the ^'s are
any (n ~ 1) independent particular integrals of the equation
•.^-J/-H--«.^=o («,,
which can be found by integrating the system
z z z
1 9 *^n
For, calling these particular integrals, 0..*0«.i, the general
value of ^ is
so that (^ satisfies the equation
that is A,^ - A,^ +...= 0.
Comparing this with (6) we reproduce (5).
Next suppose that «,.-.«„ satisfv (1), As has just been
shown, the substitutions (5) can oe made without loss of
generality, so that (1) Is equivalent to
PROF. TANNER, ON A DIFFERENTIAL EQUATION. 109
The first term of this equation vanishes by (3). The
second becomes
or ^^-^'C^u—^-i)
Substituting in (5) we get
but this is not more general than A^, because ^, is arbitrary
as well as '^. Hence (5) is equivalent to (2) when (1) is
satisfied, and (2) is accordingly the general solution of (1).
Of course many equations of a more complicatea form
can be reduced to (1) by a suitable change of variables. For
example, if ^,, B^... be functions of ss^...z^ ^^^7j ^^^ equation
- (fo, J dz^ . dz^
* A ^ * - '■ k
+ = 0... (7)
is reducible to (1) by a change of dependent variables
only, provided
A^dz^ + A^dz^ +. . .+ AJlz^ = Xrff,
Bjlz^ + B^dz^ +...+ B^dz^ ^-Xd^
(n equations).
This implies certain relations between the A\ B^s. &c. ;
when these relations are satisfied (7) can be expressed m the
form (1).
Again, certain non-linear equations are reducible to (1}, of
which the following is an example :
d{x^.x^) d{x^,x^) rf(a?„a?J
110 PROF. TANNEK, ON A DIFFERENTIAL EQUATION.
This may be written in the form
dx, dx. dx^ ^
dz^ dz dz^ *
and its solution is
^„ 6, being arbitrary functions of «„ «,, z^.
To find the general form of equations reducible to (1)
we may suppose
^«~/ml?l» 52)***5n» fi) far ••?!»]
(?n, r= 1,2,... n).
Making these substitutions in (1) and regarding ^^...^^ as
new dependent variables, |,...|^ as the new independent
variables, we shall obtain the general form of equation
sought.
It may be worth while to draw attention to the particular
case in which w =2, The identity (1) becomes
dx,dx^ dx^dx^ *
In the same case we say that if
dz^ __ ^^2
cfoj dx^
- d<f> dd)
then ,^=_,,^==_.
The equation (8) is the condition that
z^dx^ + z^dx,
should be an exact differential, and that the expression
du du
z — "—Z —
* da?, " dx^
should be an exact Jacobian ; viz. this is
d(f> du d<l> du
(8),
PROP. N ANSON ON DIFPEBENTIAL EQUATIONS, &C. Ill
Similarly (1) is the condition that the expression
du du . .^«j du
*''^~*«^/-^"^ '-^
should be an exact Jacobian, namely,
<^(^ir'-'^,-<>")
■.<^K» a',)*
October, 1877.
ON LINEAE PARTIAL DIFFERENTIAL
EQUATIONS OF THE FIRST ORDER.
By Professor E, J. Naruon, M,A,
Consider first the general linear homogeneous partial
differential equation of the first order in which the dependent
variable does not explicitly occur. This equation is of the form
^ du ^ du ^ t^\
dx, " dx.
M
when X,...X^ are any functions of the independent variables
x^,..x^. Let A denote the operation
•rr d .^- d
so that the equation (1) can be written
Aw = (2).
Introduce a new system of independent variables yj,.,.y^
in place of x^y..x^. Since
we have
du ^ du dy^ du dy^
d^" dy, ^■^■•••■^^, 5i
At^-(Ay,)|||+...+ (Ayn)^j
and the transformed equation is*
(^^.)|,+--K^^«)^=o (3)-
* Cf . Boole, Differential Equatione, Supplementary Volume, p. 69.
112 PBOF. NANSON ON PARTIAL DIFFEBENTIAL
Now choose for y^y..y^_^ a set of n — 1 independent
ntegrals of the system of ordinary simultaneous differential
equations
X z.
w,
and let y^ be any function of a5j....jr^, which is independent of
HxyVn 1- Then we have
and as y^ is independent of yi«-*y»., it cannot be an integral
of (4) so that
Ay^notssO (5).
The equation (3) then takes the form
or b virtue of (5)
v.'" • ">•
Hence w = 0(yir-ynJ ^••••(7),
when 4> is arbitrary. The process shows that this is the most
general solution. Hence we have the well-known rule for
mtegrating the equation (1).
^^ Construct the auxiliary equations (4) and integrate them
in the form
then the solution required is
when fj> is arbitrary."
Consider next the ordinary form of the linear equation, viz.
P.p,+...4Pp,«P...... (8),
when^ = -T- andP, P„.*«^» ^"^ any functions of «, aj^, x^...x^.
Let u = f{zj x^y . .«J «
be any solution of the equation (8). Since
du du ^
EQUATIONS OF THE FIRST ORDER. 113
the given equation takes the form
^.^.+-+^"a5-/^^=« (^)'
which is an equation of the kind already considered.*
To solve this, we integrate the auxiliary equations
dx. dx dz
pi=...= p'' = ^ (10)
in the form ^^ = c^, . . . m^ = c^,
then the solution of (9) is
where <(> is arbitrary, and the solution of (8) is
^(w,, w„...wj = »...(11).
The above proof is at once suggested by the passages in
Boole, to which reference is made.
The theory of the elimination of an arbitrary ftinction
may also be presented in a simnle manner by the process
of changing the independent variaoles.
Thus, suppose it required to eliminate the arbitrary func-
tion from the equation
^ = <f>iyiy"2/n-i)" (7),
where y,vyM-t ^^^ w- 1 given functions of the independent
variables a;„...a;^. Introduce in place of a:„...a?^ the new set
of independent variables y^yy^<i where y^ is any function
which is independent of the given functions y,j...y^_,- The
equation (7) gives at once
¥.-" ••■■<*'■
Now this is a linear homogeneous partial differential equa-
tion of the first order. If we change back to the original
variables the equation will still have the same character.
Thus we have a very simple proof of the proposition that
the elimination of the arbitrary function from an equation
of the form (7) always leads to a linear homogeneous partial
differential equation of the first order.
* Of. Boole, I.C., p. 68.
VOL. VII.
114 PROF. NANSON, ON DIFFERENTIAL EQUATIONS.
Let US, however, actually construct this equation. By
the ordinary processes of the Differential Calculus, we have
du ^ du dx^ du dx^ . .
¥„""^^B'^*"'^^¥n
dx dx
where -j-^ y-j^ l^^^e to be determined by n' equations of
the form
dy^ dx, '" dy^ dx, *
dy^ dx^ *" dy^ dx^
These equations give
J— =1 minor of ;^*iii «/|
where J denotes the functional determinant or Jacobian
j-l^ (13).
Substituting the values of -^,...-t-^ thus determined In
(12), the equation (6) becomes
d{x^y..x^^^xj
We have thus the theorem that If there exist between n
functions y,,. . .y^ of n independent variables x^y . ,x^ an identical
relation
then the functional determinant (13) vanishes.
Conversely, we can show from the above process of inte-
grating the equation (1) that if the Jacobian (12) vanishes,
then there must exist an identical relation of the form (14).
For the fact that (13) vanishes shows that the n equations
of the type
MATHEMATICAL NOTES. 115
are constant, i.e. that if the ratios
be determined from any n — 1 of these equations, then the
remaining equation will also be satisfied. But the n equations
express precisely the fact that y^y^y^ satisfy the equation
X ft
Hence we must have y^^some function of w,vW«-,j s-nd so
fory^, yg...y^; t.e. there must be some relation between the
functions y^yy^ of the form (14).
University, Melbourne,
AuffmtZ, IS77.
MATHEMATICAL NOTES.
On two related qiuidric functions.
Assume
<^a; = a*(c — a;)— a:(c'- b^ — cx)^
^jrx = J* (c — re) — a; (c* — a" — ca?) ,
J"
In the first of these for x write , then
c — X
il>(x).
(c" - ft- - ca;)'*
A. Caylet.
12
116 MATHEMATICAL NOTES.
OrT development in series.
Consider, in general, a function developable in a con-
vergent series proceeding according to positive powers of the
variables, and, for example, let
F{x) ^ ^ ' ^
Ae^ 4- J?€^+C6'* "■^^■^» 1 "^^^ 2 1 "*■•••■*■ ^" n 1 ^'"^
or under a symbolic form
F{x)
^e'\
Ae^'^-Be^'^-Ce'*
Denote hY/{x) any other function whatever, and by h an
increment ot a; ; we have the symbolic formula
F(hf) = Ae^^-^^^^^^ 4 Be^^'^^^^^^ + Oe***-^^^*^*^
in the development of which we are to replace
*r by/(a>),
{ahf[x + oA)}- by aji' <?!ZMjii) .
In fact, it is easy to see that this formula holds for
f{x) = G^e**, whatever O and k may be, and therefore also
for any function whatever 2 O^ of x.
We have, in particular, for -^ — : = e^*,
hf {x) = e**-^^^*) - /*-^^'\
— 2ic
which is Stirling's formula ; and for -^ — - = e'**, with
c + 1
P„ = 2 (I - 2") J?^, we have the formula
a formula due to Boole.
2
Let -^ — =5 = 6^*, E^ denoting an Eulerian number, then
2/(a:) = /*-^('+*' + e^*-^^'^^
and similarly for many other developments.
Edouard Lucas.
Paris, November t 1877.
MATHEMATICAL NOTES. 117
On a Theorem due to RodriguesJ^
Let {(« + A)"- 1}* be the series of which the general terra
is PJvr^ then
now
1 + -5— ;A + -r— i)
(o~ A" \*
also
= coeff. of p^in jl + ^rri "A- +-?:n-J:
= (a^ - 1)'*"* X coeff. of k"^ in Tl ^.-p-^k + -^-~j ,
therefore
i-m\dx) ^* *^-u-+m W ^^ ^^'
therefore
which is the theorem in question*
W. H. H. Hudson.
October 31, 1877.
♦ See Ferrers's Spherical Harmonies^ (1877), pp. 18—16, or Todhunter's
Laplaee*8 Funetiontf ^<t,, (1876), pp. 76-80.
118 MATHEMATICAL NOTES.
Proof of the Principle of the Composition of Couples
in Statics.
The following proof of the law of composition of couples
seems simpler than that usually given, and will probably
suggest Sir W. Thomson's proof of the composition of
elementary rotations in a mass of fluid moving with rotation.
It is a well-known theorem In statics that three forces
which act along the sides of a triangle and are proportional
to them compound a couple whose axis is perpendicular to the
plane of the triangle and whose moment is proportional to
the area of the triangle. Conversely, If we take any
tetrahedron and suppose the faces acted on by couples
whose axes point inwards and whose moments are pro-
portional to the faces in which they act, these couples may
be replaced by forces round the sides of the triangle, which
a little inspection will show mutually cancel each other.
The couples are therefore in equilibrium.
This result is evidently equivalent to the law of the
composition of couples.
C. NiVEN.
1
Expansion derived from Lagrange^ s Series.
Theorem. If » = o + efx and e = , n being any
quantity, then X-^ne
in which, after the differentiations, a is to be put equal to
zero, except under a functional sign.
Thus, performing the differentiations,
^ (na +»• = 2 {m. +/») (n +/'o),
^ («a +/«)» = 3 (na +/a)« (« +/'a),
^ {na -^faf = 6 (na + fa) (n +fa)'' + 3 [na +fa)y\
&c. — dfc.
MATHEMATICAL NOTES. 119
SO that the theorem gives
aj = a + e/a+ — {2/a (n -f /'a)}
e»
{6/a {n +fay + 3 (/a)"/ "a} + &c.
1.2.3
J. W. L. Glaisher.
Four Algebraical Theorems.
_ a; (a;' + 1*) (x' 4 3")... {x* + (2n - 1)'}
(2n+l)!
then
K-H
X
and A=l» ■^o = '«'»
A + i^.-, + ||A-. + &c. = (2n+l)§,
S, + i^.., + 1;! B„„ + &c. = (2n + 2) ^
For example, n = 2,
a;''(a;''4 2') ^ , «' 1.3 _ (x* + 1') (ai'' + 3')
4l "^*2!"*'2:i~^ 6! »
a; («V l*)( a!V 3") , , x(«Vl*) ^1.3 ^ ar (a!« + 2») (a» + 4')
II. If A^\i
then ^„^, + A^,,A^ + ^n-a^a- • •+ ^i(n^)An = ;^ ^„+i
if n be even, and
(the last term of the series having the factor i), if n be uneven.
For example, n = 6, the first equation gives
6* 5* 2* 4' 3' _ 6 7*
6T^5"!*2T*"ri* 3l""7 • fl'
120 MATHEMATICAL NOTES.
and, n = 7, the second equation gives
ri"*"6T*2l"^5!'3!'*"*'4!'4l"'8'8l*
III. If 8^ denote the sum of the products of the quantities
1 + 1 + 1*' 1 + 2 + 2'" l+3 + 3»'-'^**''-^'
taken n together, then
. /S, + iS, - /S,- /S,+ >&,+ >8.- 8, - 8^ + &c. ad. inf. = 1,
the terms being alternately positive and negative in pairs.
IV. If
sin(6-c) ^ sin(c-a) ^ ?m(a-i)^^^^^^^^^^^^ ^^^^^^^
where A^ denotes the terms of n dimensions in a, J, c, then
J J J A
all contain the algebraical factor [df->c}?'{'^'''bc''Ca''cihY^ and
AAA A
all contain the algebraical factor cf-^V-k-^ — hc-ea — ab.
J. W. L. Glaisher.
Elementary Proof of a Theorem in Functional Determinants.
The theorem, due to Jacobi, is as follows :
Let there be n functions ^,, ...0 of n independent variables
^1) '"^ni ^^^^ ^^ there be an identical relation
-?'(*„ -W-O ; (1)
between the function ^, the functional determinant
d[x^^ ,,.a?J
will vanish; and, converselv, if the functional determinant
vanishes, then there will be an identical relation of the
form (1).
The theorem is well known and many different proofs
have been given. Baltzer,* Boole,t and Laurent} make use
* Theorie und Anwendung der Determinanten, 3id edition p. 131.
t Differential Equations, sop. vol., p. 56.
X Mecanique RationeUe, p. 830.
MATHEMATICAL NOTES. 121
of differentials and in a proof given by Brioschi,* the method
of induction is used: the latter proof is, so far as I can
remember, veiy similar to the one originally given by Jacobi.
The following proof is direct and depends only on th
use of differential coefficients. It is the extension of the
proof given by Boolef in the case of two functions.
The proof of the first part is obvious, viz. from (1) we
have by differentiation n equations of the form
dF d<l>^ ^ ^ dF d<l>n^Q
d<f>^ dx^ *" d^^ dx^ '
and, eliminating ^-r- ,... -tt- from these equations, we get
jt^r' (')•
The converse is proved as follows :
The functions ^,,.*'^ii-i ^^^ mutually independent, or else
the proposition to be proved is granted. We can then
express ^^ in terms of 0,,...^^.^ and -^^j where ^^ is any
function which is independent of 0i,."^«-i- I^ ^^^ be done
we have by a known theorem
d{x^y..xj d{<l>,y..<l>^,^y i^J * c?(aj„...a?^„ xj
Now <^„...<^^_„ -^^ are mutually independent, so that
y"-'^->^r) not=o.
Hence, by (2), we have
which proves the proposition. E. J. Naijson.
University, Melbourne,
At*gtut 3, 1877.
♦ Theorie des DHerminarUSf p. 122,
t DifferttUial EquatiorUf p. 24.
122 MATHEMATICAL NOTES.
Froof of the formula for l** 4- 2** + 3'...+n'^*
The sum of the cubes of the first n natural numbers may
be found as follows :
Let ABj BGy CZ>, &c. (fig. 7) represent 1, 2, 3, &c. Draw
BE^ Ci\ DO^ &c. representing 1", 2«, 3", &c. Then the rect-
angles AE^ BFj CGj &c. will represent 1', 2', 3', &c. Draw
AEK at 45° to AD^ and suppose CO is the w"* rectangle,
t.e.j =n'. Then G^^=w'*— ^n(n4"l) = -HA; therefore the
figure HKOL will fit into the position HKlh. Similarly,
EHFM will fit into position EHhe^ and so. make up the
square AK^ or ;S;^3 = Un (n + 1)}*.
The squares may also be summed In the following simple
way:
n* = 1 + 3 + 5 +...+ (2n - 1),
(w-l)' = l + 3 + 5+...+ (2n-3).
2»=l+3,
l«=l;
therefore 8^^ = n + 3 (w - 1) + 5 (n - 2) +. . .+ (2n - 1),
and 2i&„^, = 2n« + 2(n-l)» + 2(n-2)*4"...+ 2.
Adding, we get
3^«,8 = (2w+ 1) {n + (n - 1) + (n- 2) +...+ 1},
^l.on.. A? n(n+l)(2n + l)
whence «,„= ^^ .
J. M. Croker.
Erpansion of products of cosines and sines.
The following note contains the expansion of
cos^j cos5j...cos5^ and of sin 5^ sin 5,^... sin 5^
In terms of the sums of certain cosines and sines.
Let a? =cos^, + t sin^,, x^ = cos0^ + i einO^ &c., and let
xjT.^...x^ be denoted by -P"(aj), cos^ cos^j...cos&^ by P" (cos^),
sin ^, sin ^,... sin 5^ by P"(sin^), and ^,4"^, + ...+ ^^ by 2^ (5),
then we have
P" {x) = cosS^e + i sin \0^
♦ See p. 60.
MATHEMATICAL NOTES. 123
therefore cos S„5 = J |F* {x) + p^J ,
sin
also 2"P" (cos 0) = P" (jB + ^ V
By the symbol 20 [A^d'\ we mean the sum of the dif-
ferent values taken by ^ [-4^^], when for ArO is substituted
the sum of any r angles, until the whole number of possible
combinations is exhausted, supposing them all unlike.
Thus, if r = 2, w = 3,
S cos {S, {d) - 2A^0] = cos [0^ ^0^+0^^2{0^ + ^J}
+ cos(^,+ 5,+ ^s- 2(5,+ ^,)} + cos(5,+ 5,+ ^3- 2 (5,+ 5,)}
c=cos(53«5,-5,) + cos(5,-^5,-5,) + cos(5,-5,-53).
Now, in expanding P" f a? + - j , we notice
first term P* (x) + last term -pirT-r = 2 cos 25,
second term, P" (a?) ] —5 H — 5 +...+ — sf
+ lastterm but one, ^prr\ K*+aj/+...+x^*}=2Scos{25- 2-4,5}.
So, adding the third term and the last term but two,
we have sum = 22 cos ( 25 -2 J^^).
If n be odd there will be no singular term in the middle.
If n be even, say 2 j, there will be a middle term.
\p^q^\^q^'"*^%q X^X^.,,Xg )
and so on with other elements which may be coupled in
the same way.
Thus, taking the whole twice over, the middle term is
S cos (2^5 -2^,5).
We are able to say therefore that generally
2"*'P*(cos5) = cosu4.5
+ 2 cos {A J - 2^,5)
+ 2 cos {A^0 - 2A^0)
+ 2 cos {A J - 2^,5).
1
1
124 MATHEMATICAL NOTES.
If n be odd = 2p + 1, last terms = 2 cos [A0 - ^A^O).
K n be even = 2 j, last terms = JS cos [A^ — 2Afi).
Practically this set of terms splits into two equal parts so
that there is no fraction.
Eocamples:
2 cosa cosiS = cos(a + /8) + ^ cos(a - ^8) 4- i cos(/8 - a)
=5 cos(a + /8) + cos (a - /8),
4 cosa cos)3 COS7 = cos (a + )3 + 7)
+cos(a+/8--7)+ cos(a+7-i8)+cos()3+7-a).
Again, with the product of sines.
If nbeodd=2p + l,
2«-» (- l)*(*-^)P"(sin5) = sm^^5- S sin(^^5- 2A^0)
+ S sin {A J - 2A^e) +...(- 1)' S sin {A J - 2^,5).
If n be even = 2 j,
2«-i (_ i)i* p« (gin ^) ^ cos^^d - S cos {A J - 2^,5)
+ 2 cos(^^5 - 2A^0)+... i (- ly 2 cos(u4^5 - 2A^0).
Practically, this last set of terms splits into two equal parts.
Eocamplei :
2 sina sin)3 = J cos(a — /8) + J cos(/8 — a) — cos(a + /8)
= cos {oL^P) — COS (a + /8),
4 sina Aafi sin7 = sin (a + /9 — 7) + sin(a + 7 — /8)
+ sin(/8-f 7-a)-sin(a + /8 + 7).
R. Verdon.
A Trtgonormtrical Identity.
cos (J — c) cos (J + c + e?) + cosa cos {a-\-d)
= cos(c — a)cos(c + a4 d) + cos& cos(& + rf)
=s cos(a — J) cos(a-f J + d) + cos c cos (c + c?)
= cosa cos (a + d) + cos J cos (J + c?) + cose cos (c + d) — cose?.
A. Cayley.
MB. LEUDESDORF, THEOREM IN KINEMATICS. 125
Extract of a Letter from Brof CayUy,
'^ I wish to construct a correspondence such as
(a? + ty)" + (oj + ty) = Zh- t r,
or, say, for gfreater convenience '^
4(aj + ty)'-.3(aj + ty) = X+tr,
viz. if oj + ty = cosu,
then X-f i F= cosSu.
Suppose Zu^ is a value of 3u corresponding to a given value
of X+tF, then the three values of x-\^iy are of course
cosUq, cos(u^±— ) , but I am afraid the calculation of w^,
even with cosh and sinh tables would be very laborious.
Writing Z+ 1 Fsa R (cos0 +f sin0), the intervals for might
be 5', 10" or even 15", those of R^ say 0*1 from to 2, and
then 0*5 up to 4 or 5 ; and 2 places of decimals would be
quite sufficient; but even this would probably involve a
great mass of calculation.
It has occurred to me that perhaps a geometrical solution
might be found for the equation X + tF=co83M."
A. Cay LEY.
October 31, 1877.
THEOREM IN KINEMATICS.
By C LevtdeBdorff If. A,
Let Aj Bj 0, P (fig. 8) be four points rigidly connected
together and moving together in any way in a plane. Let
50= a, CA = J, AB^c, PA = a\ PB^b\ PG= c\ PAB=a,
BBC^^, PGA^ri, also let A be {x,yV B{xj/), C{xj,^\
B(xy)j and let AB make an angle with Ox^ at any
moment. Thtn
aj,=a3-a' cos(^+a), aj,=a?+&' cos(^+)8-J?), 0:3=35-0' cos(^+74 -4),
Vr^y- ^' 8in(^4a), y=y^b' sin(^+^- jB), y^=y-c' sin(^+74^) ;
therefore
i [^x^Vx - Vi^d = i (^y - y^) "" i«' {cos (^ 4" a) (yds 4 dx)
•\-An[e-^a)[yde-dx)]'^ydd (i),
\ (a^,^i^a-y2^.) =i (^<y-y^-») + W {^os {0+i3-B){ydd-\-dx)
•^Bm{0 + l3-^B){yd0^dx)}-^ib"d0, (2),
i (^s^yg - ys^s) = h [^^y - y^) - ¥ {cos(^47+^)(yrf^4rfa;)
4"sin(^47 + ^)(y^^-^)}+ic'V^ (3).
126 MB. LEUBESDORF, THEOBEM IN KINEMATICS.
MaltipIylDg (1), (2), (3) by X, /i, v respectively and
adding, we have, if (-4), (J?), ((7), (P) denote the areas of the
carves traced ont by -4, Bj (7, P,
Xd{A) + fid{B)'\-vd{C)
= (X + Ai + v)i(P) + i(Xa'« + Ai5'« + vOrf5 (4),
provided only that
-W coaa + fib' cos{fi-B) + vc co8(7 + u4) = 0)
- W sin a + Ai5' sin 09-P) + vc sin (7 + u4) = 1 ^
whence
\a ifib' I vc' = sin(i3 — 7+ C) : sin(7 — a + -4) : sin(a— iS + P)
= sinPP(7 iBmCPA isinAPB.
Snbstitnting in (4), and assuming that the curves traced
out by the four points are closed, we have, by integrating
for from to 27r,
sin BPG , ., sin CPA , ^, sin^lPP , ^
— ^r— (-4) + —p (P)+ ^, (C)
/smBPG sinGPA BinAPB\,^.
+ ir(a' BmBPG+b' sinGPA + c' sin APB) (5).
The coefficient of (P) is equal to — ttt-t— ; that of ir
^ ' ^ aoc ^
if 0?, y, 2; be the triangular coordinates of P referred to the
triangle ABG^
2
= -^7^ [{aV + J*« (a; - 1) + c* (a? - 1) y} a; +...+...]
so that (5) reduces to
a:(u4)+y(P) + «((7) = (P) + 7r (a'y« + J'^aj + ^ocy\
or (P)=a:(^)+y(P)4"i5(C)
4 ir (square of tangent from P to circle round ABG).
TRANSACTIONS OF SOCIETIES. 127
If then A and B be made to move on given closed curves,
the above formula connects the areas of the curves traced
out by the two carried points G and P.
We may notice a few particular cases.
If we suppose P to lie on AB^ and to divide it in the
ratio c : c, we have z^zOj oj : y : 1 = c : c : c + c', and the
formula reduces to
i.e. we have Holditch's theorem, as given in Williamson's
Integral Calculus^ p. 200.
If A and B be made to move on the same curve (or on
curves of equal area) and if C then move on the same curve
(or on one of equal area), then
(P) = [A) + IT (square of tangent from P to circle ABO).
If A and B move on fixed circles of radii r, $ the loci of
C and P are three-bar curves, whose areas are connected
by the equation
{P) =^ z [G) + TT {xr* + ys^ — a*yz - V^zx - cVy),
and in the particular case when r = by s = aj
{P)=^z{G)-\-ir{li^x-\-a^y''d^yZ''VzX'-c^xy)
^z{G)'\^ir.Pa'.
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, November 8th, 1877.— Lord Rayleigh, F.B.S., President^ in the
chair. — The following were elected to form the Council during the Session: —
President: Lord Rayleigh, F.R.3. Vice-Presidents: Prof. J. Clerk Maxwell,
F.R.S., Mr. 0. W. Merrifield, F.R.S., Prof. H. J. S. Smith, P.R.S. Treasurer:
Mr. S., Roberts, MA. Hon, Secretaries: Messrs. M. Jenkins, M.A., and R.
Tucker^ M.A. Other members, Prof. Cayley, F.R.S., Mr. T. Cotterill, M.A., Mr.
J. W. L. Glaisher, F.R.S., Mr. H. Hart, M.A., Dr. Henrici, F.R.S., Dr. Hirst,
F.R.S., Mr. Eempe, B.A., Dr. Spottiswoode, F.R.S., Mr. J. J. Walker, M.A.
The Rey. W. ElUs, B.A., Cains College, Cambridge, was proposed for election.
ax + b
Prof. Cayley made two communications, on the function <[> (») = ^ , . and on
cx + a
the theta functions. In the first of these papers the value obtained for the nth
function is substantially of the same form as that found long ago by Babbage,
but is more compendiously expressed ; the result is
, ^ , , {\^^ -l){ax-hb) + (\" -\){^dx + b)
*P W - ^^n,i ^i>i^cx + d) + (\»» - \) {cx - a) '
128 TBANSACTIONS OF SOCIETIES.
where — ^^-^ = ^_ , . It wai amved at in a simple maimer by means of the
identity
Jf — a, 6
e
'* ^_ ^ = 0, or i/« - (a + d) if + (arf - *c) = 0,
ia,
^1
h
d
The second eommunication consisted of an account of researches npon the
donble theta functions, on which Prof. Cayley is engaged ; as an introduction, he
establishes in a strictly analogous manner the theory of the single theta functions.
Mr. Tucker read a portion of a paper by Mr. Hugh Mac Coll (communicated by
Prof. Crofton, F.R.S.^ entitled "The Calculus of Equivalent Statements.'' A short
account of this ansdytical method has been given in the July and November
numbers (1877) of the Educational TimeSf under the name of Symbohcal Languap;e.
The chief use at present made of it is to determine the new limits of integration
when we change the order of integration or the variables in a multiple integral,
and also to determine the limits of integration in questions relating to probabSity.
This object, the writer asserts, it will accomplish with perfect certainty, and by
a process almost as simple and mechanical as the ordinary operations of elementary
algebra. — The President read a paper on Progressive Waves. It has often been
remarked that when a group of waves advance into still water the velocity of
the group is less than that of the individual waves of which it is composed ; the
waves appear to advance through the group, dying away as they approach its
anterior Umit. This phenomenon seems to have been first explained by Prof.
Stokes, who regarded the group as formed by the superposition of two infinite
trains of waves of equal amplitudes and of nearly equal wave-len^hs advancing
in the same direction. The writer's attention was called to the subject about two
^ears since by Mr. Froude, and the same explanation then occurred to him
mdependently. In his work on " The Theory of Sound" (§ 191), he has considered
the question more generally. In a paper read at the Pfymouth meeting of the
British Association (afterwards printed in Nature), Prof. Osborne Rejmoids gave
a dynamical explanation of the fact that a group of deep-water waves advances
with only half the rapidity of the individual waves. Another phenomenon (also
mentioned to the author by Mr. Froude) was also discussed as admitting of a
similar explanation to that given in the present paper. A steam launch moving
quickly through the water is accompanied by a peculiar system of diverging
waves, of which the most striking feature is the obliquity of the line containing
the greatest elevation of successive waves to the wave-fronts. This wave-pattern
may be explained by the superposition of two (or more) infinite trains of waves, of
of slightly differing wave-lengths, whose direction and velocity of propagation are
so rdatea in each case that there is no change of position relatively to the boat.
The mode of composition will be best understood by drawing on paper two seta
of parallel and equidistant lines, subject to the above conditions, to represent the
crests of the component trains. In the case of two trains of slightly different
wave-lengths, it may be proved that the tangent of the angle between the line of
maxima and the wave-fronts is half the tangent of the angle between the wave-
fronts and the boat's course. — Prof. Clifford, F.B.S., communicated three notes.
(1) On the triple generation of three-bar curves. If one of the three-bar systems
u a crossed rhomboid, the other two are kites. This follows from the known fact
that the path of the moving point in both these cases is the inverse of a conic.
But it is also intuitivdy obvious as soon as the figure is drawn, and thus supplies
an elementary proof tnat the path is the inverse of a conic in the case of a kite,
which is not otherwise easy to get. (2) On the mass-centre of an octahedron.
The construction was suggested by Dr. Sylvester's construction for the mass-centre
of a tetn^edral frustum. {$) On vortex-motion. The problem solved by Stokes
may, as a general question of analysis, be stated as follows : Given the expansion
ana the rotation at every point of a moving substance, it is required to nnd the
velocity at every point. The solution was exhibited in a very simple form.
R. TUCKER, M.A., Eon, Sec,
( 129 )
ON EQUIVALENT LENSES.
By It, Pendlebury, M.A*
A LENS is said to be equiyalent to a system of n lenses
on the same axis, when bemg placed in the position of the
first lens it produces the same deviation on a given ray as
the system. The general formula for the focal length of a
lens equivalent to n lenses of focal lengths /j, ./^•••^j ai^d
separated by distances a^, a^.,,a^^_^ can be readily found.
A ray passing through a lens at a distance y from the axis
suffers a deviation ^ from the axis. Suppose a ray parallel
to the axis to be incident on the first lens of the system at
a distance y^ from the axis. Let S^ be the deviation after
passing through the first lens, ^, the distance from the
axis at which the ray cuts the second lens, 8, its deviation
after the second refraction, and so on. Then we get the fol-
lowing system of equations f writing k^^ \. . .4^ for -tt , 7 • • •'/ J i
&c.
Forming the continued fraction
1 + A, + o, + A, + o, +...+ A;„ '
it is clear that the numerator of the last convergent is S,.
But if F^ be the focal length of the equivalent lens ^=8^^
Hence -^ is. equal to the numerator of the last convergent
to the continued fraction
1 1 i 1 1
1 + A, + a, + *,+...+ *„'
VOL. VII. K
If Ae r%j, imtead <^ boa^ paraUrl to the axts before meeting
the fiiat lens, cats the axis st k ^"* ■■- ■* d from that lensj
vegct thsfixmal»
T"*- 5= """""*"' ^ IT i^ + i:+...+ ^ where fl = 5.
T&a expnsiua aMiT be fllMamed in tbe slupe of ft de-
tezmuaju. Su^iiii)^ aw xys^ A* Eaor etpiatioiis above,
^- ij-1 , « , « ,-- 0, (a»-l) tomiB.
l,-«^-l , « ,_ «, o
0. » . 1 ,-«_.- 0,
.-».
«J + i + i^i
/././/.•
Tea to oonnect die Talaes of two
fine them. Writing ^, fiw -p,
MR. NIYEN, ON SPHERICAL HARMONICS. 131
Thus, for example^
-^ = (1 + ajc;) [Jc^^Jc^^h^-^aJc^ (Z:,-f A;J +a,i, ( V^s) + ^A^iW
1 1 1 1 a, /I 1 1\ /I 1\ /I 1\
yiyi vi .7*^ JW9J3J4, /
ON SPHEEICAL HARMONICS.
By IF. D. Niven^ M,A,f Trinity CoUegei Cambridge.
§ 1. If F be the values at the point a?, y^ z oi ^ fuoction
satisfying Laplace's equation, and if V^ be the value at the
origin of coordinates, then
d d d
7=e ^ ^y *r, (1),
„ ./dV\ (dV\ , (dV\ , . ,„,
It was pointed out in a former paper in this Journal, that the
peculiarity of this expansion is, tnat any set of homogeneous
terms in a*, y, z^ say the (t + 1)^, viz.
1 Z' ^ ^ . ^Vtt- /on
of itself satisfies Laplace's equation. The expression (3) is
therefore a solid harmonic of the i}^ degree. Moreover,
since all the diflferentiations of the i^^ degree are involved in
it, and there are 2t + 1 independent ones, we may regard (3)
as a compendious form of the aforesaid harmonic of the most
general character.
§2. We shall now employ this form in the proof of
several theorems, and as a preliminary step we shall find the
value of
jj^ax+by^zjjg ^^^^
K2
132 MR. KIYEN, ON SPHERICAL HARMONICS.
the integrations being taken over the surface of a sphere
whose radius is B and centre the origin. We may clearly in
that case, by change of axes, throw the integral into the form
/:
-22
the value of which is
or 47ri?|l+ r^ i?(a«+J*+c«) +...+ -^^ i?'(a*+J*+c?^)'+...|
(5).
§3. The integral
taken over the sphere is thus seen to be
4irJPF;.
§ 4. Let F' be a second expression similar to F, and let
us put
<r d* rf[
Then jjVV'dS^SSe ^ ^ dy-^' ^ y/ dx ^^ dy +' .& y^^^g
^Jj/Kdx'^ dxj'^^Kdy'^ dy)'^^\dg'^ dz) y y'djg
where the differentiators j" > ^ j j" operate on F only,
If we assume that the integral over the sphere of the
product of two harmonics of unequal degree is zero, the two
expressions for JJVV'dS lead by expansions of the expo-
nential respectively to
//-
Li
MB. NITEN, OK SPHEBICAL HABMONICS. 133
and,8mce [^ + ^) + (^ + ^ j + U + ^ J
is really equivalent to
\dx dx dy dy dz dz) ^
by §5, to
^^^^ f2V+l ^[d^d^^'dy'dy'^^d^ ^o^o''
Since the corresponding terms of the same dimensions in B
must be equal^ we must have
d d d\\, f d c? . i'Vyr/
"" Tx^yTy-^'dzr^ \''d^-^ydry'''dzr^ ^^
G Li
| 2t + l \dx dx"^ dy dy"^ dz dz) "" ''^
§5. The simplest example of this theorem will be its
application in finaing fJQ*dSj where Qi is a zonal harmonic
having its pole in the axis of z. We shall suppose each of
the expressions inside the integral to be equal to Q^^ that is^
we shall take
//.
/ rf , c? , d\*
V'^'^yTy'-'dzJ^o
dy
Li
|2t |2t-2
and
in which it will be observed we have taken in the first case
one form of the solid zonal harmonic, and in the second case
the other. For this reason ; — On looking at the second side
of the general formula of §4 we see that the operator on V
differs from (-^^ +y ^ + ^Yz) ""^^ "^ ^"^^^^ ^d^dz
1
1B4 MR. NIYEN, ON SPHEBICAX HABMONIGEL
in place of x^ y^ z. Now. if in the first of the above har-
monics we replace x^ y^ z dj these operators on V^ we get
all the terms excepting the first disappearing by virtue of
Laplace's equation, and now the advantage of expressing
V in terms of the second form of the zonal harmonic be-
comes apparent, for all the terms disappear by differentiation
except the first, and the last written result becomes
]?A
Betuming to the general formula of § 4, we finally obtain
r
II
qm^
2i+l*
8 6. Exactly similar reasoning applies in the case of the
solid harmonics derived from the tesseral and sectorial forms,
which are
2'-L!|i^ ^^r - 2(2.'-l) * («'+y +.)+...}.
These expaoBtons are the values in different forms of
f* Y'y where
Since f = as +jy and 17 = a; - jy, we see that, according to
the explanation of the last article, f , 17 will become respec-
*''''y Ix +-^' I; *"'* L'-^'^ •'P"**"^ on F; that is,
2^ 2^
Hence (A.^ + A£ + ^£.\vr'
\dx dx dy dy dz dz) ** ^
MB. NITEN| OK SPHERICAL HARMONICS. 135
d d' d_£ . d d'V.
dydy
becomes in this case
Hence /J(r.')W= ^^ ^W^' '
§7. The work in §4 enables us to obtain easily the
value of
IIQiY.d8,
where F.-is anj surface harmonic, viz. it is
4ir5" I /d
4-n-if l_ (d\ ,y.
2t + 1 [t \dz) *" "
which is obviously the same as
iirB} 1 rf*
2t4 1 I » dh^dh^...dhi
r'Qi,
that is
47r^
2t + l^^'*^'
where ( Y^ is the value of Y^ at the pole of the Q harmonics.
§8. Since the determination of any harmonic depends
upon successive differentiations of ~, it is desirable that
methods should be devised for obtaining those differentiations
easily. Accordingly, in my former paper in the Messenger
on tnis subject, I showed how those aifferentiations, instead
of being made upon - could be made upon r Q,., where the
pole of the zonal harmonic Q^ is the point on the sphere at
which we want the general harmonic, the relation being
in fact
Now if (a, /8) be the polar coordinates of the said pole, Q^
is a function of /l6, the cosine of the distance of another point
136 Mfi. BUSSELL, ON HIGHER TRANSCENDENTS
(5, <^) on the sphere measured from (a, )8), that is, Q^ is a
function of cosa cos + sina sin cos (0 — ^8).
The substitution of this value of /i in Qi would be ob-
viously a very laborious method of determining Q^ in a
form suitable for the application of the theorem. We may,
however, expand Q^ in terms of 2t+ 1 selected tesseral ana
sectorial harmonics, as is done by Thomson and Tait, or more
simply by Ferrers {Spherical Harmonics^ p. 88). This gives,
in the notation here employed.
i+a 1 1 — (T
where r'P^ is now the second of the expressions in §5,
r*Y.<^ is the second of those in §6, and r*Z.<^ is the same as
rY/^ if we put - V(— 1) {^*^-V*^) instead of ^^ + r)*^ in the
expansion of the latter. The brackets indicate that the
values of a, 13 are to be put instead of ^, (f> in the expressions
P, Yj Z, It is obvious any differentiations upon Q^ can
now be easily determined. In other words, we have any
assigned harmonic expressed in terms of the selected tesseral
and sectorial harmonics.
ON THE OCCURRENCE OF THE HIGHER
TRANSCENDENTS IN CERTAIN
MECHANICAL PROBLEMS.
By W. H. X. Russell, FM,S.
(Continued from p. 21).
(4) A FRUSTRUM of a paraboloid of revolution rolls on a
rough horizontal plane, determine the motion when the
centre of gravity of the paraboloid coincides with the focus.
The motion is supposed throughout the investigations
in this paper (with the exception of the first) to take place
in parallel planes. Let then the line of intersection of the
plane, in which [S) the focus of the paraboloid moves with
the horizontal plane be taken for the axis of [x\ A a point
in it for the origin; let also F be the point of contact of
paraboloid and horizontal plane, and B the vertex of the
paraboloid. Let AN=^x smd NS' = i/ be the coordinates of
S : BSN= N8P= 6^ the angle which the axis of the para-
boloid makes with the vertical, also let 8B=a^ 8P=r^ then
IN CERTAIN MECHANICAL PBOBLEMS. 137
the eqnatioDS which determine the motion of the body are
as follows :
de " M' df^ ^'^M^
d'O _ Fr COS0 Pr sing
de " Jf A^ Mk* '
when F and P are the friction and pressure : the geometrical
conditions are thus found.
Let us suppose for an Instant the point P fixed while the
axis of the paraboloid Is rolling from an angle (0) with the
axis of [x) to an angle + d0^ then 8 describes a small space
perpendicular to 8P and equal to rd0^ the resolved part of
this In the direction of the axis of [x) Is rd0 cos 0] hence,
dx CL
the geometrical conditions are -^ = rcosg, y= — ^; com-
bining these equations with the equations of motion we
obtain the equation of via viva^ which proves the correctness
of our reasoning. Substituting in the equation of vis viva^
putting u = cos0y we find
e=-
V(2^)
f, f oN- W I*
n\{l^u')u'{cu^a)] •
(5) A sphere rolls down a cylinder whose base placed in a
vertical position is a parabola, to determine the motion.
Let BP be a vertical section of the parabolic cylinder
passing the centre of the sphere (7, B the vertex of the
parabola, BA the axis of the parabola supposed horizontal,
A the focus, P the point of contact of sphere and cylinder,
rthe point of Intersection of CP and BA, Take A for the
origin, AB for the axis of («), and let ANj NP be the
coordinates of P. Also let GTA=^0, AB=^a^ CP=^r^ then
AN=2a ^, -NP=2atang. Now let (x) and (y) be
cos f/ \ / u/ /
the coordinates of (7, ^ the angle through which any radius
of the sphere rolls, then* the equations of motion are
d'x _ Fsin0 .Bcosg
d? W^ M ^
d'j/ Fcos0 BBm0
df M '^ M ^'
d^<t> _ Ft
^ df " Mk''
138 MR. RUSSELL, ON HIGHER TRANSCENDENTS, &C.
where F and R have their usual significations} also we have
dd rj^cos'^r
To explain this equation we remark that it is well known
that if a sphere (radius p) roll upon a sphere (radius p), and
if Q be the angle through which any radius p rolls, while the
line joining the centres of the two spheres describes an
angle 5, then ^ = , Q. In the present case instead of p
r
we make use of the radius of curvature of the parabola at
the point of contact, we take the angles infinitely small, and
since is measured in a direction contrary to that of the
rolling sphere we take dd negative.
The other equations of condition are
x=sr cos^ + 2a —
cos»5 '
y=^r sind+2a tantf.
In order that the equation of vis viva may hold, we
mast have
• adx . /%dy ^ dd> ^
these equations are easily seen to be identically true, and we
are therefore able to write down the equation for vis viva
and we obtain finally
^(7^+^*) r (rcos»tf + 2a)c?tf
e=
/;
s/i^lg) J r cos'5 ^{c -r sin 6 -'2a tan 0) *
which may be reduced to an algebraical form by putting
w = tani5.
(6) A heavy rod fastened to a hinge at one extremity
presses on a semi-elliptic cylinder moving on a smooth
horizontal plane containing the hinge, to determine the motion.
As the rod descends, we suppose the flat surface of the
semi-elliptic cylinder, made by a plane passing through the
M. LUCAS, OK EULERIAN NUMBEBS. 139
axis of the elliptic cylinder so as to cut off the greatest area,
to ^lide along the horizontal plane. Let C be the centre
of the ellipse, S'the hinge, HP the beam pressing at P upon
the semi-cylinder. Let HC=x^ 6 be the angle which the
normal at P makes with the major axis, 2r the length of
the beam to the plane, and the equations of motion are
d^ _ ffrsing _ ^ itang
cU'x P ^
-— = — cose/,
df m ^
where
_ J* sin tan a cos^ a V(l - c" sin'tf)
^"" a V(l-i" sin''^) ■*" V(l-i"8in''5) *" cos^ '
r ^^ dx a(l — e*sin5)
from whence ^ = ^3.g^(i_,.,i^,g) ;
by aid of this equation we immediately deduce the equation
of vis viva from the equations of motion, and we obtain finally
'1 Cd0 { m¥ cos*g (1 -6* sin'g)-f inV (1 -e')' sin^g ] *
V(27w^r)jcos'g| (l-e*sin*g)(cosa-cosg) j *
which may be reduced to an algebraical form by putting
cosd = u.
(To be continued.)
ON EULERIAN NUMBERS.
By M, Edouard Luca$»
1. If we put
secaj = 1 + Q^x* + a^cc* + a^x^ +... &c.,
we have, on multiplying the left-hand side of this equation
by coso;, and the right-hand side by the series for cosa?, the
relation
(z CI a 1
«.« 21 ^ 41 •••*(2n-2)!*(2w)!""^*
From this relation Mr. Glaisber has deduced an expressioq for
a^ as a determinant of the n^^ order {Messenger ^ vol. Vi., p. 52).
140
M. LUCAS, ON EULERIAN NUMBESS.
The Eulerian numbers are, In absolute valuei given by
the formula
-Ew = (-ir(2n)la^.
We thus haye, changing x into aciy the symbolic formula
Sx
e +6
= 6
in the development of which the exponents of E are to be
replaced by suffixes, and E^ by unity. Getting rid of the
denominators, we find, for n positive, the recurring relation
(^+ !)•+ {E- 1)* =
leading to the determinant
(1),
^«.=(-i)'
1,
1,
0,
0,
0,...
1,
6,
1.
0,
0,...
I,
15,
15,
I,
0,...
1,
• ••
28,
70,
28,
1,...
•••••••
(n rows)*.
This determinant is formed of lines of even rank and of
columns of uneven rank of the arithmetical triangle.
We have also the symbolic formula
2(-r + 3'*-5" + 7"+.,.+ (4aj-l)*} = (4a5 + J?)*--Er;
and, in addition, the formulae
I** 3««'^5«" 7«"'^"—- 2"+»(2n)l '
L
X dx
E.
e'^ + e"" * 2
9%
2. Eulerian numbers are integers and they are uneven.
Sherk has demonstrated that they end alternately in the
figures 1 and 5. These properties can be proved as follows:
We deduce from the relation (I) for p prime the congruence
^1,-1 + J^p_,+ ^,..5+...+ ^8 4 J5;= 0, (mod. J?) ;
whence, denoting by A^ the sum of the first p Eulerian
numbers taken with their proper signs,
-4f-, = 0, (mod. p).
* This value of ^2« ^ & determinant was given by Mr. Hammond in his paper
'On^'the relation between Bernoulli's numl^rs and the Binomial coefficients,'
Proceedings of the London Mathematieal Society, toL tii., p. 18, (1875) .—Ed.
M. LUCAS, ON EULEBIAN NUUBEBS.
The first values are given by the formulas
E^-\- 15^^+ 15^, + jr, = o,
141
whence, starting from ^y_,.
(mod. J}).
The comparison of these two systems of formulas giyes
Buccessively
\ (mod. p).
We have, in general,
Kn = ^2»+Kp-i)j (mod. J}),
whatever value the positive integer k may have, and con-
sequently :
Theorem. The residues of the Eulerian numbers, for any
prime modulus whatever, reproduce themselves periodically
m the same order, just as the residues of powers.
These considerations are applicable, in general, to the
differential coefficients of a rational fraction of e'^ but under
certain conditions, as in the case of
When ^(1) is zero, as in the development of- 5, the
X ^ e
theorem does not hold; the differential coefficients are no
longer integers and contain in the denominators an indefinite
series of prime numbers; it is so, for example, with the
Bernoullian numbers.
Fariiy November 1 1877.
( 142 )
AN EXTENSION OF ARBOGAST'S METHOD OF
DERIVATIONS.
By J, J, Thomionf Trinity College, Cambridge.
By Arbogast's method if we have a series
cx^ dot?
V as a + 6ar-f To + Tq + *'^*
We are able to calculate the successive coefficients in the
expansion of any function of v in ascending powers of x.
The rule is, each coefficient is derived from the preceding by
differentiating it with respect to z and putting
da J db ^ J p
dx
dx ' dx
in the result. The object of the following paper is to
demonstrate a similar rule when there are any number of
variables in the series to be expanded.
Lot w = c^j + c^jc + c^ +
'««
[2
L!L
+ &C.,
it is required to find the coefficients in the expansion of <f>{u)
in ascending powers of aj, y.
We notice that when x and y both = 0,
dV" fd\
\dx) \jy) "'"''^^ny (!)•
Let ^M = -^(«,y)«
By Maclaurin's theorem we have
^(w)=^(0,0) + aj
dF
dx
+y
dF
dy
+ 2xy
Hk^
the coefficient of
n{n-l)...{n-m+l) a"-"y" .
d'F
<£F
d'Fl
dxdyl'^^ |rfy
[ + &c.,
[m
L?
18
^F
cbT'^dy
IxaO^V^k
MR. THOMSON, ON ARBOGAST'S DERIVATIONS. 143
Let us denote
Wr«,,„;thenby (1) w,«,^ = c,^,^
u when x and y are both zero by
If we assume
^^(^fW,*y)"^(r4-0«,
dx
tv
(2),
da;"-"eZy'
a>^f»*0
will evidently
because
O (O
,f»-m 7 « >
l«U.-=<'.*°*
but
«-0,|f*0
is the coefficient of — -^ -
m
«tM),|f*0
'"^'y=Sk^^ ^'^'
[m
— j-^ in the expansion of ^ (m) ; hence the coefficient of
91 iTl"??! I (fTl ^ TItt "4" X) •<)? 1/
— ^ '-^ ^ — j-^ is got by differentiating ^ (c^) - w
times with respect to x and m times with respect to y,
remembering the conventions (2).
An example may make this clearer : — suppose the coef-
ficient of 2xy in the expansion of sin u is required.
By the theorem it = j-r
d sin Cg
dy
[2 dxdy ^
dx
dc.
cosco^'=cosc..c^,
dc^ dc
dx
•J- (cosc,0=-sinc.;Tfc +COSC,
dx ^
= - sin c^.c^.c^H- cos c,.c^;
therefore the coefficient of 2xy in the expansion of sin u
= j-g {^^8 ^0- c^n, - sin c, .c.c^l.
It is evident that a similar theorem will hold for any
number of variables.
( 144 )
ON A FORMULA IN ELLIPTIC FUNCTIONS.
By J. W. L. QlaUher.
, 1. Fbou the equation
@[x+y)@(x-y)= ^^-^ (1 - i' sn'* sn*y),
we have
02 (m4 v) 02 (a-») = Q'^"®*^^ (i _ A« 8n»2w sn*2v)...(l),
02(« + t,)=®^|±i:) {l-*'8n*(tt + i;)},
02 (« - 1,) = ®^^ [1 - A« sn* (M - 1;)},
02«-g^(l-A?sn««), 0(2»)=^(l-i»8nH
whence substituting for ©2(m + v), 02(w — v), 02m, 02t;
their values in (1), we have the formula
l-A'sn*2Msn'2v
{l-A'sn*(w+t?)}{l-A' sn*(M-t?)} ,^ ,, , ,.4 .^.
= ^ 7i — 7^^ 4 N2/1 — 7)1 4 ,u — - (l-i"sn'wsn't?)\..(2J.
(1 — A sn w) (1 — kr sn v) ^ / n /
This equation may also be obtained by a double application
of the formula
^ ts «/ N «A X (1 - A* sn*w) (1 - A* sn*t?)
l-A; Bn'(» + t>)8n'(«-t>) = ^ (!-&» ,n»„ bii««)' >
which is readily proved.
2. Writing
/(w, v) = 1 — A" sn' w sn'v, ^m = 1 — A" sn*w,
we see from (2) that
^2 (u+ 1;) ^* («+ 1>) . «^2 (m - V) <^* (m - «) ,„ , ,
(^2M^*M.^2t)^*t;) -^ ^' "
/(8«,8t;) = &c. = { l/'^Kv),
kfi. WHITWORTH, SUB-FACTORIAL N.
145
Thus, starting with/(w, v), and expressing it successively in
terms of/(iw, ^^)jf{i^i iv)j •••» ani observing that when n is
infinite,
we obtain the theorem, that if the infinite product
(1 -&« sn*u) (1 - ;fc« sn*iu)* (I - A« sn^")'* (1 - &' sn^iu)**...
be denoted by x (^)j ^^^^
SUB-FACTORIAL JV;
By IF. ^/fen Whitworth.
f
1. A NEW symbol in algebra is only half a benefit unless
it have a new name. We believe that the symbol In as an
abbreviation of the continued product of the first n integers,
was lon^ in use before the name factorial n was adopted.
But until it received its name it appealed only to the eye
and not to the ear, and in reading aloud could only be
described by a periphrasis.
In introducing the symbol ||72, we propose at once to call
it suh'factorial w.
2. The close alliance of In and \\n will be immediately
seen from the following, which may be taken as the defini-
tion of the sub-factorials. It will be observed, that the
omission of the words in italics will give us the factorials.
Write down 1, and subtract 1 ; the result is ||1.
Multiply by 2, and add 1 ; the result is ||2.
Multiply by 3, andsuhtract 1 ; the result is ||3.
Multiply by 4, and add 1 ; the result is
Multiply by 5, and subtract 1 ; the result is
Multiply by 6, and add 1 ;, the result is
And so on.
VOL. VII. L
4.
5.
6.
146 MB. WHITWORTH, SUB-FACTORIAL K
And generally [fn may be obtained from |[n-l by multiply-
ing by n and adding (- 1)*.
3. We have ljw = n ||n-l±l,
and ||n~Jl = (w - 1) jjn-2 T 1.
Therefore, by addition,
\\n + ||n-lr= ijn-J + (n - 1) |[n - 2 ,
or l!n = (w-l)(|K-1+|jn--2),
which shews that |jn m a?w;aya divisible % n - 1.
4. Again, we have
Divide by [^w, and we get
lln ^ |[n-rl (« i)«
[n ""[^31+ "[^»
Similarly t^ Jt^ ^VT
[w-1 [ n-2 ^ [ n-1 »
l|n-2 ||w--3 (-!)"-«
|n-2 "^ n-3 [n-2 *
L
and so on,
FinaUy, 1 = ^+1
and
^
LE LI LE'
Li Li'
Therefore, by addition,
6. If « be the base of Napierian logarithms we have bv the
exponential theorem
1-111-
^ = 1 -- -- + _ . &c..„to infinity.
MR. WHITWORTH, SUB-FACTORIAL N.
147
\\n 1
Therefore, if w be even, — > - 5
if n be odd
||n
1
and if n be increased indefinitely
,\n I
limit = = ~ .
6. The rapidity with which the ratio of the factorial to the
flub-factorial converges to equality with e is seen by a glance at
the following table, in which the values of the first twelve
factorials and sub-factorials are registered, together with the
quotients [n -=- \\n and J|n -^ f n to seven places of decimals.
n In-j- n
n
n ||»n- [n 1 w
1
CO
1
1
2
2-0000000
2
1
0-5000000
2
3
3-0000000
6
2
0-3333333
3
4
2-6666666
24
9
0-3750000
4
5
2-7272727
120
44
0-3666666
5
6
2-7169811
720
265
0-3680555
6
7
2-7184466
5040
1854
0-3678571
7
8
2-7182623
40320
14833
0-3678819
8
9
2-7182836
362880
133496
0-3678791
9
10
2-7182816
3628800
1334961
0-3678794
10
11
2-7182818
39916800
14684570
0-3678794
11
12
2-7182818
479001600
176214841
0-3678794
12
• ••
CO
2-7182818
00
• • •
eo
0-3678794
00
7. Sub-factorials chiefly occur in connexion with permu-
tations.
For example, if n terms are to be arranged in order, this
can be done in \n ways. If they are again to be arranged in
order so that no term shall be where it was iefore^ this can be
done in \\n ways {Choice and Chance^ Prop, xxxil). If they
are to be arranged so that no term ma^^ be followed by the
term which originally followed it, ' this can be done in
n + ||n — 1 ways {Choice and Chance^ Prop, xxxili).
l2
( 148 )
ON THE PORISM OF THE RING OF CIRCLES
TOUCHING TWO CIRCLES.
By H. M. Taylor, M.A.
It is easily seen that the radius of a circle which touches
two concentric circles of radii a and fi is ^ (a — )3), and
that the distance of its centre from the common centre
of the two circles is J (a + jS). It follows therefore that if
^ = sin - , a complete ring of tangent circles could be
described between the two circles, each tangent circle touching
those on either side of it (fig. 9).
Now by inversion we can deduce the condition that such
a ring of tangent circles can be described in the case of a
pair of non-concentric circles; and it follows, at once, that
if one such ring is possible, an infinite number of such rings
are possible, or, in fact, that we can begin a ring with any
tangent circle.
Now let us assume that two concentric circles of radii oty
fi satisfying the relation a — /3=(a + i8) sin- , be inverted
with respect to a pole distant d from the common centre ;
then if A be the constant of inversion, and the radii of the
inverse circles be a and i, and c be the distance between
their centres ; and if the line through through the centres
of these circles eut them in 2>, (7, JB^ A (figs. 10 and 11); then
0A = ■^^ = x-^2a (1),
IS
05=^— g = aj + a-c + J (2),
00=j^'>'x + a-c-h (3),
^^=aT^=- W'
, a— )8 . 7r
also — --3 = sm — ,
1 + sm —
^w Aft
or 3 = «m suppose (6).
1-sin-
n
5=J -w,
MB. H. M. TAYLOR, ON THE RING OF CIRCLES^ &C. 149
To find the required relation we must eliminate from
these equations the 5 quantities A, d^ a, 13 ^ Xj which are
only equivalent to 4 quantities.
In fact we may put h^l and then eliminate the re-
maining 4 quantities.
Ffom (1) and (4)
rfCv^"* ^^^»
rfrr^«=^+« (^);
from (2) and (3)
d'^/S
^—^x^a-'C ...(9);
fix)m (7) and (9)
d d _
Therefore by substituting from (6) and (8)
a b ^ c
or c=5-(a — w5) (10),
similarly ""^fii^^^) (^^)-
Therefore from (6), (8), (10), and (11)
nth ^ g i?'^{a^mhY
a <?nf? — (a - mi)'
ftu? [a - hm) == [am - 6) (a — Jm)*,
or, since a is not equal to &m,
fiic'=:(a- Jm) (am — i) (12).
The only remark which it is necessary to make to
or — = m -i— ^ — -, Y\ty
150 MR. ELLIOTT, A THEOREM IN AREAS, &C.
complete the proof that this relation between the radii and
the distance between the centres of two circles is suflScient
for the existence of a ring of tangent circles is that any two
non-intersecting circles can always be inverted into two
concentric circles, if we take either of two particular points
for the pole of inversion. The distances of these points
from the centre of the circle whose radius is b are the roots
of the equation
which are always real, if the given circles do not intersect.
In fig. 9, 91 = 6 and a = 3)3. Figs. 10 and 11 are obtained
by inverting fig. 9 with respect to a pole, where rf = 5/8;
whence it follows that AB : BG : CD^Q : 2 : 1.
It may be added that, if we write w — for - in the above
. n n
equations, we obtain the condition for the existence of a
ring of n circles encircling the inner of the two given ones
m times.
A THEOREM IN AREAS INCLUDING
HOLDITCH'S, WITH ITS ANALOGUE IN THREE
DIMENSIONS.
By ^. B. Elliott, M,A. Queen's College, Oxford.
I. Holditch's theorem in the extended form given in
Williamson's Calculus establishes a relation connecting the
areas of the closed curves traced by three given points in a
rod of constant length as that rod moves in one plane through
any cycle of positions back to its original one.* It is easy,
as follows, to connect the areas passed round by three tracing
points in a varying straight line in the more general case,
when, instead of being at fixed distances, they are only at
distances whose ratios remain constant, and which return to
their initial lengths and positions after a complete cycle. In
other words, we may replace the rod by a uniform elastic
string.
Let [x^y^ {^Jf^ ^^ ^^ coordinates referred to rectangular
axes of two pomts A^B in their plane, and let {xy) be those
* The limitBtion that in thia motion it must hftye rotated through an angle 2^
is unnecessary.
ME. ELLIOTT, A THEOREM IN AREAS, &C. 151
of G which divides AB m the constant ratio m\ n i\t may, of
course, be either internal or external to AB)\ ana give the
positions a slight change in the plane of reference. Then
= {my^ + ny^ [mdx^ + ndx^
= rn^y^dx^ + n*y,dx^ + mn [y^dx^ + y^dx^
= wi (w + w) y^dx^ + » (w + n) y^dx^ -mn{%/^- y,) d[x^ - ajj
(1).
Now suppose A to travel all round the perimeter of a
closed area {A)^ and B simultaneously all round that of
another closed area (-B), the two motions being quite inde-
Eendent and subject to no restrictions whatever, except that
oth be continuous, having no abrupt passage from one
position to another finitely diflfering from it. C will then
also travel simultaneously and continuously all round the
perimeter of another closed area, which call [C). Integrating
over a complete circuit we have then
fy^dx, = [A), fy^dx, = (5), fydx » ( C).
Also J /(y,— yi)rf(aj, — ajj equals the area swept out by -45
relatively to A^ that is to say, the area enclosea by the path
of a point always situated with regard to a fixed point, just
as B is with regard to A. Call this relative area 8\ then it
follows from (1) that
(wi + n)' ((7) =s w (m + n) [B) + w («i + n) {A)^mn8^
a result which may be stated thus. Through any fixed point
in the plane of a closed area 8 let radii vectores be aravm
to all points of its perimeter ; and let chords ABj parallel
and equal to these radii vectores^ be placed with one extremity
A in each case in the perimeter of a closed area (-4), and the
other B on that of another [B). The perimeter must be such
that the points AjBso placed pass all round them respectively,
and do not in either case return to their first positions from
the same side as that towards which they left them. If in
either case, that of B say, this is done, the area {B) must be
replaced by zero. Then {C) being the area enchsed by the
trace of a pointy alioays dividing ABin the constant ratio m : n,
{A)j {B)y (C) are connected by the formula (2).
J
152 MR. ELLIOTT, A THEOREM IN AREAS, &C.
Areas described In opposite senses of rotation must of
course be taken as of opposite signs.
This relation may be expressed symmetrically in terms of
any position, by writing m : n : m + n ^ AC : CB : AB.
Doing this it becomes
which, paying attention to sign, may be written
BC{A)-^CA[B)-\-AB[C)^^BC.CA.AB^ (3),
which is entirely symmetrical, -j-^ being the symbol of no
linear dimensions, which operating on AB^^ BC* or GA^
produces in each case the area swept out by the corresponding
segment relatively to its extremity.
In the special case where AB is of constant length, and
divides it mto two constant parts a, b so that 8 being a
circular area described about its centre is 7r (a + hf ; (2) and
(3) become the known equivalent relations,
^ ' a + 6 1 (4).
BC{A)-\-GA[B)^AB{C) + irBG.CA.AB^e]
Numerous examples are easily deduced from (2) and (3),
, as for instance i If U^ G divide the one internally and the other
externally in the same constant ratio mi n a tine AB whose
extremities move simultaneously and independently all round
the perimeters of two closed areas {A) (-Bj, the areas enclosed
by meir traces satisfy the relation
(w + n)"((7) + (m- w)«((7') = 2 {m« (5) +n«(^)|,
II. Froceedinff now to three dimensions, suppose two
closed surfaces of volumes (-4) [B) respectively. Suppose
also a series of points A to cover the whole surface of {A)^
and a series of corresponding points B to cover the whole
surface of (j5), in such a way that to positions of A at
infinitely small distances all round any specified position
correspond positions of [B) at infinitely small distances all
round the corresponding position, the two systems subject to
this one restriction having any independent laws of distribu-*
MR. ELLIOTT, A THEOBEM IN AHEAS, &C. 153
tion over their respective surfaces whatever.* It is required
to express as simply as possible the volume enclosed by the
locus of a point dividing A Bin any constant ratio m : n.
Taking rectangular axes, call A (a3iy,«j), B {x^ji^ and G
the point dividing AB in the ratio m : n {xyz). Then
(m + n)' zdxdy = [mz^ + nz^ [mdx^ + ndx^ {^dy^ + nc?yj.
The right-hand side of this containing, when expanded,
four distinct sets of terms, cannot be expressed more simply
than as a sum of four volume elements, whereas in the
analogous plane formula (1) three area elements were
sufficient. The equation is then best written
{m + n)' zdxdy = m (m* - n*) z^dx^dt/^ — n (m* — n') z^dx^dy^
- ^(^-w) («.-^i) {dx^-dx,) {dy^'dy,)
4M41
+ -y (m + n) («, + «J {dx^ + dbj (Jy, + JyJ (5).
Now zdxdy is the volume element of {G) the space bounded
by the locus of (7, z^dx^dy^ the volume element of {A) and
z^dxjiy^ that of (-B). Also (;5, — «^) d [x^ - a?,) c? (y, — ^ J is the
volume element of the locus of B relative to A^ that is to say,
of the space which would be included by the locus of a point
always situated with regard to a fixed point precisely as B is
with regard to A ; and ^ [z^ 4 «,) d {x^ + x) d{y^ + y J is the
volume element of the locus of the middle point. of AB.
Writing then [M) for the volume enclosed by the locus of the
middle point and V for the relative volume, it results from
the integration of (5) over a complete series of positions that
— i wiw (w — n) F+ 47wn [m + n) (Jf ),
+ (m + «)' ^^J 2 (». + «)• ^-l^J-
In the special case where AB is of constant length a + &
and C7 diviaes it into two given parts a, &, this becomes, since
* See howeyer the concluding paragraph below.
154 MB. ELLIOTT, A THEOREM IN AREAS, &C.
V^ being now a spherical volmne described about Its centrei
equals ^w (a + 6)',
(^= (^t-^^) -*(^)}+(^«(^) - ^aJ(a-J)...(7).
The relation (6) involving as it does [M) as well as (-4),
(J5) and (C7) is one connecting the volumes corresponding to
four points on a varying straight line, whereas the analogous
relation in a plane (2) considered only three points. It is
of course easy by eliminating of (M) from (6) and the similar
relation connecting [A)^ (5), {M) and (D) the volume
surrounded by the locus of a fourth point 1>, which divides
AB in another constant ratio m' : n\ to obtain the general
relation as to four points on the line, neither of which is of
necessity the middle point of the segment limited by any two
others. This may be conveniently found in a symmetrical
form in terms of any position, as follows.
In any position we have AC : CB : AB=m : n : m + n so
that (6) may be written
A.AO.CB ,^, AG^dB{AC-CB)
"*■ AB' ^^> 2ZAW '^»
(O fAO X (^ fCB X {A)
AC.CB~ \0B J AB' ^ \AC J AB*
In like manner
jD) _ IAD N {B) (DB \ (A)
AD.DB ~ \DB J AB- "^ \AD J AB*
^.^-liAD-m^-
Thus, subtracting, we see that A^ j5, (7, D being any four
points on a varying straight line at distances in constant
ratios to each other throughout the motion, which return
after a complete cycle to their initial positions
{C) {D) AG.DB-AD.GB (B)
AC.CB AD.DB'^ CB.DB ' AB*
AD.CB^AC.DB [A) . ^^
■f A ^ TT^ •~TTm +C/X/.
AG. AD ' AIP^ A^
MB. ELLIOTT, A THEOREM IN AREAS, &C. 155
or since
AC.DB-AD.CB={AD - CD)[CB- CD)- AD.CB
=CD{CD-AD-CB)
= -AB.CD,
JC) {D) (.g)
AC.CB.CD AD.DB.CD~ CB.DB.AB
, {A) V
"^ AC.AD.AB"^ Aff'
which, paying due regard to the signs of the segments, may
be written
M) . {B) ^ [0)
AB.AC.AD ' BA.BC.BD ' GA.CB.CB
(8),
^ DA.DB.DG AB^
V
a relation that is entirely symmetrical^ for if -j^ be denoted
by V, this is of no dimensions in length, so that v.AG%
V. Alf^ V. BG^y &c., represent the volumes swept out rela-
tively to their first written extremity in each case by -4(7,
ADy BGj &c., just as v,AB^ denotes that swept out by AB
relatively to A.
In the special case where ABGD is a rod of constant
length, so that AB^ AGy AD^ BG, &c., are all constant, and
consequently V is the spherical volume ^irAB^^ this becomes
(^) , (^) , JC)
AB.AC.AD "^ BA.BC.BD '*' CA.CB.CD
^ DA^DB.DC ^^-^-'' (')'
the analogue of the second form of the special result (4) just
as (7) is of the first.
In practically applying these formulae to examples, caution
is necessary in order to attach the right sign to each of the
volumes involved. It will have been noticed that, throughout
the above, the volume included by the locus of one point P
relatively to another Q has been considered the negative of
that included by the locus of Q relatively to P. From con-
siderations of this kind, it will be clear that in formuleo (6) to
(9) volumes generated by points of a shifting straight line on
156 MATHEMATICAL NOTES.
one side of the InstaDtaneous centre of rotation in anj of
the possible infinitesimal motions (I, for convenience, use
language strictly applicable only to the special case of the
kinematics of a rod of fixed length) being considered positive,
those whose generating points are on the other side must be
taken as negative.
It thus further appears, that for the formulas to hold
without modification the four generating points A^ B^ (7, M
or A^ Bj (7, 2>, as the case may be, must be so restricted in
position on their straight line, that in every shifting of this
considered, the instantaneous centre lies between the same
two of them : a restriction which has none analogous to it in
the corresponding theorem of plane description first con-
sidered.
MATHEMATICAL NOTES.
An Elliptic Function Identity,
The following equation is true, identically,
sn(a — i8)snasni9 + sn(i8 — 7)sni9sn7-i-sn(7-a)sn78na
+ sn (a- /8) sn (/8- 7) an (7- a) = 0.
M. M. U. Wilkinson,
Note on the Calculus of Functions.
I do not know whether attention has been drawn to the
somewhat obvious fact that in solving functional equations
we may replace the arbitrary constants by symbols of
operation. The operations must not be algebraical, such
as V(«*)» V(l — w*)» tor these are taken into account in solving
the equation; but they may be differentiations (positive or
negative) which do not change when the subject is changed.
For instance, take
a solution of which is
if) {x) == Cx.
This, however, is not a general solution, for ^ includes all
operative symbols which satisfy the distributive law, and in
MATHEMATICAL NOTES. 157
the solution C may be replaced by any integral function of
such symbols. Thus
A being constant, iyj integers, is a solution.
Again, a solution of
is ^ (oj) = C7 logo;.
Here, again, G may be replaced by any integral function
of -7- , rr- , &c For instance
, / % a , c dx
is a solution, since
c d{xy) c dx c dy
Qcy* du x' du y'du*
But C may not involve -7-, an operator which changes
with the subject. There is not a solution, for example,
of the form
*(^) = c^logaj = ^;
for the equation
c _ c c
xy^ X y
is not identically true.
January t 1878.
H. W. Lloyd Tanner.
Arithmetical Note.
The continued product
1.2...W- l.w, =n(n),
cannot be a power of any integer, n being ^eater than 1.
For \ip be a prime IT Ip) cannot be a power since it contains
a prime factor i>, once and only once. For the same reason
n(p + 2) cannot be a power, if jf be less than p. Nor can
158 MATHEMATICAL NOTES.
even n (2p) be a power (square) if between ;> and 2p there
is a prime, ^^7 p\ ^^^ ^^^ ^ [^p) would contain the factory'
once and only once. Hence a necessary condition that 11 (n)
should be a power for some value of n, is that some prime
number^ should be followed by a sequence oip composite
numbers. This is obviously impossible when p is large;
for small values of p, the following sequence of primes will
shew the impossibility, since in it each prime {p) is followed
by another less than 2p,
2, 3, 5, 7, 13) 23, 43, 83, 163, ....
H. W. Llotd Tanner.
January^ 1878.
Note on Arhogasfs Method of Derivations.
It is an injustice to Arbogast to speak of Ilva first method,
as Arbogast's method ;* there is really nothing in this, it is
the straightforward process of expanding
^ f a + Ja5 + T-r^ca?" +... j
by the differentiation of ^u, writing a, &, c, J... in place of
du dru au p . I ^ t It in p
w, -r- J -j-i , ;T-i , &c. or say m place of w, w , w , w , &c.
respectively; thus
^, ^'a.&, i {<l)'a.c+ ^"a.y}, ^ Ua.d + <^"a. he \
\ -|.f'a.2ic + ^'"a.i')
= i {^'a.d+ ^"a.3be + ^"'a.i'}, &c.,
and in subsequent terms the number of additions necessary
for obtaining the numerical coefficients increases with great
rapidity.
That which is specifically Arbogast's method, is his
second method, viz. here the coefficients of the successive
powers of x in the expansion of ^{a-{-hx + cx* + dx*+,..)j
are obtained by the rule of the last and the last but one ; thus
we have
where each numerical coefficient is found directly, without
an addition in any case.
A. Catlet.
* See pp. 142, 148.
( 159 )
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, Deoemtier 18th, 1877.— 0. W. Merrifield, Esq., F.R.8., r.-P., in the
chair. The Bey. W. Ellis, B.A., was elected a Member, and Mr. F. B. W. Phillips,
B.A., Balliol Ck>llege, Oxford, was proposed for election. Mr. S. Roberts, M.A.,
read a paper on Normals, which contained theorems depending on t^e inyariants
and coyariants of the qnartic equation representing a pencil of four normals to a
conic, and drew attention to the remarkable cubic locus of the points of possible
concurrence of three normals at the yertioes of a giyen inscribed triangle.
Dr. Hirst, F.B.8., and Mr. J. J. Walker spoke on the subject of the communi-
cation.— Prof. Cayley, F.K.S., read a pax)er on the Geometrical Representation of
Imaginary Variables by a Real Correspondence of Two Planes. The communi-
cation was related to a former paper by the same writer, entitled " A G^metrical
Illustration of a Theorem relatmg to an Irrational Function of an Imaginary
Variable'* {Proceedings of London Mathematical Society^ t. vill., pp. 212—214).
Prof. Cayley was under 1^ impression that the theory was a known one, but he
has not found it anywhere set out in detail ; he remarked that it was noticeable
that, although intimately connected with, it is quite distinct from (and seems to
g> beyond) that of a Kiemann's Surface. — A set of four models, presented by
r. ^uthen, was exhibited. Their title is ''Quatre Modules reprSsentant des
emrf aces d^yeloppables ayec des renseignements sur la constructions des moddlea et
Bur les singiilantfis qu'ils repr^ntent par V. Malthe Bruun and C. Crone ayec
quelques remarques sur les surfaces deyeloppables et sur T utility des meddles par
M. le Dr. Zeuthen (Copenhag^e)." Their construction is founded uppn the
description of a moael giyen in Dr. Salmon's Geometry of Three Dimensions
SK 289, 8rd ed.), the inyention of which is ascribed by Prof. Cayley to Prof,
ladcbum. These models exhibit many of the chief singularities of deyelopable
surfaces.
Thursday, January 10th, 1878.— Lord Rayleigb, F.R.S., President, in the
Chair. Mr. F. B. W. Phillips was elected a Member and Mr. R. R. Webb was
admitted into the Society. The following papers were read: Mr. J. Hammond
''On the meaning of the Differential Symbol 2>**, when n is Fractional."
(Prof. Cayley gaye a few references to papers on the subject by Riemann,
Schroeter^ and others, and expressed his opinion that the matter had not yet
been satisfactorily settled). Prof. Lloyd Tanner ''On Partial Differential
Equations with seyeral Dependent Variables." Lord Rayleigh "On l^e
Relation between the Functions of Laplace and Bessel, in §788 of Thomson
andTait's Natural Phihstmhyy a suggestion is made to examine the transition
from formulas dealing wiw lAplaee's spherical functions to the corresponding
formulsB proper to a plane. It is eyident at once, from this point of yiew, that
Bessel's functions are merely particular cases of Laplace's more general functioni,
but the fact seems to be yery little known. Mr. Ferrers in his elemental^
treatise on Spherical Harmonics, nudces no mention of Bessel's functions, and
Mr. TocUiunter in his work on these functions states expressly that Bessel's
functions are not connected with the main subject of the book. The object of
the present paper was to point out briefly the correspondence of some of the
fbrmtdso. Tha Author showed that the Bessel's function of zero order, (/^),
is the limiting form of Legendre's function P« (/i) when n is indefinitely great
and ft (= cos 6) such that n sin 6 is finite, equal (say) to t. This was proyed
by taking Murphy's series for P« (Todhunter, §28) In like manner Bessel's
functions of higher order are limits of those Laplace's functions to which
To^unter giyes the name of Associated Functions. A theorem was found for
the general functions corresponding to the relation subsisting between three
consecutiye Bemel's functions [yiz. ^z {Jm-\ {z) + J«h-i {z)) = mJm («)] ; Prof. Cayley
stated that the results obtained were yery interesting. Mr. S. Roberts gaye
some results bearing upon his paper read at tiie December meeting. Prof. Cayley
«iye an expression for the suHace of an ellipeoid communicated to him by
Prof. Tait. The Chairman, Profs. Cayley and Tanner, and Mr. Webb spoke
upon the subject.
R. TUOKBR, M.A., Eon, See,
( 160 )
ON A CLASS OF DETERMINANTS.
By J. W. L. GlaUher.
§14. The present paper relates to determinants of the
forms
«8) ^t5 «i» 1) —
«45 ««) «•> «i» —
and
a, — 6,, 1 , . , . , •••
«4-*4> «8l «t> ^D —
This is the class of determinants that occur in a previons
paper, ^'Expressions for Laplace's coefficients, Bernoullian
and Eulerian numbers, &c. as determinants'' (vol. vi. pp.
49—63), and as this may be regarded as a continuation of
that paper, the numbering of the paragraphs is made con-
tinuous.
§ 15. It is shown in § 10 that
1
1 + a^x + ajx? + a,sc' + &c.
= 1 - -4 ^aj + A^x^ -A^x^-\ &c.
(1),
where
^.=
«I»
1,
• >
•>
• • •
«,.
«.>
1,
• >
. • •
«..
««>
On
1,
.• •
• • . •
. • • •
«..
«u
1.. •
[n rows),
and that
1 + a.a + a,aj* + a^a;" + &c. * * "
(2),
where
■p.=
Oj — i„ 1, . , • , ...
«,-Ja» «t) 1> •! •••
«4-*4) ^a> «t) ^i> •••
(w rows).
These are the fundamental formulae. Of course, in the equa-
tions (1) and (2) the sign of x may be chapged.
MR. GLAISUER, ON A CLASS OF DETERMINANTS. 161
§ 16. Changing the sign of a; in (1), we have
^ . . 5 — 5-7—1 — a — 5 — =a 1 — a,a; + aV — a^ -f &c.,
whence we see that if
(n rows),
then
A=
«..
1, . , ...
«..
^il ^1 •••
«.J
Oj, Oj, ...
«»=
A,
1 , • 1 ...
....
-^a) -^i» •••
(w rows).
§17. Let
1 + JjOJ + 6,aj* + J,aj' + &c.
so that
6„ 1, ., ... I ,
J„ ij, 1, ...
= 1 - jB.oj + 5^0?' - 5,aj* + &c.,
5.=
»•=
*8) *2) *1) —
then
1 4- 6^a; + &,x'+ &3x'+&c.
and therefore by (2)
Oj— 6j, 1 , . , ...
««-*8) ««) «i> —
-Bj, 1 , • , ...
-^s> A) A> —
— l-'-^ia?+ A^—A^-^- &c . . .
=(-r
^8- A- ^1) A) ••
each determinant having n rows.
For example, put n = 2, and the theorem gives
< - «A - («, - ^) = -B.* - ^ A - [B, - A),
viz. a,' - afi, - o, + J, = 6," - a,h, - (5/ - 5.) + a* - a„
which is true.
VOL. VII. M
162 MR. aLAISHER, ON A CLASS OP DETERMINANTS.
§ 18. It IS to be noticed that A^ and P^ being defined as
in § 15, then
n n 1 n— 1 S n— ■ •*■ n
This is readily proved either by multiplying (1) by
1 + J,a; + J,aj' + JgCc' + &c.
and comparing the right-hand member with that of (2), or
by observing that in the determinant P^ the coefficient of
J, is ±A„,,.
§ 19. Since (3) may be written
1 - B^x -\- B^x"" - ^3a?"+ &c. _ 1 - ^,0? + ^^a;' - ^3a;' 4- &c.
1 + a^x + a^x^ + a^x^ + &c. ~ 1 ■\- b^x -{■ b^a? + b^x^ + &c. '
it follows that
^4 --^45 «8» ^25 «!)•••
ftj — -4j, J„ 1, . , ...
*4~ A) ^) *i? *i) •••
each determinant having n rows.
A corresponding equation, in which a^ and -4^, J^ and 5^,
&c., are interchanged, also holds good.
§20. If.
P =
a, — 6j, 1, . , ...
1 •
then
<?.=(-)"
-T,) 1 ) . , ...
•^2) -^15 ^ ) •••
-^8) -^2) -^U •••
<?„=
6j — ttj, 1, . , ...
^8-^8) *2? ^) •••
ii=(-y
V|) 1 > . , ...
v»j Vjj 1 , ...
Vj) Va? Vi) •••
each determinant having n rows.
This result is at once seen to be true, for P^ is defined
in § 15, and Q^ is given by the equation
1-f a,a; + aa;'+a3aj' + &c. , /^ « /^ s . r
-— -^^i — . , ' » ' 8 ^ p = 1 - ^,aj + ^.a;'- ^30;'+ &c.,
MR. GLAISHEB, ON A CLASS OF DETERMINANTS. 163
SO that
1 - P,a! + P^' - P^x^ + «&c. "
- X — Vi**' ~ Va'^
[/ — Vft*^ "^ ^*'
As an example, put n = 2, and the theorem gives
J,-a„ 1
=
P., 1
P.-, P.
= («.-*/-
are
§ 21. Other relations that follow at once by means of (2)
(i) K = «« - -P.««-i + -Pa-. - • • •* -Pn-
(") «n =
(••") ^.=
P,-i,, Pi, 1, ...
(n rows).
(n rows).
§22. Since
= 1 + i,aj 4 \x^ 4 Jj^' + &c.,
it follows that
1 _ 1 4- ^ifl? + i^a^' + i^sg?' + &C.
1 4- il/^a; 4- M^x^-\- M^o^ 4 &c. " 1 + a,aj 4- ajOs^ + ag^H &c. '
where JIfj = a^ — P^,
^8 = «8-«aA + «A-A>
•» »
so that
a, — 6j, 1, . , ...
a^-Jj, «!, 1, ...
«8"-^> ^2> ^1) •••
each determinant having n rows.
jfj, 1 , . } ... 1 1
if,, Jf,, 1 , ...
-3^8> ^2) ^1) -
m2
164 MB. GLAISHER, ON A CLASS OF DETEEMINANTS.
§ 23. Starting with the equation
1
1 - a^aj + a^x - a^x + &c. i i a i
and multiplying hj l—qx + jV — &c., and also by its equi-
valent ^-- , we find that
1 — yd? + q*x^ - q^x* + &c.
1 — a^x + a^x^ - a^x^ + &c.
1
1 — (a, — Jj a; + (a,— qa^) x^— (a, — qa^) x* -f- &c.
= 1 + (^,- q)x+ (^ -2^,+ j») JC*+(^3-2^,+2»^,-2>'+&c.
= 1 + CjX + C^x^ + CjOj'* + &c. suppose.
Thus we have
a, — J, 1 , . , • I •••
«8-2*> ^a» ^1) ^) •••
«4~2*) ^8) ^2) ^1) —
(n rows)
a.—
a.—
a. —
2 ) 1 , •,•,...
ja„ Oj- J , 1 , . , ...
qa^j a^-qa,^ a^-q , 1 ,
. . 4
(n rows)
and also A^=C^'\-q (7^.,.
The equality of the determinants is easily proved by
multiplying the first row of the second determinant by q and
adding it to the second row, multiplying the second row of
the determinant so formed by q^ and adding it to the third
row, and so on.
§ 24. As an example, consider the series in § 2, viz.
X
where
log (1 + x)
= 1+ V,x+ r^x^+ F,a;'* + &c.,
ii 1, . , ...
iy ai '■} •••
t) 8) i) •••
(n rows).
MR. KEMPE, THEOREM IN KINEMATICS.
165
X
(1+0?) log(l +0?) * * • ' J
80 that, dividing by a;, and integrating
log (1 + a?) = logaj + O.oj + i G^x^ + i0^x'' + &c.
Then, by § 23, since j = 1,
<?.=-
and also = (— )'
Further
and
2) 1) •) •) •••
ii i} ^} • ) •••
4) 4) 4» ^) •••
f) i) 8) i) •••
{n rows).
"*" A « • « • • •••
1
1.2' "^' • »
-L -L _i
2.3» 1.2' ^' ' '•••
» ~ 1 ) •••
3.4' 2.3' 1.2
1111
4.5' 3.4' 2.3' 1.2'"*
(n rows).
NOTE ON MR. LETTDESDORF'S THEOREM IN
KINEMATICS.
By A, B, Kempe, B.A,
There are two points in Mr. Leudesdorf 's paper, p. 125,
which should, 1 think, be noticed.
(1) It is not necessary that curves traced out by the
points A^ By Cy P should all be closed curves in the usual
sense in which the term is used, that is, visibly closed curves.
It is enough if the points return together to their starting
places. This either involves their passing round visibly
closed curves or their reciprocating on any curves. An
examination of points on the moving plane close to the
reciprocating pomts shows that the latter really describe
closed curves, of which the width vanishes, and consequently
the area also.
166 MR. KEMPB, THEOREM IN KINEMATICS.
Thus in the case of the three-bar motion referred to at
the end of Mr. Leudesdorf 's paper, in certain cases the radial
bars r, 8 do not completely rotate, but reciprocate; in this
case the areas of the curves described by the points A and B
are not wr', ir«' respectively, but vanish.
(2) The integration of is not necessarily from to 27r.
but may be from to any angle and back to 0, so that the u
term vanishes, and the equation (5) takes the simpler form
{P) = x{A)+y[B) + z{0).
The following example illustrates the two modifications.
Consider the curves described by points on the connecting rod
of a beam engine. Here does not pass through 27r but
returns to ; also the area of the curve described by that
point of the connecting rod which is connected with the beam,
IS 0. Thus, if r = radius of the crank, fP) = &.7rr^, where
k = ratio in which P divides the distance oetween the beam
pin and the crank pin.
The formula of Mr. Leudesdorf may be arrived at in the
following very simple manner.
Let P be connected with A^ Bj C hj the straight lines
a', b'j c', and let
CPA^I3',
APB=^y']
then a' + )8' + 7' = 27r.
Now anv small displacement that the system admits of
majr be made by rotating it about P and displacing it parallel
to itself.
Let 8s be the linear displacement of P, and let it make
an angle a- with PA,
Let the displacements of PAj PB^ PC^ normal to their
directions, be Swj, Sw^, hn^^ then
hn^ = hs sin <r,
8w, = Zs sin (7' + <r),
S/ig = is sin (/8' — cr) ;
therefore, since a' + )8' + 7 = 27r,
hn^ sin a' + h\ sln^S' + in^ sin 7' = ;
MR. W. W. TAYLOR, ON THE RING OF CIRCLES, &C. 167
therefore n^ sin a' + n^ sin /3' + n^ sin 7' = 0,
where n„ w^, n, are the total normal displacements.
If the system starts from one position and returns to it
again, there not being a complete rotation about P, but a
partial rotation and a return,
{A) = {P)-a\, (B)={P)-h\, {0) = {P)-c\',
therefore ^-^ sina' + ^ sInyS' + ^ siny
a c
,„. Fsina' amis' sin 7'"!
This may be put into the form
{P) = {A) X ^ [B) y -^ {C) z.
If the system rotates once the extra term will be added.
There is no difficulty in obtaining the modified formula.
Western Circuit,
March I2th, 1873.
ON THE RING OF GIRCLES TOUCHING TWO
CIRCLES, AND KINDRED PORISMS.
By W. W. Taylor, M.A.
In the February number of the Messenger (p. 148) it has
been proved by my brother, Mr. H. M. Taylor, that the con-
dition for a ring of circles each touching its next neighbours on
either side and every one touching each of two primary circles
is tchen one of these tioo circles is included within the other ^
mc^= (a — mb) [am — b) (1),
where a and b are the radii of the two circles, and c is the
distance between their centres, and
• ' ''^
1 4 sm -
n
.m= — ^— — ,
1- sin -
n
where n is rational^ but not necessarily integral.
Fractional values of n should be reduced to their lowest
terms; the numerator then indicates the number of circles
that compose a ring, and the denominator the number of
complete cycles these circles make with respect to the two
original circles.
If one circle is external to the other the sign of b must
be changed in the above formula, as is easily seen by writing
168 MR. W, W. TAYLOB, ON THE RING OF CIRCLES
down the^original equations (1), (2), (3),''(4)*in 'the paper
referred to (p. 148), in the case where the point of inversion
lies between the two concentric circles.
Hence, when a and b are equal, the condition becomes
wic* = a* (1 + m)\
or V(^) c = a (1 + m).
1 + sin- cos -
•KT n n
JN ow m =5 — ^— = , ;
1 — sm - ( 1 - sm - 1
n \ nj
1 — sm -
therefore c^a ■ x — ^— = — (2).
. TT TT TT ^ '
1 — sm- cos— cos-
n n n
Now if A and 5 are two of a ring oi p equal circles, such
as are described above, circumscribing another circle C, then
. TT
sm-
? = _£ (3),
2 sm^ ^ —
P
where r is the number of circles in the ring intervening
between A and B^ a and c retaining the same values as above.
The^condition for the identity of equations (2) and (3) is
. TT
sm-
/ .\v =cos- (4).
sm^ —
P
Now i£ p is an even number, we can satisfy this equation
by putting r + 1 » ip, and finding a relation between p and n
from the equation
sm— =»cos - = sm ( — ) •
p n \2 nJ ^
P
or therefore - + - = ^
n p
this being the only solution, because both n and p must be
greater than 2.
Equation (4) can also admit of the same solution when
^ is an odd number, by supplying a circle which shall be
TOUCHINa TWO CIRCLES, AND KINDRED PORISMS. 169
opposite to J, which can he done hy the following con-
struction.
Let G and D (fig. 12) be the two original circles and A one
of the ring of circles touching them ; describe a circle to touch
A and G at the point of contact a and also to touch D 2Xb\
describe a circle B to touch D bX. b and also to touch (7.
A and B will be the required pair. This proves the truth
of the following porism.
If two concentric circles form such a pair as to admit of
a ring of circles being described to touch each member of the
ring and the circle amoining on each side, then anj op{)osite
pair of any one of such rings will also fulfil the same condition.
The number of circles in the two rings bebg connected
by the formula
1 1^1
n"^p ""2'
in which fractional values of n and p must be interpreted
as above.
Upon inversion the only alteration that this porism
undergoes is that the word concentric may be omitted.
From the formula
1 + 1 = 1
n p 2*
we can obtain any number of pairs of values of n and p ;
it is easily seen, by putting the formula into the form
that the only pairs of which both members are integral are
w = 3, p = 6,
n = 4, 2^ = 4,
w = 6j 2^ = 3.
It is curious that when n » 4 and i? = 4 the same figure
(fig. 13) represents both systems of circles, when any pair
that are not in contact are taken as the two primary circles.
The other figures (figs. 14 — 17) represent the cases when
In these the points of contact of successive circles are
marked with the numbers 1...8, in order to show how
170 MR. W. W. TAYLOR, ON THE RING OF CIRCLES, &C.
many cycles are completed ; and the circles that have been
chosen from the first series to form the nucleus of the second
system, have been marked with darker lines.
It is hardly necessary to remark that n and p are
interchangeable.
Now let us consider an anchor ring containing a ring of n
spheres, each sphere touching the anchor ring along a circle,
and also touching the consecutive spheres on either side ; the
principal transverse section of this will give a figure such as
(14, 16) ; and any axial section will give a figure such as (15,
17), the radii and distances of the centres of the two circles
being related to one another in such a manner that there
will be a ring of p circles where - + - = ^ ; or, considering
the symmetry of the figure, there will be a ring of p spheres
touching the anchor ring externally along a circle, and each
touching one another.
Conversely, if there be a ring of p spheres touching ex-
ternally, there will be a ring of n spheres touching internally.
We are now in a position to prove the following porism.
• If in any binodal cyclide a ring of n spheres can be de-
scribed, each sphere touching the two adjacent ones, and also
touching the cyclide along a circle, then any number of such
rings of spheres can be described starting with any sphere
that so touches the cyclide, whether internally or externally,
the number of spheres in every ring of one set (internal
or external according as first ring was internal or external),
being n and the number of spheres in every ring of the other
set being 'p (where - -f - = J as before ] .
To prove this, we have only to take a pair of spheres
touching each other, and each touching the cyclide along a
circle on the side opposite to that on which the ring of spheres
is situated, and invert with respect to the point of contact
of these two spheres ; the cyclide is thus inverted into an
anchor ring with an internal ring of spheres, and the proposi-
tion has been proved for such an anchor ring, and therefore
on re-inversion we see that it is true for the cyclide.
The condition for the anchor ring is clearly the same, as
for the original ring of circles in my brother's paper ; there-
fore on inversion, the condition for the cyclide will be the
same as his condition mc^ = {a — b7n) [am — i), paying regard
to proper signs of a and i, if for a and b we take the values
of the radii of the greatest and least of either set of generating
circles.
( 171 )
ON LONG SUCCESSIONS OE COMPOSITE
NUMBERS.
By J, W. L, Olaisher.
(Continued from p. 106).
§ 4. This section contains lists of sequences of 99 or more
consecutive composite numbers occurring in the seventh,
eighth, and ninth millions. They were formed in exactly
the same manner as those in § 2 (pp. 104, 105), viz. all the
instances in which the enumeration showed that there were
0, I, 2, or 3 primes in a century, were looked out in Base's
tables, and the cases noted down in which the sequence was
about 50 or upwards. From these, which were very
numerous, all the sequences of 99 and upwards were selected,
thus forming the following lists :
SEVENTH MILLION. — Sequences of 99 and upwards.
Lower Limit.
6,012,899
Upper Limit.
6,013,013
Sequence.
113
6,027,283
6,027,383
99
6,034,247
6,084,977
6,034,393
6,085,103
145
125
6,085,441
6,085,561
119
6,158,563
6,158,681
117
6,242,263
6,242,363
99
6,333,799
6,347,059
6,371,401
6,376,193
6,333,907
6,347,177
6,371,537
6,376,303
107
117
135
109
6,385,619
6,385,733
113
6,429,223
6,429,323
99
6,613,631
6,613,753
121
6,646,049
6,646,153
103
6,655,423
6,726,821
6,655,531
6,726,947
107
125
6,752,623
6,752,747
123
6,789,793
6,789,901
107
6,808,273
6,808,397
123
6,812,233
6,812,339
105
6,826,159
6,836,867
6,845,417
6,851,863
6,877,109
6,879,683
6,826,271
6,836,969
6,845,519
6,851,963
6,877,219
6,879,797
111
101
101
99
109
113
172 MR. GLAISHEB, ON LONG SUCCESSIONS
Lower Liniit. Upper Limit. Sequence.
6,958,667 6,958,801 133
6,972,127 6,972,233 105
6,983,843 6,983,947 103
6,991,147 6,991,253 105
EIGHTH MILLION. — Sequences of 99 and uptoards.
Lower Liinit. Upper limit. Sequence.
7,129,877 7,130,003 125
7,130,579 7,130,687 107
7,160,227 7,160,347 119
7,227,551 7,227,667 115
7,230,331 7,230,479 147
7,293,989 7,294,093 103
7,309,427 7,309,529 101
7,345,967 7,346,069 101
7,445,047 7,445,159 111
7,494,763 7,494,869 105
7,565,191 7,565,303 111
7,621,259 7,621,399 139
7,629,371 7,629,487 115
7,662,517 7,662,617 99
7,683,131 7,683,233 101
7,702,573 7,702,687 113
7,743,233 7,743,371 137
7,753,679 7,753,787 107
7,771,307 7,771,411 103
7,784,039 7,784,159 119
7,803,491 7,803,611 119
7,826,899 7,827,019 119
7,881,373 7,881,487 113
7,950,001 7,950,109 107
NINTH MILLION. — Sequences of 99 and upioards.
Lower Limit. Upper Limit. Sequence.
8,001,359 8,001,491 131
8,027,699 8,027,807 107
8,073,647 8,073,749 101
8,166,107 8,166,209 101
8,172,713 8,172,821 107
8,208,449 8,208,553 103
8,211,227 8,211,331 103
8,294,021 8,294,123 101
8,303,957 8,304,061 103
8,332,427 8,332,529 101
8,350,483 8,350,583 99
8,367,397 8,367,517 119
OF COMPOSITE NUMBERS. 173
Lower limit. Fpper Limit. Sequence,
8,409,721 8,409,829 107
8,421,251 8,421,403 151
8,441,869 8,441,969 99
8,454,959 8,455,063 103
8,470,927 8,471,053 125
8,487,421 8,487,527 105
8,514,949 8,515,063 113
8,648,557 8,648,677 119
8,752,871 8,752,987 115
8,793,881 8,793,991 109
8,806,891 8,806,991 99
8,841,529 8,841,629 99
8,850,671 8,850,773 101
8,889,031 8,889,143 111
8,905,199 8,905,321 121
8,917,523 8,917,663 139
8,939,597 8,939,699 101
8,947,217 8,947,319 101
8,981,461 8,981,563 101
It will be noticed that the two longest sequences in these
three millions are 151, in the ninth million, and 147, in the
eighth million, while a sequence of 147 was met with at
the beginning of the third million (2,010,733—2,010,881).
The longest sequence found in the first million was 113, in
the second 131, in the third 147, in the seventh 145, in the
eighth 147, in the ninth 151, so that the lists for the seventh
and eighth millions do not contain longer sequences than had
already been found in the third million, and the ninth million
only produces an extension of 2, Attention, however, should
be directed to the remarks in §2 (p. 105), where it is pointed
out that the method adopted does not give all the sequences,
so that the lists are necessarily incomplete. A sequence
exceeding 151 might, of course, escape detection if each of
the centuries over which it extended contained more than
3 primes ; but even supposing each of the centuries to contain
only 4 primes, it is clear that a sequence exceeding about 170
would be impossible, and probably a lower limit could be fixed.
The sequence of 151 was obtained from a 0-prime century,
while that of 147 was obtained both from a 3-prime and a
2-prime century, there being 3 primes in the century 7,230,300
— 7,230,400, and 2 in the century 7,230,400—7,230,500.
§ 5. Having by me in MS. a complete list of primes, with
their differences, up to about 30,000 (which was n)rmed some
174 MR. GLATSHER, ON LONG SUCCESSIONS
time since for a special purpose), I selected all the instances
in which the difference between two consecutive primes was
equal to or exceeded 20 [i.e, in which the sequence was equal
to or exceeded 19), so as to obtain the relatively long
sequences that occur early in the series of natural numbers.
It was subsequently found on trial to be almost as easy to
pick out the sequences from Chernac's Grtbrum Arithmeticum
as from the MS., and I accordingly had all the sequences of
19 and upwards looked out from Chernac up to 100,000.*
The results are contained in the following table :
1 — 100,000. — Numbers of Sequences of 19 and upwards.
Length of
Number of
Length of
Number of
Sequence.
Sequences.
Sequence,
Sequences.
19
238
43
5
21
223
45
4
23
206
47
3
25
88
49
5
27
98
51
7
29
146
53
4
31
32
55
1
33
33
57
4
35
54
59
1
37
19
61
1
39
28
63
1
41
19
71
1
Total number of sequences =1,221.
The most remarkable feature of this table is that it shows
that there are no less than 146 sequences of 29, although
there are only 98 sequences of 27, and 32 sequences of 31.
It seemed at first that there must have been some error in
the work ; but the whole enumeration was performed again,
and with the same result. I have myself looked out the
146 sequences of 29 and verified them.f
The meaning of the next list is as follows : between 11 and
23 there is no sequence exceeding 3 till 5 occurs at 23—29 ;
the first sequence that exceeds 5 is a sequence of 7 at
89—97, the first sequence that exceeds 7 is a sequence of 13 at
113-127, and so on. Thus between 19,661 and 31,397 there
is no sequence that exceeds 51, but, then, the longest
sequence, 71, occurs. Perhaps the most remarkable of these
♦ One half -column (68,303 — 68,399) in Chemac is erroneous, in consequence of
a bar having fallen out from the bottom and been replaced at the top, as was
pointed out by Burckhardt in the preface to his first million. Account has been
taken of this m the enumeration of the sequences.
t Since this was written a third enumeration, from Burckhardt's tables, has
been made. The result entirely confirms the above table.
OF COMPOSITE NUMBERS. 175
sequences is that of 33 which occurs as early as 1,327—1,361,
for beyond this no sequence so long as 33 is met with until
8,467—8,501, where thejre is another sequence of 33 ; but there
are 10 sequences of 29 in this interval.
Lower Limit.
Upper Limit.
Sequence.
7
11
3
23
29
5
89
97
7
113
127
13
523
541
17
887
907
19
1,327
1,361
33
9,551
9,587
35
15,683
15,727
43
19,609
19,661
51
31,397
31,469
71
The following table
contains the
sequences of 51 and
upwards that occur in the first 100,000 natural numbers.
1 — 100,000. — Sequences of 51 and upwards.
Lower Limit.
Upper Limit.
Sequence.
19,609
19,661
51
25,471
25,523
51
31,397
31,469
71
34,061
34,123
61
35,617
35,671
53
35,677
35,729
51
40,289
40,343
53
40,639
40,693
53
43,331
43,391
59
43,801
43,853
51
44,293
44,351
57
48,679
48,731
51
58,831
58,889
57
69,281
59,333
51
74,959
75,011
51
79,699
79,757
57
82,073
82,129
55
85,933
85,991
57
86,869
86,923
53-
89,689
89,753
63
It should be specially noted that the method adopted for
obtaining the sequences given in this section (viz. those
between 1 and 100,000) should yield all the sequences, so
176 MR. GLAISHER, ON COMPOSITE NUMBERS.
that if there are any omissions, they are due to errors.
The work was performed in duplicate, and is, I believe,
correct, but I have only verified a portion of it myself.
§6. In §1 "the hundred numbers between lOO/i and
100 (w+ 1)" are defined as the (n + 1)"* century. The words
between inverted commas should be " the hundred numbers
between lOOn - 1 and 100 (w + 1)"; so that, for example,
the third century consists of the numbers 200, 201,... 299,
while the first century only consists of the 99 numbers
1, 2, ...99 (unless be considered a number). Of course, in
matters relating to the enumeration of primes, it is a matter
of indifference whether the first century be defined to be
1, 2, ...100, the second 101, 102, ...200 and so on, or whether
the first be (0), 1, 2, ...99, the second 100, 101, ...199 and
so on, as the numbers 100, 200, &c. are even numbers and
do not appear in the factor tables at all, so that practically,
a century consists of the 99 numbers between lOOn and
100 (n + 1). As a rule, in the arrangement of tables, it is
more convenient to adopt the latter system, so that all the
numbers of the same century may have the same figure in
the hundreds' place.*
§ 7. There are many instances near the beginning of the
series of natural numbers of two primes being separated by
only one number, such as 29 and 31, 41 and 43, 59 and 61, &c.
These may be called prime-pairs. A glance at Burckhardt's
and Dase's tables shows that prime-pairs are of tolerably
frequent occurrence at all parts of the tables, although they
become less numerous as we advance higher in the series
of natural numbers. To exhibit this diminution I have had
all the prime-pairs that occur in each chiliad of the first
hundred chiliads of each of the six millions counted and
arranged in tables. These are complete and will form a
separate paper.
* The same iincertainty exists with regard to a oentniy of years, yiz., as to
whether the nineteenth century began on January 1, 1800, or January 1, 1801.
It is only a question of usage, but there is no ancient nsage, as the term century,
in this sense, is modem. ^'A century is any collation of one hundrea;
its restriction to collection of years is modem... We hold it clear that no
usage canr exist except one of very modem times. The present practice
of astronomers and chronologers is to make the fint year of the reckoning
to be the first year of a century, so that a.d. 1— 100 is the first centunr,
A.D. 1801—1900 is the nineteenth century.*' (De Morgan, Companion to the
AlmanaCy 1850, pp. 24 — 26). In tabular matter, nowever, where either or 1 may
be taken as the starting point, the former seems more adapted to our system of
numeration. The tendency to begin from 1 is probably a reUc of the Roman sptem
i, ii, iii, dec, in which zero has no place as the commencement of the series of
numenJs.
( 177 )
rg.
Is
FORMULA INVOLVma THE SEVENTH ROOTS
OF UNITY,
By Professor Cayley,
Let » be an imaginary cube root of unity, a)'+6)+l=0, or
Bay 6) = i {- l-f tV(3)} ; a' = -7 (l + S©), /3^ = -7 (l + Sw*)
(values giving a')8* = 343), and the cube roots a, fi bemg such
7
that ayS = 7 ; then a + /8; =!a+ — ,i8a three-valued function
(since changing the root o) we merely Interchange a and - j ;
and if r be an Imaginary seventh root of unity, then
3(r+r»)= a+ ^8-1,
3(r' + r")= <»a + ©")8-l,
8(r* + r*) = «*a+ a))8-L
Any one of these fonnul« gives the other two; for observe
that we have a* = - ayS (1 + 3a)), /S" = - ayS (1 + 3a)''), that is
a* = — )8 (l-f 3a)), yS' = - a (1 + 3a)*) ; hence, starting for In-
stance with the first formula, we deduce
9(r* + /f 2)= a*+2a)8 + )8*-2a-2)8 + l,
=s-)8(l + 3a)) + 14-a(l + 3a)*)-2a-2)8+l,
= -a(3 + 3a)')-)8(3 + 3a)) + 15,
= 3a)a + 3a)*)8 + 15,
that Is
3 (r* + r') = a)a + a)*/8 - 1,
and In like manner by squaring each side of this we have the
third formula
3(/ + r*)=a)*a+ a))8-l.
The foregoing formula apply to the combinations r 4 r*,
t^ + r*. T^-\-r* of the seventh roots of unity, but we. may
investigate the theory for the roots themselves r, r', r', r ,
r*, r*. These depend on the new radical V(— 7) or *V(7)|
Introducing Instead hereof X, F, where
X=i{-1 + »V(T)},
r=i{_i_»V(7)},
rOL. VII. N
178 PROP. CAYLEY, PORMULiB INVOLVING
then if ^'^G + awX + (l+3»»)r;
£• = 6 + 3©"X+ (1 + 3(0 ) r,
where -45=iV(7))
we have (Lagrange, Equations Numertquesj p. 294),
3r = Z+^ + J?,
and I found that In order to bring this into connexion with
the foregoing formula, 3 (r + r*) = a + )S— 1, where as before
a' = - 7 (1 + 3a)), )8' = - 7 (1 + 3fl)"), a/S = 7, it was necessary
that Bj A should be linear multiples of a, /3 respectively, the
coefficients being rational functions of <o, X] and that the
actual relations were
£=? {4-tt) + X(l-2ft))},
^ = .^{5 + a) + Z(3 + 2ft))};
in verification of which, it may be remarked that these
equations give
viz, in virtue of the equation ©* -f o + 1 = 0, the term in
{ } is =2lH-2lXf7-r', =7(Z«+3Xf3), or since X»+Xf2=0,
this is =7(2X+1), =7iV(7), the equation thus is lAB=aP.i^(l)y
which is true in virtue of AB = i V(7) and a/3 = 7. The same
relations may also be written
-a=5(ft>« + X)
I found in the first instance
3r =-X'+ A ^ B ^
3r» = -l-X+^(oi*- -X:) + 5(a>- Z),
3r« = X+a)M+©5,
3r* = -l-Z+^(a)-a)«Z)+5(a)«- a)Z),
3r* = X+a)^+®'5,
3r' = -.l-X+^(l-«Z) + 5(I -««X),
which in fact gave the foregoing formulie
3 (r + O = - 1+ a + )8,
3 (r* + r") = - 1 + ©a + oi'^^
3(r*+r")=-l + ft)*a+ ©)8.
THE SEVENTH ROOTS OF UNITY.
179
But there is a want of symmetry in these expressions for
r, r", &c., inasmuch as the values of r, r", r* are of a different
form from those of r®, r*, r"; to obtain the proper forms we
must for A^ B substitute their values in terms of a, /8|
and we thus obtain
3r =
8r'=:-
Zr^=
^7
a
7
3/= X+ ?
3r*=-l-X+^
7
4- ft)+Z( l-2a>)}+
)8
3+ a)+Z(-H2a>)}+^
l+5a)+X( 2+3ft))}+Y
-l-f2a)+X(-2-3»)}+|
-5-4a)+Z(-3- 0))+^
^2-3a)+X( 3+ (o)}+^
5+ o)fX( 3+2(»)},
2- o)+X(-3-2a))},
-4-5fl)+X(-l-3©))},
-3-2ft)+Z( l+3ft>)},
-U4a))+Z(-2+ 6))},
l+3tt)+Z( 2- 0))},
viz. each of the imaginary seventh roots is thus expressed aa
a linear function of the cubic radicals a, ^ (involving a> under
the radical signs) with coefficients which are functions of oi, X.
Kecollecting the equations a*=— i8 (I + Scd), /8*=— a (l+3ft)),
a/3 = 7; ft)* + ft) + l = 0, Z* + X+2 = 0; it is clear that,
starting for instance from the equation for 3r, and squaring
each side of the e£[uation, we should, after proper reductions,
obtain for Or* an expression of the like form ; viz. we thus in
fact obtain the expression for 3r^ ; then from the expressions
of 3r and 3r^, multiplying together and reducing, we should
obtain the expression for 3r': and so on, viz. from any one of
the six equations we can in this manner obtain the remaining
five equations.
At the time of writing what precedes I did not recollect
Jacobi's paper "Ueber die Kreistheilung und ihre Anwendung
auf die Zahlentheorie, Berliner Monatsber.^ b. 1837 and Crelle^
t. 30 (1846), pp. 166-182. The starting point is the following
rgf ^ 1
theorem : if a; be a root of the equation ^=^0^ j> & prime
number, then if ^ is a prime root otp^ and
JF'(a) = a?-f ax^-\'0?x^\..+ ar'^x^\
N2
180 PROF. CAYLEY, FOEMULiE INVOLVING
a'"* - 1
where a is any root of — = 0, we have
where y^ (a) is a rational and integral function of a with
integral coefficients ; or, what is the same thing, if a, fi be
any two roots of the above-mentioned equation, then
where '^ (a, 0) is a rational and integral function of a, 13
with integral coefficients ; as regards the proof of this it may
be remarked that writing x* for a?, i^(a), F{l3)j and F[a0)
become respectively or^F{a)y p'^F(P)^ [a0y^F\a0)] hence,
F{a) F{0)-T'F{a0) remains unaltered, and it thus appears
that the function in question is expressible rationally in
terms of the adjoint quantities a and 0. With this ex-
planation the following extract will be easily intelligible :
"The true form (never yet given) of the roots of the
equation a/* — 1 = is as follows : The roots, as is known, can
easily be expressed by mere addition of the functions F{a).
If \ is a factor of ^— 1 and a^= 1, then it is further known
that {-F(a)]^^ is a mere function of a. But it is only necessary
to know those values of -F'(a) for which \ is the power of a
prime number. For suppose W\"... is a factor o{ p—lj
further let \, V, V... be powers of diflferent prime numbers,
and a, a', a"... prime \th, Vth, Vth roots of unity, then
where y^ (a, V, a"...) denotes a rational and integral function
ofa, a', a"... with integral coefficients. Hence, considering
always the {p - l)th roots of unity as given, there are
contained in the expression for x only radicals the exponents
of which are powers of prime numbers, and products of such
radicals. But if X. is a power of a prime number, =/**,
suppose, the corresponding function F{a) can be found as
follows: Assume
F{a)F{a*)=iri{a)F{an,
then F(a) =:7(^, (a) ^. (a) ...ir^,{a) F{af^ )},
and so on up to
THE SEVENTH ROOTS OF UNITY. 181
[so that the formulae contain ultimately /x-th roots only. It is
remarked in a foot-note that when n = 1, the /i — 1 ranctions
can always be reduced to one-sixth part in number, and that
by an induction continued as far as /a = 31, Jacobi had found
that all the functions yjr could be expressed by means of the
values of a single one of these functions].
The /A — 1 functions determine not only the values of all
the magnitudes under the radical signs, but also the mutual
dependence of the radicals themselves. For replacing a by
the different powers of a, one can by means of the values so
obtained for these functions rationally express all the /a" — 1
functions ^(a') by means of the powers of F{a) ; since all
the /*"— 1 magnitudes {F{a)Y-7-F{a*) are each of them equal
to a product of several of the functions -^(a). Herein
consists one of the great advantages of the method over that
of Gauss, since in this the discovery of the mutual dependency
of the different radicals requires a special investigation,
which, on account of its laboriousness, is scarcely practicable
for even small primes; whereas the introduction of the
functions yfr gives simultaneously the quantities under the
radical signs, and the mutual dependency of the radicals.
The formation of the functions '^ is obtained by a very
simple algorithm, which requires only that one should, from
the table for the residues of 5^"*, form another table giving
5r"' = l + ^"*(mod.^), [see table IV. of the Memoir]. Ac-
cording to these rules one of my auditors [Rosenhain] in a
Prize-Essay of the [Berlin] Academy has completely solved
the equations a^— 1=0 for all the prime numbers p up
to 103."
J am endeavouring to procure the Prize-Essay just
referred to. As an example — ^which however is too simple
a one to fully bring out Jacobi's method, and its difference
from that of Gauss — consider the equation for the fifth roots
of unity, a5* + a:' + aj" + a; + l = 0. According to Gauss, we
have a;+a;* and aj*+a;', the roots of the equation w*+m— 1 =0 ;
say ic + oj* = i {- 1 + V(5j}, a?' + a?" = ^ {- 1 - V(5)}. The first
of these combined with x.x*=l gives a;— aj*= V[— i f^H- V(5)}] ;
and thence 4aj = - 1 + V(5) + a/[- 2 {5 + V(5)}] ; ii from the
second of them combined with aj".a5'=l, we were in like
manner to obtain the values of x^ and a?', it would be
necessary to investigate the signs to be given to the radicals,
in order that the values so obtained for a;* and aj' might be
consistent with the value just found for x. For the Jacobian
process, observing that a prime fourth root of unity is a = t,
and writing for shortness F^^ i^„ F^y F^ to denote ^(a),
182 PROF. NANSON, NOTE ON HYDRODYNAMICS.
F{a*)j -F(a'), F{a*) respectively, these functions are
F, = aj + a:*- (aj'4a'),
viz. we have -F;=-1, i^,*=5, or say -F- V(5), J;"=- (l+2/)i?;,
=-(l+2t W(5) ; and similarly ^3*=-(l-2t)^, =-(l-2t) V(5) ;
but also P^F^ = - 5, so that the values F, = y {- (1 + 2i) V(5)},
i^jsVl— (1 — 2i) V(5)}, must be taken consistently with this
last eauation FF^ = V(5). The values of F^^ i^„ i^„ i^^ being
thus known, tne four equations then give simultaneously
a?, a?*, aj*, a?', these values being of course consistent with each
other. It may be remarked that the form in which x
presents itself is
4aj = - 1 + V(5) + V{- (1 + 2t) V(5)} + V{- (1 - 2t) V(5)} J
with the before-mentioned condition as to the last two
radicals ; with this condition we in fact have
V{- (1 + 2») V(5)} + V{- (1 - 2i) V(5)} = V[- 2 {5 + V(5)}],
as is at once verified by squaring the two sides.
NOTE ON HYDRODYNAMICS.
By Professor J?. J. Nan8on, M,A,
Three first integrals of the ordinary hydrodynamical
equations were obtained by Cauchy in the case of an incom-
pressible fluid, and have since been extended by Stokes to the
case of any fluid in which the pressure is a function of the
density. These integrals may be written in the form
P Pf^da p^db p^dc
P Po da Po dh p^ dc ) (I))
^ = ^ ^ + Ts !^ -i. ^ :^
P Poda p^ db p^ dc
PROF. NJLKSON9 NOTE ON HTDRODlTlfAMICS.
183
mnd may be fonnd at p. 182 of Besant's Hydromechanics^ 3rd
edition ; from them Uelmholtz^s laws of vortex motion may
be deduced. The laws of vortex motion for any fluid in
which ^ is a fraction of p may also be deduced from the
following equations, which are an extension of those originally
given by jStokes and Helmholtz.
d /f \ _f^,5^, K ^^
dt \pj p dz p dy p dz
p dy
dt \p) p dm p dy p dx
d^ /f \ _ f ^ n dw ^dw
dt \p/ " p dx p dy p dz
(H).
These equation may be at once deduced from the equa-
tions I. For, differentiating the first of equations I with re-
spect to /, we have
^* \p) Po ^ Po^ Po^
— fe J^ dx du dy du dz)
p^ [dx da dy da dz da)
5p f^ ^ du dy du dz)
p^ [dx db dy db dz db)
5j idu dx du dy du dz)
p^ \dx do dy dc dz de]
— i du V du t du
p dx p dy p dz *
Thomson has, however, given a more general theorem,
which includes the above. This theorem is that the value of
J{udx + vdy + todz)
taken round any drcuit, moving with the fluid, is constant
with regard to the time. If the circuit be inmiitely small,
and Aj l8, C denote the areas of its projections on the three
coordinate planes, the theorem becomes
AS^Bn+ OS=A,S, + £,v,+ CX (in),
or as we may write it
|(uif + 5v+(7S) = 0.
(IV).
184 PBOF. NANSON, KOTE OK HTDRODTNAMICS.
Professor H. Lamb has shown that the equations II follow
from IV, and vice versd ; suprd p. 42,
The equations I may be deduced from UI in the following
manner.
Let x + Xj y+ Tj z + Z he the coordinates of any point
on an infinitely small circuit surrounding the point Xj y, Zj
and let a, by Cj a, ^3, 7, be the values when ^ = ot x^y^ z^
X, r, Z] then
•^ dx , dx Q dz
and therefore we have
2^= ^YdZ-ZdY)
and eimllar values are fonnd for B^ O. Hence III becomes
+ ^.
U(c, a) *^^ d{c, a) ^^d{c, a) ^ V
and since this must hold for any infinitely small circuit
surrounding the point a?, y, «, the coefficients of -4^, B^^ C^ in
this equation must separately vanish ; this, with the equation
of continuity,
^rf(a,6,c) ~^<»
M. E. LUCAS, Jl passage IN MERSENNE'S WORKS. 185
gives the equations I. Conversely, from the equation T, III
follows at once.
Hence Thomson's theorem, in the case of an infinitely
small circuit, stated in the form IV, is precisely equivalent to
the Helmholtz differential equations II, and stated in the
integral form III, it is precisely equivalent to the Cauchy
integrals I.
TJniyersity, Melbourne^
September 26, 1877.
ON THE INTERPRETATION OF A PASSAGE
IN MERSENNE'S WORKS.
By M, Edouard Lucas,
M. Genocchi has recently called attention, hpropos^oi
a paper of mine, to a passage in Mersenne's Works, from
whicn it results that numbers of the form 2" — 1 are com-
posite, except when n has the values
2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257, ... .
I may observe that, in order to verify by known methods
the last assertion of Mersenne, viz, that 2 — 1 is a prime,
the whole population of the glooe, calculating simultaneously,
would require more than a million of millions of millions of
centuries.
Numbers of the form 2* ± 1 can only be prime, with
the sign — if the exponent be prime, and with the sign +
if the exponent is a power of 2; and it is known that the
J>rimes of the latter form are those for which the circum-
ference of a circle may be geometrically divided into equal
parts. We have, then, to consider the three distinct classes
^=2*^»-l, i/=2*^'-l, (7=22" + l.
I may add that Format has given the decomposition
2'' -1=223x616318177,
and Plana has given
2" - 1 = 13367 X 164511353.
By means of a new method M. Landry has lately found
the decompositions
2" - 1 = 431 X 9719 X 2099863,
2" - 1 = 2351 X 4513 x 13264529,
2~ - 1 = 6361 X 69431 x 20394401,
2«» - 1 = 179951 X 3203431780337,
186 M. E. LUCAS, ON A PASSAGE IN MEBSENNE^S WORKS.
and I have myself remarked that
2'»- 1=0 (mod. 439),
and proved the theorem : —
If the numbers 4^ + 3 and Sq + l are prime, then
2^«-l = 0(mod. 8j + 7);
and therefore the numbers
2~-l, 2^"-l, 2"»-.l, 2^»^-l, 2«^-l, 2-»^-l, ...
are not prime.
En riaumS all these results seem to indicate that Mersenne
was in possession of arithmetical methods that are now lost.
I shall now indicate a new method of verification for each
of the forms A and B.
V. Numbers of the form ^ = 2**** - 1.
Form the series of numbers
1, 3, 7, 47, 2207, 4870847, 27325150497407, ...,
in which each is equal to the square of the preceding one
diminished by 2, and retain the residues to modulus A ; the
calculation of the residues is easily performed bj successive
subtractions, the first ten multiples of A having been first
calculated.
If no one of the 4^ + 3 first residues is equal to zero, the
number A is composite ; if the first zero is comprised within
the limits 2^ + 1 and 4; + 3, the number A is prime ; in fact,
if a, < 2 j^ + 1, denotes the position of the first zero residue,
the divisors of A belong to the form 2"^ ± 1, and to the quad-
ratic form a? — 2y'.
Example. For -4 = 2' — 1 , we have the residues
1, 3, 7, 47, 48, 16, (mod. 127),
whence the number is prime.
For -4 = 2" — 1 we form the residues
1, 3, 7, 47, 160, 1034, 620, -438, -576, 160;
and A^ = 2047, is not prime and the residues reproduce them-
selves periodically. Thus 2" — 1 is composite,
2"- 1 = 23x89.
2% Numbers of the form 5= 2*"+' - 1.
Form the series of numbers r^,
1, -1, 7, 17, 5983,
such that r,^, = 2r^ - 3**"',
MATHEMATICAL NOTES. 187
and take the series of residues to the modulus B. The
number B is prime if the first zero residue has a position
comprised between 2q and 4; H- 1 ; it is composite if no one
of the 4^ + 1 first residues is equal to zero ; and, if a, = 2^,
is the position of the first zero residue, the divisors of B
belong to the linear form 2'^k±l^ combined with those of the
quadratic divisors of the form 2a? — y*.
Thus r, = 598a = 193 x 31 ; therefore 2* - 1 is prime.
MATHEMATICAL NOTES.
A Problem in Partitions.
Take for instance 6 letters ; a partition into 3's, such as
ahcj def contains the 6 duads a5, ac^ bc^ de^ df^ ef. A partition
into 2's such as ab.cd.ef contains the 3 duads ah^ cd^ ef.
Hence if there are a partitions into 3^s, and /3 partitions
into 2's, and these contain all the duads each once and only
once, 6a + 3/8 = 15, or 2a + /8 = 5. The solutions of this last
equation are (a = 0, /8 = 5), (a=l, /8 = 3), (a = 2, ^^\)^
and it is at once seen that the first two sets give solutions of
the partition problem, but that the third set gives no solution ;
thus we have
a = 0, )8 = 5
ab .cd ,ef
dc .be.ef
ad , bf, ce
ae, bd.cf
af* bc.de
a = l, ^ = 3
abc.def
ad.be .cf
ae.b/.cd
af.bd.ce.
Similarly for any other number of letters, for instance 15 ;
if we have a partitions into 5's and )8 partitions into 3's. then
if these contain all the duads 4a + 2^ = 14, or what is the
same 2a + /8 = 7; ifa = 0, /8=7, the partition problem can be
solved (this is in fact the problem of the 15 school-girls), but
can it be solved for any other values (and if so which values)
of a, /8? Or again for 30 letters; if we have a partition9
into 5's, /8 partitions into 3's and 7 partitions into 2's ; then
if these contain all the duads 4a + 2/8 + 7 = 29 ; and the
question is for what values of a, /8, 7, does the partition-
problem admit of solution.
The question is important from its connexion with the
theory of groups, but it seems to be a very difficult one.
188 MATHEMATICAL NOTES.
I take the opportunity of mentioning the following
theorem: two non-commutative symbols a, ^, which are
such that ^a = ct^^ cannot give rise to a group made up of
symbols of the form c^^. in fact, the assumed relation gives
^a'' = af^a?^^ ; and hence if ^a'' be of the form in question,
= a"/S^ suppose, we have al"^' ^a\oL'l3^.fi\ ^oT*^'^^] that is
1 =a*)8*, and thence i8a= 1, that is j3 = a^j viz. the symbols
are commutative, and the only group is that made up of the
powers of a.
A. Caylet.
New Demonstration of the Fundamental Property of Linear
Differential Equations,
The following demonstration is shorter than that which I
have given in vol. IV., pp. 177, 178 (April 1875). Let
Dry^A,Dr-y^A^ir-Y--^A^Py^rA^^o
be a linear differential equation, the coefficient being any
functions whatever of a;, and let is? be a solution of this
e(][uation. Put y==zjtdx and the transformed equation in t
will be, as is known from a remark due to Alembert, a
linear equation of the order n - 1 in ^, viz.
D^-'t + B^D'^t + B^D'^H. . .+ B^^Dt -{B^A^- 0.
Herein put t=^uz'^ or u^zt The transformed equation
in u will be of the (n — 1)^ order ia m, viz.
Now, from the definition of ^,
Dz
Dy = zti'Dz.Jtdx = u + — y,
whence u = Dy y.
It follows that we have identically
ITy + A,ir'y + Ajr^y.. .+ A,.J)y + Ajf
= {ir-^+0,ir*^ 0,iy-'...+ C^D+ G,.,) (2>y- ay)...{l),
if Dz-ae = (2).
MATHEMATICAL NOTES. 189
Reciprocally^ from (1) and (2) we can deduce th$it Is a
solution of tbe given equation ; in fact, since the latter maj
be written in the form
if (Ij and (2) hold good simultaneously, we see that it is
satisned hj y^-z.
This proposition, with that of which it Is the reciprocal,
constitute the fundamental property of linear equations.
Paul Mansion.
Antwerp.
On Sylvester* 8 Kinematic Paradox*
In his Royal Institution lecture,* Sylvester showed how
by a linkage of 78t bars, the following problem termed by
him the kinematic paradox might be solved : ^^ Eequired to.
construct a link work fixed or centered at two of its points,
such that (when the machine is set in motion) some other
point or points therein shall be compelled to move in the
line of centres."
Such a linkage may be very easily obtained by means of
the relations connecting six points A^ E^ i>, 0, jF, B^ lying
in order on a straight line, and such that
AB.AO^a% BC.BD^^a\ EB.ED = a\ FA.FE=2a\
For, let ^B=a^^, then ^0=a^^, 50= ^,
' x- a^ aj + a x— a'
£2>=?Jl^=a;--, EB^x,
X x^ '
. ^ a: 4 a 2a* , .
AIfj = a x= (x — a)^
x^a ar-a ^ ^
whence FE^x-a^ and therefore FB=ay and is therefore a
constant quantity.
If, therefore, -4, -B, G be connected by a reciprocator
whose modulus is a^ ; -B, (7, D by a reciprocator of modulus
4a' ; -B, 5, 2> by a reciprocator of modulus a*, and Fj -4, E
by a reciprocator of moaulus 2a', then in the free motion of
* January 23, 1874, published in the Proceedings of the Royal Institution,
t It should be noticed, that by the use of four-bar reciprocators instead of
Feaucellier's cells, the number 78 can be reduced to 52.
190 MATHEMATICAL NOTES.
this linkage, consisting of only 16 bars, F and S, althongh
not rigidly connected, will always remain at the same distance
a apart.
This result I stated at the meeting of the London Mathe-
matical Society on May 10, 1877 (see p. 55 of the present
volume)^
Harry Hart.
B. M. Academy, Woolwich.
A Theorem in Kinematics.
The following remarkable theorem may readily be deduced
from that of Mr. Leudesdorf, ante p. 125, on which I have
already at p. 165 made some brief observations.
^^ If one plane sliding upon another start from any position,
move in any manner, makmg any number of rotations, and
return to its initial position, then a circle can be found on the
moving plane, every point on which describes on the fixed
plane a curve of area ; and if any other, circle concentric
with this zero-circle be taken on the moving plane, the areas
described by all the points on this circle are the same, and are
proportioned to the area enclosed between that circle and the
zero-circle.
" If the moving plane returns to its initial position without
having made a complete rotation, the system of concentric
circles is replaced by a system of parallel straight lines, the
area described by a point on any line being proportional to
the distance of that line from the zero line."
The existence of so singular a point as the centre of the
concentric circles in the first part of this theorem is note-
worthy.
A. B. Kempe.
WcBtem Circuit,
March SO^A, 1878.
I. Arithmetical Note.
Write down a 5, divide it by 2 giving 2 with 1 over, divide
12 by 2 giving 6, divide 6 by 2 giving 3, divide 3 by 2
fiving 1 with 1 over, divide 11 by 2 giving 5 with 1 over,
ivide 15 by 2 giving 7 with 1 over, and so on till the figures
repeat. We thus obtain the figures 52631578947368421, and
these with a cipher prefixed are the period of 3^, viz.
-^ = -052631578947368421.
MATHEMATICAL NOTES. 191
If we start with 50 and halve In the same manner, pre-
fixing two ciphers, we obtain the period of j^^, viz.
j|^ = •00502512562814070351758793969849246231155778894
472361809045226130653266331658291457286432160804020i.
Similarly, If we start with 500 and halve as before, we
obtain, after prefixing three ciphers,
T^? = •0005002501250625312656328164082041020510...,
and, generally, the process gives the reciprocal of 1 followed
by any numbers of 9.
If we start with 20, 200, 2000, &c., and divide continually
by 5 instead of by 2, prefixing one, two, three, &c., ciphers,
we obtain the periods of 49, 499, 4999, .... For example,
-^ = -620408163265306122448979591836734693877551
,^^ = -0020040080160320641282565130260521042084... .
The process is very expeditious, the figures of the periods
being obtained as fast as the hand can write them.
II. Euler^s Formula in Trigonometry,
Take a chord AB (fig. 18), in a circle subtending an
angle 20 at the centre 0, and therefore an angle 6 at the
circumference, so that Z A CB= 0. Bisect the arc AB in JIf,
then /.ABM=/LACM=^d^ and the chords AM^ MB are
together equal to AB sec^&. Similarly, itAMj MB be bisected
in N^j -AT,, the four equal chords AN^^ -^,^j ^^^^ -^,^ are
together equal to AB nec^d sec J^. Proceediag in this way, we
see that if Ay B be joined by 2*" equal chords inscribed in the
arc ABj then the perimeter of these chords
= AB &ec^0 sec J^... sec ^ 0,
Thus the ratio that the chord AB bears to the arc AB
= cos^ff cosj^ cosj^...,
where 2d is the angle subtended by AB at the centre, and
thus
—^ =cosJd cosjd cosjd... .
It is, I think, interesting to notice how directly by the
above process, the expression asec^dsec^dsec^d... admits of
geometrical interpretation ; but the process is, of course, not
new; in fact, Vieta, Van Ceulen, and others who used the
192 TRANSACTIONS OF SOCIETIES,
method of polygons were In effect acquainted with Euler's
formula, though it did not then admit of expression in its
general form, as the cosine had not at that time been
introduced.
III. Numerical value of a Series.
The value of the series
111 1 1 I o -i . J.
2^ ■*■ F "^ 5^ "^ r "^ TP "^ 13^ "^ ^^ ^"-^ •
(the denominators being the squares of the prime numbers),
is, to 24 places,
0*4522 4742 0041 0654 9850 6546,
the last figure being uncertain.
Euler, in the Introductio in Analysin Injmitorum^ 1. 1. § 282,
gives the value to 15 places as
0-4522 4742 0041 222,
so that his last three figures are erroneous. As far as I
know, Euler^s calculation has not hitherto been verified or
extended.
J. W. L. Glaishek.
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, FebruMV, 14tlij 1878.— Lord Rayleigh, F.R.S., Presidentf and
Bubeojuently Mr. 0. W. Memfield, F.R.S., Vice-Prendent. in the chair. The
following commnnicationB were made to the Society: ^On a General Method
of Solving Partial Differential Equations," Prof. H. W. Lloyd Tanner j "On the
G onditions for Steady Motion of a Fluid," Prof. H. Lamb, Adelaide ; " On a Pro-
perty of the Four-piece Linkage, and on a curious Locus in Linkages," Mr. A. B.
Kempe, B.A. ; " On Robert Flower's Method of computing Logarithms," Mr.
S. M. Drach; "On the Pluckerian characteristics of Uie Modu£ur Equations,"
Prof. H. J. S. Smith, F.R.S. Mr. Drach also exhibited drawings of Tridicloids
made seyeral years ago for Mr. Perigal.
niursday, March 14th, 1878.— Lord Rayleigh, F.R.S., President^ in the chair.
Mr. Artemas Marfcio, M.A., Erie, Pa, was proposed for Section. The Hon. Sec.
read portions of a paper by Prof. J. Clerk Maxwell, F.R.S., " On the Electrical
capacity of a long narrow cylinder and of a disc of sensible thickness." Prof.
Gayley, Mr. J. W. L. Glaisher, Mr. S. Roberts, and the President, made short
communications to the Society.
R. TuoKBB, M.A., Eon, 8eo,
THE END OP VOL. VII. JJ^
Ilai;t>l. Yolm. [jfew Series]
^€^d
1/
/^n'
^«:
FlcUi.2, M.TTT.' [miseries]
i1^£c^jBfif t Son. J.itA-j
*^
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