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l\
THE
MESSENGER OF MATHEMATICS,
EDITEP BY
W. ALLEN WHITWORTH, M.A.,
FRr.T.OW OF HT. JOHM*S COT.T.BOR, OAMBHIDOK.
0. TAYLOR, M.A.,
PBLLOW or 8T. /0HM*8 COIXKOE, CAMBRIDOK.
R. PENDLEBURY, M.A.,
PELT.OW OF ST. J0HM*8 COT.LBOB, OAMBBIDOB.
J. W. L. GLAISHER, M.A., F.R.S.,
FRI.LOW OFTBIKITT OOT.T.ROB, OAMBBinOK.
VOL. VIIL ^
[May, 1878— April, 1879.] /
MAOMILLAN AND 00.
Uontron aniy C£ambrilige«
EDINBURGH: EDMONSTON & CO. GLASGOW: JAMES MACLEHOSE.
DUBLIN : HODGES, FOSTER & CO. OXFORD : JOHN HENRY AND J. PARKER.
1879.
V
t
'-\
-' ••v'^
CAMBRIDGE :
PBINT£D BY W. METCALFE AND SON, TRINITY STREET.
CONTENTS.
ARITHMETIC, ALGEBRA, AND TRIGONOMETRY.
PAOB
On a rale for abbreviating the calculation of the number of in- or co- variants
of a given order and weight in the coefficients of a binary quantic of
a given degree. By Professor /. /. Sylvester .
Arithmetical note. By Professor H. W, L, Tanner .
An enumeration of prime-pairs. By /. W. Z, Glaisher
On some theorems in determinants. By R. F. Scott . .
An algebraical identity. By Professor Cayley
Note on the above. By /. W, L. Glaisher
On certain sums of squares. By J, W, L. Glaisher . . ,
On a class of algebraical identities. By J. W. L. Glaisher ,
On long successions of composite numbers. By Edouard Lucas '
A theorem in combinations. By W, Allen Whitworth
Generalisation of Prof. Cayley's theorem on partitions. By /. W. L. Glaisher
Arrangements of m things of one sort and n things of another sort under
certain cdnditions of priority. By PF. Allen Whitworth
Cauchy's theorem regarding the divisibility of {x + y)" -I- (— x)* + (— y)\
By Thomas Muir . . . . . .
Note on Cauchy's theorem. By /. W. X. Glaisher . .
A formula by Gauss for the calculation of log 2 and certain other logarithms.
By Professor Cayley .......
Value of a series. By /. W. L. Glaisher . . . , .
Note on " Choice and chance." By W, Alien Whitworth
Note on an inequality. By W, Walton .....
On some symmetrical forms of determinants. By R. F, Scott , 131,
Note on certain determinants connected with algebraical expressions having
the same form as their component factors. By Samuel Roberts
Theorems in algebra. By /. W, L. Glaisher . , . ,
Note on a theorem of Professor Cayley's. By R, F. Scott
On a class of determinants, with a note on partitions. By /. W» L. Glaisher
Notes on determinants. By R. F, Scott .....
Note on continuants. By Professor /. /. Sylvester ....
1
13
28
33
45
46
48
53
81
82
83
105
119
121
125
127
129
130
145
138
140
155
158
182
187
GEOMETRY OF TWO AND THREE DIMENSIONS.
On the relations between the angles of five circles in a plane or of six
spheres in space. By Edouard Lucas .... 37
On the deformation of a model of a hyperboloid. By Professor Cayley , 51
Geometrical demonstration of a known theorem relating to surfaces.
By Professor Mannheim . . . . . * 122
On triangles self -con jugate with respect to a parabola. By A, F, Torry . 122
Theorem relating to a system of conies. By R. Pendlebury , . 130
Equation of the wave-surface in elliptic coordinates. By Professor Cayley , 190
IV CONTENTS.
DIFFEBBNTIAL AND INTEGRAL CALCULUS AND
DIFFERENTIAL EQUATIONS.
Elementaiy demonstration of Taylor's theorem for fonctions of an imaginary
variable. By Prof essor PauZ Jfannon . . . . .17
Example illnstratiye of a point in the solution of differential equations in
series. By J, W, L. Glaisher ..... 20
Note on certain theorems in definite integration. By /. W, L. Glaisher . 68
A theorem relating to pfaflians. By Professor H. W, L, Tanner . 56
The intrinsic equation of the elastic curve. By Professor A, G. GreenhiU 82
Note on a definite integral. By Professor Cayley • . . .126
THEORY OP ELLIPTIC FUNCTIONS.
On the occurrence of the higher transcendents in certain mechanical
problems (continued from voL vii. p. 139). By W, H, L, Russell . 8
New formula for the integration of -rr? + -j^ = 0. By Professor Cayley . 60
The intrinsic equation of the elastic curve. By Professor A, G, GreenhiU . 82
Note on a definite integral. By Professor Cayley • . . 126
On a formula in elliptic functions. By Professor Cayley . ; .127
KINEMATICS, MECHANICS, HYDROMECHANICS, OPTICS,
AND ELECTRODYNAMICS.
'On the occurrence of the higher transcendents in certain mechanical
problems (continued from vol. vir. p. 189). By W, E. L, Russell . 8
Note on the theorem in kinematics (vol. yii. p. 125). By C Leudesdoif 11
Note on Thomson's theory of the tides of an elastic sphere. By G. H, Darwin 23 |'
Proof of the theorem in kinematics (vol. vir. p. 190). By A. B. Kempe 42
Fluid motion in a rotating semicircular cylinder. By W. M, Hicks . 42
On the caustic by refraction of a circle for parallel rays. By J, W. L, Glaisher 44
Centres of pressure. By T, C. Lewis ..... 49
On a case of wave motion. By Professor C. Niven « . . .75
On Ampere's electrodynamic theory. By T, C, Lewis ... 84
Fluid motion in a rotating quadrantal cylinder. By Professor A, G, GreenhiU 89
On centres of pressure, metacentres, <&c. By T, C Lewis, . . 114
Note on the centre of gravity of the frustrum of a pyramid. By /. W. Sharpe 124
Note on the theorem in kinematics (vol. vii. p. 190). By A, B, Kempe 130
Solution of a mechanical problem. By Professor A, G, GreenhiU . .151
On the stresses caused in an elastic solid by inequalities of temperature.
By /. Mopkinson . . . . . . . . 168
Yortex motion in a viscous incompressible fluid. By /. /. Thomson . 174
TRANSACTIONS OF SOCIETIES.
London Mathematical Society. By R, Tucker . . 27, 62, 121, 168, 192
Meeting of the British Association at Dublin. By J. W, L, Glaisher , 88
[Note. Pp. 1—16 were published in May, 1878 ; pp. 17—82 in June, 1878 ;
and so on, pp. 177 — 192 being published in April, 1879 ; so that the month of
publication of any paper may be readily ascertained.]
I
MESSENaER OF MATHEMATICS.
k
ON A RULE FOR ABBREVIATING THE
CALCULATION OF THE NUMBER OF IN- OR
CO-VARIANTS OF A GIVEN ORDER AND
WEIGHT IN THE COEFFICIENTS OF A BINARY
QUANTIC OF A GIVEN DEGREE.
By Professor J. J, Sylvester,
If i is the degree of a quantic we know now by apodtcttc
easoning that the number of its in- or co-variants of order ?'
" weight w in the coefficients is (w : ijj) — {{w — 1) : tjJU
here in general (a? : i^j) denotes the number of modes of
omposin^ x withy numbers each having any value from to i
both inclusive) or (what is the same thing) with i numbers
iach having any value from to^. The object of this note
3 to show how to calculate the difference between the two
denumerants above given without calculating each of them
separately, whereby the actual amount of calculation required
will be reduced to a small fraction of what it would
otherwise be. I shall not stop to draw theoretical con-
equences from this theorem, but present it to the readers of
he Messenger in the way it has occurred to me as a rule for
bbreviating labor.
It is founded on the exhaustive method of representing
partition systems by following a dictionary order of sequence,
and will be best understood by beginning with an example.
Suppose then that i^? = 7, « = 5,y = 4, we may find (7 : 5, 4)
by setting out and counting the arrangements where 4 is the
number of parts and 5 the limit to each part, viz. 5.2, 5.1.1,
4.3, 4.2.1, 4.1.1.1, 3.3.1, 3.2.2, 3.2.1.1, 2.2.2.1.
For brevity the zeros required to fill up the number of
parts to 4 are omitted in this table.
To find (6 : 5, 4) we may consider
1** Those arrangements which begin with 5.
2" Those arrangements which begin with a number less
than 5.
VOL. YIII. B
2 PROF. STLYESTEB, RULE FOR ABBREYIATTMa, &C.
To obtain the latter also arranged in dictionary order of
Bequence, we may (subject to an exception to be stated
immediately below) proceed by diminishing each initial
number in the above table by unity.
The exception to be made is where 2 initial numbers are
alike, as in 3.3.1; 2.2.2.1. These arrangements must not be
counted in as the arrangements 2.3.1, 1.2.2.1 will already
have been obtained from 4.2.1, 3.2.1.1 respectively.
Hence the number of an*angements in the above table to
be preserved is less by 2 than the total number.
On the other hand we shall have the arrangement 5.1, to
which there is nothing corresponding in the table for (7 : 5, 4).
Hence the difference required is
2-1, i.e. (7 : 5, 4) - (6 : 5, 4) = 1 .
Let us take as a second example w (the weight) 12^ i
(the limit to each part) 6, andy (the number of parts) 4.
Let A be the table for (12 : 6, 4) in dictionary order, and
let A be the part of the table for (11:6, 4), also arranged in
dictionary order, for which 6 is nowhere the initial term.
Let A be what A becomes when each initial number is
diminisned by unity.
Then, by the same reasoning as above, we must have
j1'-^, = 6.6, 5.5.2, 5.5.1.1, 4.4.4, 4.4.3.1, 4.4.2.2, 3.3.3.3,
7 in number.
Also calling B the part of the table for (11:6, 4), beginning
with 6 we have £-6.5, 6.4.1, 6.3.2, 6.3.1.1, 6.2.2.1, 5 in
number.
Hence (12 : 6, 4) - (11 : 6, 4) = 7 - 5, = 2.
To verify this, let us interchange the values 6 and 4,
this by a well-known theorem leaves the value of each
denumerant unaltered.
We have now ^'-^, = 4.4.4, 4.4.3.1, 4.4.2.2, 4.4.2.1.1,
4.4.1.1.1.1, 3.3.3.3, 3.3.3.2,1, 3.3.3.1.1.1, 3.3.2.2.2, 3.3.2.2.1.1,
2.2.2.2.2.2, number is 11.
Also 5= 4.4.3, 4.4.2.1, 4.4.1.1.1, 4.3.3.1, 4.3.2.2, 4.3.2.1.1,
4.3.1.1.1.1, 4.2.2.2.1, 4.2.2.1.1.1, number is 9, and thus
(12 : 4, 6) - (11 : 4, 6) = 11 - 9 = 2 as before. Evidently this
identity between the two forms of [w : i, 5) - {[w - 1) : t, 5}
given by this method and also the incapability of this difference
becoming negative when w is not greater than ^y , which I have
elsewhere demonstrated, may be made to yield arithmetical
properties of a new kind, and not unlikely to prove very
valuable in certain parts of the theory of numbers ; but what
has impressed itseli on my mind is the enormotM saving of
PBOF. STLVE8TEB, BULE FOB ABBBEVIATING, &C. 3
labor In the actaal business of calculating Invarlantlve
formulsB, which this method confers. The existence of a
perfectlj definite table exhibiting an exhaustive arrangement
of ruled partitions (as I call partitions subject to the two
indices i^j) in Itself constitutes a theorem (however simple) ,
and the method above given Is a further and more recondite
theorem deduced from it, combined of course with other
intuitional propositions.
Let us take as another example w = 20, « = 13, y = 3.
Here ^'-^, = 10.10, 9.9.2, 8.8.4, 7.7.6. 5=13.6, 13.5.1,
13.4.2, 13.3.3. Therefore (20 : 13, 3) - (19 : 13, 3) = 0.
Again let us calculate (40 : 20, 4) - (39 : 20 : 4).
Here ^-^=20.20, 19.19.2, 19.19.1.1, 18.18.4, 18.18.3.1,
18.18.2.2, 17.17.6, 17.17.5.1, 17.17.4.2, 17.17.3.3, and simi-
larly 16.16 with 5 duads, 15.15 with 6 duads, 14.14 with
7 duads. Also 13.13 with 13.1, 12.2, 11.3, 10.4, 9.5,8.6, 7.7,
12.12 with 12.4, 11.5, 10.6, 9.7, 8.8, 11.11 with 11.7, 10.8, 9.9
10.10.10.10. Thus the number of terms In A' — A^ Is
(1 + 2 + 3 + 4 + 5 + 6 + 7) + (7 + 5 + 3+1) = 44.
And B Is composed of arrangements containing 20, to-
gether with the number of triads into which 39 — 20, ix. 19
can be decomposed, none greater than 20, i.€. the number of
terms In 5 Is 19 : 20, 3, which Is the same as the absolute
number of modes of resolving 19 Into 3 parts or fewer,
which is
l + l + 2 + 2 + 3 + 3 + 4 + 4+(10:9, 2) + (9: 10, 2)
+ (8: 11, 2) + (7: 12, 2) = 25+ 5 + 5 + 3 + 2 = 40.
Thus (40 : 20, 4) - (39 : 20, 4) = 44 - 40 = 4,
which is easily verified, for the difference between the above
two denumerants is the number of linearly Independent In-
Tariants of the 20*^ order to a quartic, i,e. Is the number of
ways of composing 20 with 2 and 3 (the orders of the funda-
mental Invariants) which Is 4 as found above.
The method thus simply and almost Intuitively^ deduced,
may be expressed in the form of a theorem as follows :
2^(^-22: 2,i-2)-(u;-.t-l:t,y-l)
= (t£?:t,y)-(w?-l li^j)
= 2^'(w^-2j:j,t-.2)-(i(7-y-l:y, t-1).
The Inferior unit Is taken zero for the purpose of theo-
retical simplicity. Let the effective value of this limit be
called [^], and consider the first of the above three equals.
b2
4 PROF. STLYESTER, RULE FOB ABBRETIATINa, &C.
The valae of [;] is given by the condition that
to-'2[q] shall be not greater than (J — 2) [jj]^
to
%.e. [;] not less than -r |
J
%.e. [j] is T if T is an integer, — + 1 if -r is fractional,
[E standing as nsaal for the integer part of the quantity
which it precedes).
The nomber of actual terms differing from zero under the
sign of summation is therefore
J J
similarly the number of terms under the sign of summation
in the conjugate form will be l-^-E ^ . .
Thus the first or second expression will be the best to
employ, according as j' is greater or less than t.
Again, since {w : V, j) = («;- w : t, j),
we may in place of
employ {w' : t, j) — {w' + 1 : «, y),
which Is — [{w' + 1 : i, j) - {w' : t, j)].
Hence, we may always secure in the application of this
method, that the numerator in E *^ . or in E *^ . shall j
not be greater than i^ij. Supposing / to be greater or not less
than i, so that the first formula is applied, it will be found
most convenient, so long as q Is less or not greater thanj- 2,
to consider q the number of the parts In any of the quantities
{w-2q: qyj — 2)j andj — 2 the limit to the magnitude of \
each part, and until q becomes equal to «- 1, this hypp- *!
thesis win always be the case. When q = i or when j = «' ;
and j = t — 1 In the respective cases of J being only one unit '
greater than i or equal to t, the two indices q ij- 2 ma'^^ ^ ;
with advantage be reversed. For any other values of ^—f ^ J
PROP. SYLVESTER, RULE FOR ABBREVIATING, &C. 6
the order of the indices need not be disturbed. It maj bo
"worth while to call attention to the two independent theorems
of reciprocity made use of in the preceding discussion, in-
dicated by the equations
{w : t, j)
= {w :j\ t)
= (ij-w:y,t),
both of them of importance in the theory of invariants after
the English methoa.
Johns HopkinB Uniyersitf , Baltimore,
November 26, 1877.
Addition.
Notwithstanding what has been stated above as to the
choice between the two formulaQ representing A {w : t. j)j th«
advantage of diniinishing the smaller of the two indices t, j\
will simplify the calculations to a degree that far more than
outweighs the disadvantage of increasing the number of
terms under the sign of summation. Let us suppose then
that / is less than io^ and that A {w : t, j) is positive, repre-
senting in fact indifferently the number of linearly indepen-
dent covariants of order i to a quantic of degree j\ or of order
J to a quantic of degree i. Then, unless these covariants are
invariants, we must have w<i^ij.
Consequently, the best formula to apply in such case will
be obtained by writing A [to : t, J)
= (t; - w : t, J) - (y - w + 1 : «, J)
= -A(j;-w+l:f,y)
The number of terms other than zero under the sign of
to
summation will then be 1 + jE -7 .
For the case of invariants we may with at least equal
advantage use the formula
Let OS apply this to the case of finding
A (i|^ : 18, 5) , le. (45 : 18, 5).
6 PROP. SYLVESTBiB, RULE FOfl ABBRETIATING, &C.
In the work below I use, whenever usefal, the formala of
transformation {x : i^J) = (^'— a? : t,y), and employ ^ to .
denote the number of ways of breaking up fi into three or
fewer parts, which we know is the nearest integer to ^ - ■ ■ ;
and in like manner - for the number of ways of breaking up
V into two parts : also in place of {x : A:, 3), whenever k is at
least as great as ar, I use the obviously equivalent value - .
Let us then first calculate
S™[45-2j:j,3],say5.
The values of g inferior to 9 will give quantities in which
3^^ < 45 — 2^, and which will therefore be zero.
We have thus
5= (9 : 18, 3) + (11 : 17, 3) + (13 : 16, 3) + (15 : 15,-3)
4 (17 : 14, 3) + (19 : 13, 3) + (21 : 12, 3) + (23 : 11, 3)
+ (25 : 10, 3) + (27 : 9, 3)
= I + V + ¥ + ¥ + (17 : 14, 3) + (19 : IS, 3) + (15 : 12, 3)
+ (10 : 11, 3) + (5 : 10, 3) + (0 : 9, 3).
Also
(17:14,3) = (17:17,3)^f-|-f=V-l-2-2=V-5,
(19 : 13, 3) = (19 : 19, 3)-i-J^f-f-f-f =(19:19, 3)-15,
(15 : 12, 3) = (15 : 15, 3)-^ - |- f = V - 5.
Thus ;8f=H. V + V + V + V+ V + V^+¥ + f+*-25
=*+f + f + ¥ + V + V + 2.V + ¥ + V-25.
Again let /8' = (44- 18 : 18, 4) = (26 : 18, 4).
Then
fif'=(8: 18,3) + (9: 17, 3) + (10: 16, 3) + (ll : 15,3)
+ (12 : 14, 3) + (13 : 13, 3) + (14 : 12, 3) + (15 : 11, 3)
+ (16 : 10, 3) + (17 : 9, 3) + (18 : 8, 3) + (19 : 7, 3)
=l+l+¥ + V + y + V + (V-3) + (y-8)
+ (V-8) + (V-l) + f + |-20
= * + f + f + f+2.V^ + V + V + ¥+2.V + V-20;
PROF. SYLYESTEBi RULE FOR ABBREVIATING, &C. 7
therefore
= 1-2 + 5-7-10-14-19-484-27 + 33 + 40-5
= 106-^105 = 1,
which is right, there being just one invariant to the quantic of
the eighteenth order in the coefficients, so that A(45 : 18, 5) = I.
It appears from the tables given in M. Fa^ de Bruno^s
valuable ThSorte dea Formes Btnaires^ Turin, 1877, that this
Invariant contains 848 terms. Therefore the value of
(18 : 8, 6) is very considerably greater* than 848.
Thus, by the direct method of calculating A (45 : 18, 5),
many more than 1695 terms would have required setting out.
There is one case which deserves special consideration,
viz., vhen one of the indices i or J becomes infinite.
The function A (k? : ^ oo ) then represents the total
number of in- and co- variants of weight w of any given order
not less than w to sl quantic of the /i^ degree.
The two formulas for this case become respectively
S^ [t(j-2j: j,/a].
>*
,«•/*
and S^ [^-^'ii Qo]-[t£?-/A-l :/i, qo].
n
or if we agree to understand in all cases by — the number of
ways of making up n with the integers 0, 1. 2, 3...m, or, what
is the same, the number of ways of breaking up n into m or
fewer parts, the second formula becomes
ni t£? — 2 J w —fi" 1 ^
of these two the first is by far the most expeditious.
Let us take as an example A (20 : 6, oo ), t.e. ^ — V .
The first formula (neglecting the values of q which make
ti? — 2 J negative and those which make 43 < («? — 2^), will give
for the value of A
(0:10,4) = (0)
+ (2:9,4) +(2)
+ (4:8,4) +(4)
* I sa^ yery considerably greater than, because only a certain number of the
terms which satisfy ^e required conditions of order and weight actually appear in
the octodecimal invariant in question. Thus ex. ^r. there is no/^, no/*, and of
the (10, 11, 6) i. e, V> terms which might contain /', only six, yiz. the terms
0(mtained in a (ac - P)^ actually make their f^ppearance in it.
8 MR. RUSSELL, ON THE HIGHER TRANSCENDENTS
4(6:7,4) +f
4(8:6,4) +(8:6,4)
+ (10:6,4) +(10:5,4)
+ (12:4,4) +(4:4,4), t.6. (4),
where In general (m) means all the modes of breaking up m
into parts. The value of (10 : 5, 4) will be easily found to be
9 of (8 : 6, 4) 12 and of |, 9, also of (4) is 6. The value of
V - V tl^^s becomes 1 + 2 + 5 + 9 + 12 + 9 + 5 = 43.
By the second formula the value of the same quantity
would be f+Y + Si^ + V + V+¥-Vj which would be
exceedingly tedious to calculate.
In like manner if w is odd we shall have a series of
denumerants of the form
(l ' ^ 1 f^)i (3 : ^^ , /*) , (5 : ^ , /i), &c.
Thus ex. qr. ^ — ^^ [i.e. lAi% number of in- and co-
variants to a sextic of weight 11 and of any given order not
inferior to 1 1, or, if we please to vary the expression, the
number of in- and co-variants of weight 11 and the sixth
order to any quantic of a degree not imerior to 11) will be
(1:5,4)] { (1)
+ (3:4,4)1 +(3)
+ (5:3,4) "" +(5:3,4)
+ (7 :2, 4)J [ + (1:2, 4)i.e. (1)
= 1 + 3 + 4+1 = 9.
November 28, 1877.
ON THE OCCURRENCE OF THE HIGHER
TRANSCENDENTS IN CERTAIN
MECHANICAL PROBLEMS.
By W. H, L. RusBell, F.R.8.
(Continued from vol. yii., p. 139).
(7) A CYLINDER is placed with its axis horizontal upon a
smooth inclined plane, a string one end of which is attached
to a fixed point is woond round the cylinder; determine the
motion.
When the distance of the fixed point from the plane
is equal to the radius of the cylinder, the problem is much
simplified. Under this condition it has been discussed by
IN CERTAIN MECHANICAL PROBLEMS. 9
Euler, whose solution will be found in Walton^s Mechanical
Problems. I here investigate the general case which is much
more complicated, and I think possesses considerable interest.
Let A be the fixed point, K the centre of a circular
section made by plane through A perpendicular to the axis
of the cylinder ; C the point where this section touches the
inclined plane ; let the plane of the circular section be the
plane of (ccy), A the origin -4X, A F, the axes respectively
parallel and perpendicular to the inclined plane.
Let E be the point in which the string from A meets
the cylinder, D the point where the axis of (y) meets the
inclined plane. Also let KF be a line drawn from K per-
pendicular to the axis of [y) and meeting it ia F^ O the
mtersection of AF and KF.
Let -4Jf=a?, MK^y be the coordinates of K^ also let
KO radius of cylinder =a, AF=^ c, then we shall find
An^ oca; c^ fs/{ai' •{• c* - a^)
a ^(T a --C '
a*^(? a^^c^'
jra ^^ — ^^ V(^* 4 c* — g*)
a — c ar — c '
Fn — ^ V(^' -f c* — g*) ^ c^x
a*-(r o'— c*
Hence also we shall have
cosACZ-Crs ^ ., , . a — sr*
aoj — c v(ic + c — a)
Consequently, adopting the usual notation, the equations
of motion arc
d^x ^a\/(ic' + c'-a*) — ca;
^-33 =^5^ cosa- r — ^-i^ — jj-^ — ^ — - ,
df ^ ax-c»J{a?-{'C^'-a*y
^ df " ^"^^
when a is the angle of inclination of inclined plane ; there
is no equation in y since y = c. The equation of vis viva
manifestly implies that
a
dd _ a V(a;' -f c» ~ g") -. ca;
dx " ax- c^ {x* ■{- c^ — a^) *
10 MB. RUSSELL, ON THE HIQHEB TRANSCENDENTS, &C.
and the interest of the problem mainly consists in proving
this equation by geometry, which may be effected as follows :
Let the centre of the sphere move through a small space
from K to K\ and the point of contact of string and cylinder
from E to E\ yet so that the length of string from fixed
point to point, of contact may remain the same, in other
words, that AE^AE'. This implies that E' is the com-
mencement, E the termination of the first element of contact
in the two cases. Draw EKD to meet the axis of {x) in i>,
EK'n to meet the axis of [x] in i>'; JO/, K'M' per-
pendicular to the axis. Then EK^ E'K' = a, KM=^ K'M'^ c :
dd is the angle between ED and E'D'^ also dx is represented
by iOf'.
Then, since K'A? = (a: + dxf + c', we have, retaining the
first order of infinitesimals,
conR'AM-^"^ - * I "^^
'
cozKAM ^.^ ^^^.^^.j 1 ^^^^^^,
and, similarly.
t:
. 1 . » — — c cxdx
BmKAM = -TT-i — sr — 1,
coaZ'it'jl- ** I ^'^ as' + ei'-a*
CXjaJCj SL JUL-— '• ..• ji ' t\ r • i •
V(a! +c*) a (ai' + c')*
*
whence sin ^X'^-^^f.t'^ /) ^c& V(a:» + c" a')
also Bm^i?^=^^(^ + '^-°^-°^
cos J^'iy^l' = cosiT'^ Jf taiEK'A + sId ^^4 Jf sm E'K' A j
_ oa!+cV(g* + c'-o') 1 f oc-a!V(a!' + c'-a'') )* J
ST? "^ a t i^T? j
where cob j:i>Jl = ax + 0^(0^ + 0' -a')
as'+c*
Hence sahtracting these expressions
But i(-D+i>')=^^^, i(^--^) = i(<'^;
MB. LEUDESDORF, THEOREM IN KINEMATICS. 11
andy therefore,
dO X t^{x* + c* — a*) — ere a »J{x* 4- c* — a*) - ca?
dx x^i-c^ aa; — c VlaJ^ + c' — a') '
as Is immediately proved by multiplication.
Hence, the equation of vis viva is demonstrated geometri-
cally, and the solution of the problem is given by an equation
of the form
when <!> (x) and % (x) are rational functions of [x).
By means of an integral of the same form we may
determine the motion of an elliptic cylinder rolling on a
rough horizontal plane, of an elliptic cylinder gliding in
contact with two smooth planes, one horizontal the other
vertical, and of a sphere rolling down a rough elliptic cylinder.
We arrive at the general conclusion that an immense
number of dynamical problems^ when the motion is in parallel
planes, can be resolved by means of hyperelliptic integrals,
and the higher algebraical transcendents; and, moreover,
that a number of curious geometrical theorems are naturally
suggested by the application of the principle of vis viva to
the equations of motion in complicated cases.
NOTE ON THE THEOREM IN KINEMATICS
. (VOL. VII., p. 125).
By C. Leudesdorfi M,A.
After sending this result to the l£essengerj a shorter
5 roof than the one I had given was suggested to me by
Ir. Elliott's paper, published in the February number. It
seems even simpler than that given by Mr. Kempe at
fi. 166. If the coordinates of -4, Bj (7, P be [p^qX (p^^j
P«?i)j iPi) respectively, it is seen without diflSculty tnat
p = asp. + aP. + «P.» ? = «2. + M, + «?«>
and that
qdp = {xq, + yq^ + eq^ (ar<^, + ydp^ + adp^
= ^ix^Vi. + yi%^¥% + «? A - y^ (2. - 2 J (<^jp. - <^jPt)
so that, integrating over a complete circait,
(P) = « (4) +y (5) + « ((7) -y« /(2,- J,) rf (ft -ft) -...-... .
^"* liSa ~St)^ iPt ~Pt) '8 ^^ *"'®* traced out by
12 MR. LEUDESDORP, THEOREM IN KINEMATICS.
relatively to jB, which, since BG^a always, is equal to
nircf if the system make n complete rotations, and to zero
if the system make a partial rotation and a return, as
described by Mr. Kempe. Similarly for the other two
integrals; so that
(P)-aj(-4)-y(^)-«((7) = -n^(a'^y« + &*«a; + c*a;y) or
= nirf or (1),
t being the length of the tangent from P to the circle round
ABC.
The formula may ako be proved by supposing the motion
of the four points to be produced by tneir being carried
with a closed curve which rolls on another in its plane. It is
not necessary to give this proof, but its form has suggested
to me the existence of a theorem of a form analogous
to (1), connecting the areas of the pedals of any closed
curve with regard to four points -4, jB, (7, P in its plane
taken successively as origin. Denoting these areas by (^),
(jB), ((7), (P), that of the closed curve by (5), and the angle
made with a fixed direction by the tangent to this curve
by 0, we have
2(P) = (;S) + i/0P»rf<^,
and three similar equations, being any point in the plane.
Thus
[P)^x[A)^y{,B)^z{G)
^iJ{OP'-^xOA^''yOB^--zOC^)d<f>
(by an easily proved geometrical relation)
-\nirf or (2),
according as the tangent turns through 2n9r, or through any
angle and back again to its initial position, as its point of
contact moves round the curve. If P lie on -4P, dividing it
in the ratio c : c',
/m c[B) + d{A) . , ^
a result resembling Holditch's theorem in form.
On making (P) constant in (2), the locus of P is found
to be a circle: thus Stelner's well-known theorem follows
from (2) in tne same way that the theorem given by
Mr. Kempe at p. 190 follows from (1).
Pembroke College^ Oxford.
/
(^
^
{ 13 )
r
AEITHMETICAL NOTE,
By Prof. JT. W, Lloyd Tanner, M.A.
This note relates to numbers which have multiples of the
form 100... 001. If r be the radix of the scale of notation,
N a given number, we seek the values of m and r which
satisfy the congruence '
r^ 4- 1 =0 (mod. N) , (1).
Clearly (1) is impossible unless N be prime to r.
Assuming this to be the case, we have
r^^-.l=0(mod.J\r) (2),
where 4>^ ^^ ^^ number of numbers less than N and
\ prime to N. Put
where M is odd. Then (2) may be written ^
(r^_ 1) (,^+ 1) (^2^+ 1) (^^'if ^ i)...(^2P-^+l) = (mod. N)
(3).
The factors on the left of (3) are prime to each other;
save for a factor 2 common to all when r is odd.
I. Suppose, in the first place, that iV^is an odd prime or a
power of an odd prime. Then only one of the elementary
congruences implied in (3) can be satisfied ; and, since (3) is
) true, one of these elementary congruences must be satisfied.
If r^- 1 = (mod. N) (4),
then (I) is impossible. For writing (1) in the form
r"' = -l(mod. JV]
and raising it to the odd power if, we have
r«^ = -l(mod.i^
which is inconsistent with (4).
If (4) be not satisfied then some one of the other con-
gruences implied in (3) must hold, say
y2'if+ lHO(mod.-»0 (5),
I
14 PROP. TANNER, ARITHMETICAL NOTE.
and this gives a multiple of the required form. It is to be
noted that 2«Jf is not necessarily the smallest value of m
which satisfies (1). It is, however, easily proved that if (I)
be true as well as (5), m must be an odd multiple of 2?, and
the smallest value of m is the product of 2, into some factor
of M, To prove the first of these statements observe that if
then, since Jf is odd,
r =- 1;
But raising (5) to the power M^ we get
2«ifJf, -- J. 1^ . odd
r =T 1, accordmg as M^ is
even'
^
which 16 Qonsistent with the former result only if M^ is odd.
In the same way if ^' be less than g^ and M^ be odd, the
congruence r ' + 1=0
makes (5) impossible.
The second statement is proved when it is shown that
two congruences
2«J!f I , - _ 2«Jf2 , 1 _ A
r *+ l=r *+ 1 =0
imply a third r + 1 =
where M is the G.C.M. oi M^yM^. Now from the first pair
we get r ■= ± i.
By properly dioosing ^, k^ we can make
This done, the lower sign only is consistent with either of
the first two congruences.
II. Next take -ST to be a power of 2. Here r must be odd.
Say it is 2 V - 1 where r' is odd. Then N must be either
2 or 2** (where g[ is not greater than q) according as m is even *t
or odd. t
For if m be even (1) becomes !
(2V-lf+l=A:2'+2, \
and this is divisible by no power of 2 but the first.
1
V
PROP. TANNER, ARITHMETICAL NOTE. 15
If m be odd,
(2V- 1)- - 1 = ^2»'+ mr'2^- 1 + I,
and since 9n, / are both odd, this is divisible by any power of
2 not greater than 2q.
III. Lastly, if -AT be a composite number, say
N=2'.a.fi...
where ol^ Py.. are powers of odd primes.
Of course N must be prime to r^— 1, save as to the
factor 2', for if
r^- IE (mod. a) (6),
then a has no multiple of the form r^^ 1, and therefore N
cannot have one.
When (6) is not true, then for some value of q we must
have r +1 = (mod. a).
In this ease if r** + 1 be a multiple of a, m must be an odd
multiple of 2*. Therefore if r*" + 1 be a multiple of N^ m
must be an odd multiple of 2^ Hence it follows that q must
be the same for all the factors a, ^... .
As regards the even factors of N. If r be even, a must
vanish. If r be odd a may be or 1, unless q vanishes, when
-A/" must not be even in a higher degree than r + 1.
When ^ is a power of an odd prime, and M is small,
it is easy to see whether N has or has not a multiple of the
required form, by trying whether N divides r^— 1 or not.
For example, take the prime numbers of the form 2* + l.
Here <^-^''= 2^, M— 1 ; so that unless 2^+1 happens to be a
\ factor of r — 1, it has a multiple of the required form.
^ But this test gives no information as to the value of j,
and is therefore useless in the case of composite numbers.
To supply this deficiency observe that if (6) be satisfied, then
r is a residue, with respect to N^ of numbers raised, to the
power 2"''*, but not to the power 2''"'. For suppose r is a
residue of the powers 2^ , or
T^cf (mod. N)\
therefore r = a (mod. N) ;
therefore /'~"'^= a''^= a^^ = 1 (mod. N).
Gomparlng with (5),ive see that r is a residue of the powers
16 PSOF. TAlfKEB, AfilTHHETICAL NOTE.
^ l{p-~p'=q+l iorp'=p-~q-l)f but not when p-p' = S
{orp=p~q).
Tneee reBidaea can at once be written down if only the
quadratic residues of N are known. For if a, is the quad-
ratic residence of a^, and a, of a,, then a, is the btquactatic
residue of a, ; nnce if
a'=a^j and a,'=a„
then a*=a^
In the examples which follow, the numbers are written in
the denary scale unless otherwise stated.
[Emmple 1. N= 7, <)>N= G, M=3, p = l,
j=jj-l=sO. 1,6:2,5:3,4,
1 : 4:2.
Here 3, 5. 6 are non-residues of the squares, but of course
residues of tne first powers. For these numbers then q is
^ — 1 or 0. In each of these scales 7 has a multiple r" + 1 or
1001. In the scale of 6 there is also a multiple r -i- 1 or 11.
Example 2. N= 13, i>N= 1 2, Jf = 3, i. = 2,
q=p-l = l, 1,12; 2,11; 3, 10; 4, 9; 5, 8; 6,7,
2=j,-2 = 0. 1 ; 4 ; 9 ; 8 ; 12 ; 10 ,
1; 3; 8;9;1;9.
For 2, 5, 6, 7, 8, 11, q=l, and in each of these scales
ther« is a multiple r'+l or 1000001 of N. In the scales
5, 8 there is a smaller multiple r* + 1 or 101.
For 4, 10, 12, 2 = 0, BO that r' + l or 1001 is a multiple of
^in these scales. In the scale of 12, r+ 1 is also a multiple.
In the scales 1, 3, 9 there is no multiple of the required
form:
ExamfU 3. N= 17, ^J?= 16, M= 1, p = 4. .
y=p-I=3. 1,16:2,15:3,14:4,18:6,12:6,11:7,10:8,9
q=p-%=^. 1 : 4 : 9 : 16 : 8 : 2 : 15 : 13
j=p-8=l. 1 : 16 : 13 : 1 ; 13 : 4 : 4 : 16
«=«_i=n. 1 : 1 : 16 : 1 : 16 : 16 : 16 : 1
3 a multiple in every scale less than N bat that
le of 16, where j = 0, there is a multiple 11. In
t
PROP. MANSION, TAYLOR'S THEOREM, &€• 17
the scales of 4 and 13, 101 is a multiple. In tbe scales of
2, 8, 9, 15, for which ^ = 2, there is a multiple 10001. And
in the remaining scales (less than 17) r^+l is a multiple.
Moreover, since Jf is 1, these are in each case the smallest
multiples of the required form.
Example 4:. N=7.1^y <f}N= 72^ Jf=9, j? = 3.
Reference to Ex. 1 shews that for 7, q has only one
value 0. The scales corresponding to this value are
3, 5, 6, 10, 12, 13.,.7A + 3, 7A + 5, 7* -f 6, ... •
For -Ar=13, Ex. 2 shews that for j = the values of r
are
4, 10, 12, 17, ...13A: + 4, 13A: + 10, 13^+12, ....
Hence, N will have a multiple r^+ 1 or 1000000001 in the
scales of 10, 12, 17, &c. However, if we observe that for 7
and 13 Jf is 3, it is clear that we have in all of these scales a
smaller multiple r^ + 1 or 1001.
In a similar manner it may be found that -^ = 7,13.17
has a multiple of the required form in the scale of 101 ; the
lowest scale in which there is one.
Jpnl, 1878.
;ELEMENTART demonstration of TAYLOR'S
THEOREM FOR FUNCTIONS OF AN
IMAGINARY VARIABLE.
"By Professor Paul Mansion,
I. Fundamental formula. Let Fz^<f> (a?, y) + {yjr (a?, y)
be a function of an imaginary variable z = x-h yi^ continuous,
as also its derived function, from «o "= ^o "*" y©*' ^^ Z^X-\- Yi*,
when we pass from z^ to Z by values of oj + yi such that a?, y
are the coordinates of a point of a curve y = ^oj, continuous
also between the points a (a;^, y^ and h (X, Y). We know
that
Sx 8y ' Sy Sx * Sx Sx '
1 say that
FZ^Fz^^ (^ F'zdz (1),
J Zq
VOL. YIII. C
18 PROP. MANSION, TATLOB^S THEOEEM FOK
the second member denoting tbe limit of the sum of all the
expressions F'zdz. when ar, y take successively the values
co^pondlDg to the diffei^nt points of the curve al, or
y =s j^a?, so that dy = x'db. We have, in fact,
(:
S^4)^'
whence / Fzdz^l (-^-ht-^ldSr
IL Taylor^a theorem. Putting in this formula
we have Taylor's, or rather Cauchy's, theorem,
»•*
U. 1.2...(n-l)'^ •
III. Forms of the remainder given hy MM. Folk and
Darhoux. Denote by p^ »„ jp^, three positive arbitrary
quantities not superior to w, oy &, 5^, 6^ quantities comprised
within and ], by /& a quantity not negative and not
superior to unity, by v an indeterminate angle.
Write also, with Weierstrass, R (a + Pi) for the real part
a of an imaginary expression a + /9i, /(a + fii) for the imap^i-»
nary part fiu Taking for the curve y^yp the straight Ime
k
FUNCTIONS OP AN IMAGINARY VARIABLE, 19
which joins the points a and hj we can put the remainder
r^ in either of the following forms :
^.^,ui [z-^z,r (i-gr ,ff.^ .-1
^
z^^z^+0^{Z^z^), z^^z^^0^[Z^z^).
The first is due to M. Darboux (Liouville's Journal,
continued by Resal, September, 1876), who obtains it hj
a different method. The second is due to M. Falk (memour
presented tb the Society of Sciences of Upsala, July 19, 1877) ;
it leads more easily, than does M. Darboux's form, to the
ftindamental formula relative to the true values of expressions
that assume the form - •
IV. Applications. By means of the last forms of tie
remainder we readily establish, in an elementary manner, in
the cases where they subsist, the developments in infinite
series of the functions e", log (1 -f «), (1 + «)**, z being of the
form X + y t.
V. Demonstration of M. FaWs formula^ hy Homersham
Cox*8 metlwd. Having recourse only to Rollers theorem
we can demonstrate without difficulty tne formula
JZ- Fz, « B [(Z- z,) F\-\ + /[(Z- z,) i^'«J...(2),
analogous to (1), but less precise. Applying this relation
to the function
Fz^fZ- ^fo + ^fz + i^f'z+...
-^ IX^£l)H - ^^- ^^"''
where P is such that Ji^ = 0. and^ successively equal top^
and to »,, we obtain again Taylor's theorem, with M. Falk's
form of the remainder.
This demonstration does not essentially differ from that
of the Swedish mathematician, for he obtains Taylor's formula
C2
20 MR. GLAISHER, ON A POINT IN THE SOLUTION
ty adding tbe development of the real part of the function
to that of the imaginary part. And the formula (2) has been
obtained by tbe same procedure.
VI. Conclusion. It follows from what precedes that we
can deduce the complete theory of the elementary functions ^
(functions expressible algebraically by radicals, singly periodic
transcendental functions and their inverses), t?ie variable
heing real or imaginary^ from EoUe's theorem or from the
geometrical lemma equivalent to it, " If a plane curve AIB^
continuous between the points AB and whose tangent changes
continuously between their two points, is cut by the secant
AB^ there is an intermediate point / where the tangent is ^.
parallel to AB^ "^
Antwerp, ifay,' 1878.
EXAMPLE ILLUSTRATIVE OF A POINT IN
THE SOLUTION OF DIFFERENTIAL
EQUATIONS IN SERIES.
By J", W. L, Olaisher,
In a paper " Note on the integration of certain differential
equations by series,"* Prof. Cayley has directed attention to
a curious point in the solution of differential equations in
series, viz. that if we have a solution
("•^^-'-^i^^+H' 1
then if one of the factors in a numerator, say a^, vanishes,
we may stop at the preceding term, the finite series
V 6, 0^0^.., o^_^ J
* Messenger of Mathematics, Old Series, vol. v., pp. 77 — 83 (1869),
1
being a particular integral ; but that if we continue the series
notwithstanding this evanescent factor, and if a factor in a '
denominator, say i,, {s > r) vanishes, then the series re-
commences with the term involving af^'y and we have another
particular integral
A' ^ ^af^+ |f±! a^^' + &c.) , ^
in which A' - may be replaced by a new arbitrary constant
B, Thus if the law of formation of the coeflScients of the
i
OF DIFFERENTIAL EQUATIONS IN SERIES. 21
terms is such that a factor in a numerator first vanishes and
subsequently a factor in a denominator, we have a particular
integral consisting of a finite number of terms, and the series
recommences and gives another particular integral.
Prof. Cayley illustrates this in his paper, and in what
follows I merely propose to give another example of a
differential equation whose solution in series involves the
point in Question in a simple form.
Consider the differential equation
To integrate this in series, put
and by substitution in the differential equation we find that
the relation between A^ and -4,^^, is
(m+r+l)(/n+r-p+l)^^j-(2m+2r-p)(2?n+2r-;?-|-l)J.^=0.
Let -4_j, A^... be each equal to zero, and putting r = — 1,
the values of m are given by the equation
viz. m = 0, or m =^. Taking the root m = 0, we have
_ l?.j?-l .
^» 2.i>-2 ^»'
-4- = ——z — '^ ^ -4_, &c.,
whence the particular integral corresponding to the root
m = is
p-l p-l.p-'2 2!
and, similarlyi the particular integral corresponding to
m^-p is
[ p\ 1 ^ + l.p-h 2 2 1 j -
Now suppose p not = ± an integer; these series are
independent, neither terminates, and the general integral
♦ The algebraical identity P = P' — Q is readily proved by putting p = n + h,
dividing out the factor h when it occurs in both numerator and denominator, and
making h indefinitely small. It is this consideration that justifies the dividing
out by or » — n above.
I
22 MR. GLAISHER, ON A POINT IN THE SOLUTION, &C. .^
of the differential equation is AP-\-BQy A and B being N
arbitrary constants.
Suppose p = a positive Integer, n, then P consists of a
finite portion
-,, . w.n — 1 w.w— 1.71-2.W — 3a5* ^A
n-1 n-l.n-2 2l *
the last term being
I J w-l.n-2...iw [\n)V ^ ^ ^^ »
if n be even, and (— )*^'*"^^/iic*^*"^^ if n be uneven. ^ '
There are then ^w — 1 or ^ (n — 1) zero terms according ^y
as n is even or uneven, and the series recommences with the
term in oj", and is
, .„n! 0.- 1.-2...- (n-1) aj** f, ^.n+1 ^ )
H n-l.n-2...1.0 T! j^ + ITm-^ + ^'^ j
if we divide out the factor orn — w from the numerator and
denominator. Thus, when the series recommences, it gives
the other particular integral, and the general integral may
be obtained from P alone and is AP' + BQ.
Similarly \ip = a negative integer, Q = Q' — P, where Q
denotes the finite part of the series continued up to the terra
immediately preceding the first term that contains a zero
factor in a numerator, and the general integral is -4 Q' + BP,
Kp be unrestricted we have the two particular Integrals
Pand Q, but if p be a positive Integer P=P- Q^ so that
we may take the two particular integrals to be P and Q,
and the point to be specially noticed is this : Knowing that,
for all values of p^ P, extending to infinity. Is a particular
integral, it might seem that when P stops and recommences,
it would be necessary to include the whole series up to
infinity, so that the particular Integral would be P + J?, and
we should have, not a finite integral, but an infinite series
with certain terms omitted ; but in reality we may take the
finite portion P as a new particular integral, and It must be
of the form, const, x Q,
' it
4
4
^ KB. DARWIN| NOTE ON THOMSON'S THEORY, &C. 23
In this case the two Infinite series P, Q admit of sum-
mation, for
and, as is well known,* when ji = a positive integer,
P+(2 = l-;w+Ml-?aj*-&c.,
. the series extending to^o + l or Hp + l) terms accordmg
f as j7 is even or uneven, aod if p = a negative integer,
P+e = af|l+;?aj+^^|i^«« + &c.l,
the series extending to — J^^ + 1 or J (— 2^ + 1) terms according
as p is even or uneven.
^ HP and Q had not admitted of summation, we should
have had the more usual case of the differential equation not
possessing a finite particular integral except in the cases
when. one of the series terminates through the presence of
a zero factor in a numerator. This happens in tne solution
a*u
of Eiccati's equation, -7-5 — a5^'*ti = 0, and its numerous
^ transformations, where the series terminate when q=^± the
reciprocal of an uneven integer, and we have two finite
particular integrals in each case, so that the equation is
nnitelj integrable for these forms of ;, and for these forms
only, as is well known. But the above example illustrates
equally well, and in a simpler form, the points in the theory
t already alluded to, and also how it happens that an equation
may admit of finite integration in certain cases and not
in others.
NOTE ON THOMSON'S THEORY OF THE TIDES
OF AN ELASTIC SPHERE.
f By O. H. Darwin, M,A,
The results of the theory of the elastic yielding of the
earth would of course be more interesting, if it were possible
to fully introduce the effects of the want of homogeneity of
elasticity and density of the interior of the earth ; but besides
^ * Todhunter's J)ifer§ntial CcUculus, chapter ix.
i
1
24 MR. DAnWIN, NOtB ON THOMSON'S THEORY
the mathematical difficulties of the case, the complete absence
of data as to the nature of the deep-seated matter niakes
it impossible to do so. It is, however, possible to make a
more or less probable estimate of the extent to which a
given yielding of the surface will affect the ocean tide-wave, ^
when the earth is trieated as - heterogeneous. And as we
can only judge of the amount of the bodily tide in the earth
by observations on the ocean tides, this estimate may be
of some value.
The heterogeneity of the interior must of course be
accompanied by heterogeneity of elasticity,* and under the
influence of a given tide generating force, this will affect
the internal distribution of strain, and the form of the surface ^t
to an unknown extent. The diminution of ocean tide which
arises from the yielding of the nucleus is entirely due to the
alteration in the form of the level surfaces outside the
nucleus. But it is by no means obvious how far the potential .
of the earth, when its surface is distorted to a given amount,
may differ from that of the homogeneous spheroid considered
by Sir W. Thomson ; and in face of our ignorance of the
law of internal elasticity, the problem does not admit of
a precise solution.
I propose, however, to make an hypothesis, which seems
as probable as anjr other, as to the law of the ellipticity
of the internal stram ellipsoids, when the surface is strained
to a given amount, ana then to find the potential at an
external point.
Suppose, then, that under the influence of a bodily har-
monic potential of the second degree the earth's surface-
assumes the form r = a-\' a^ where tr is a surface harmonic of
the second order. Then I propose to assume that the
ellipticity of any internal strain ellipsoid is related to that of
the surface by the same Jaw as though the earth were
homogeneous, elastic, and incompressible, and had its surface
brought into the form r = a + cr by a tide generating poten-
tial of the second order. If fi be the coefficient of rigidity
of an elastic incompressible sphere under the action of a
bodily force, of which the potential is wr^8^ then Sir W.
Thomson's solutionf shews that the radial displacement at
any point r Is given by
* That is to say, if the earth is elastic at all.
t ThoDBBon and Tait's Natural Philosophy^ § 834, equation. (14).
OF THE TIDES OF AN ELASTIC SPHERE. 25
5a'
Putting r = a, we have o* = — • S^. And If r = a' + o*' be
the equation to a strain ellipsoid of mean radius a', we have
by our hypothesis
^_o^_ 8a* " 3a *
a' ' a" 5a» '
and <r'= ^ |f - I {^^ | ^=/(«') o- suppose.
Now the potential of a homogeneous spheroid r=^a' ■\-af[a)^
of density 2, at an external point is
|,j ^\ ^ ay (a-) .,
and therefore the potential of a spheroidal shell of density q^
whose inner and outer surfaces are given by r = a' +/(«') o*
and r = a' + ha -^/{a' -I Ba) cr, is
4^2 ^ Sa' + f^ ^. {ay (a')} cr8a'.
If then we integrate this expression from a' = a to a' = 0, and
treat ^ as a function of a'j we have the potential of the earth
on the present hypothesis. The integral is
-^ J qa''da' + ~ j i'da! ^'^'^^^^^^^ ^'
a*
The first of these two terms is clearly g — (where ^^ is gravity),
and is the same as though the earth were homogeneous ; and
it only remains to evaluate the second. Now, according to
the Laplacean law of internal density of the earth, if D
be the mean density, and / the ratio of D to the surface
density, and d a certain angle which Is about 144^
_^D a a
^ f mid' a
Substituting this value for j, and for /(a') its value, we have
. a'e
, j^ sin — ,5
Then If we change the variable of integration by putting
26 MR. DAKWINi NOTB ON THOMSON'S THEORY, &C.
« = — , and put 4tirD^-^. we get for the second term
a ' ^ a ' °
of the earthed potential
Now f^(-) <^ would be the potential of the surface layer
given by r = a + o-, tf the earth were homogeneous and had
a density i), and the rest of the expression is a numerical
factor (which may be called JST), by whiclT this potential
must be reduced in order to get the potential of the hetero-
geneous earth on the present hypothesis.
If the integration be effected it will be found that
K^rr ./iR 76 2161 ^r32 2161 4 216
i/2r=-cottf^g+g^--^J-cosec5[^^ + -^-J-g,H.-^
= 5-1442, when 5- 144\
Whence K= — ^ — .
Also /= 3 [^ - -^) = 2-1178 by Laplace's theory.
Therefore K^ |^ - -972.
2-1178
Hence, on the present hypothesis, the potential of the earth
at a point outside its mass is
9^ + (-972) y (?)V.
This differs by very little from what it would be if the earth
were homogeneous; for in that case '972 would be merely
replaced by unity.
Therefore, if at any future time it should be found that
the fortnightly tide* is less than it would be theoretically on
a rigid nucleus, it will then be probable that the surface of
the earth rises and falls by about the same amount as would
follow from the theory of the bodily tides of a homogeneous
elastic sphere whose density is equal to the earth's mean
density. This investigation being founded on conjecture,
cannot claim anything better than a probability for its result ;
but without calculation, I, at least, could not form any sort of
guess of what the result might be, and the question is of
undoubted interest in the physics of the earth.
* Sir W. Thomson relies principally on observation of the fortnightly ocean
tide for detecting bodily tides in the earth. '
1
( 27 )
TRAl^SACTIONS OP SOCIETIES.
London Mathematieal Society,
Thursday, April llth, 1878.— 0. W. Merrifielcl, Esq., F.R.S., Viee-Presideni. in
the Chair. Mr. Artemas Martin, M.A., Erie, PennsylyanuL was elected a
xnemt)er: Mr. W. M. Hicks, M.A., and Mr. T. R. Terry, M.A., were proposed
for election. The Chairman, on the recommendation of the Council, nominated
ProfessoiB Brioschi (Milan), Darbonx (Paris), Gk)rdan (Erlangen), Sophus Lie
(Christiania), and Mannheim (Paris), as Honorary Foreign Members of the Society,
the Council having decided to increase the number of foreign members from seyen
to twelve. Prof. H. J. S. Smith, F.B.S., read two papers, " Second notice on the
characteristics of the Modular Curves " and a ** Note relating to the Theory of the
Division of the Circle.'* Prof. Cayley spoke on the subject of both papers, asking,
in the course of his remarks, if a solution had been effected for the inscription of
a regular heptagon, assuming the trisection of an angle. Mr. Tucker read a letter
from Prof. Tait, in which he stated that by a simple physical process he could
easily obtain any number of definite integrals similar to the following :
ffemflrf^de r'-a«-X« + iaXco»e
J J • [r*fa«+X*-2aX cos 0- 2r\ cos a+2flrr(cos a cos d-sin o sin 6 cos «^)1»
2ir{f^^ X«)
(r« + X« - 2rX cos o) ♦
Prof. Tait said he wished to know whether the solution could be easily
effected by direct analytical processes. Mr. Tucker then read an abstract of a
paper by Prof. Minchin, ^^ On the astatic conditions of a body acted on by given
forces." When a bod^ is acted on by given forces applied at given points in the
body, if it is in equilibrium, it will under certain conditions remain so when it
is displaced in any manner, each force retaining its magnitude, direction, and
point of application in the body. The requisite conditions are called the astatic
conditions. The investigation of them by ordinary Cartesian methods is given in
Moigno's StatiqvM at great length. Prof. Minchin s paper treat^s them by elemen-
tary quaternion methods, and adds a few geometrical results not noticed by
Moigno. The paper also contains a proof of Minding's theorem, viz. that in
certain positions of displacement of the body, the given forces reduce to a single
resultant force; and when this happens, the line of action of the single force
intersects both focal conies of a certain quadric, having for centre the centre of
the central plane, and this plane for one of its principal planes. He then read
part of a paper by Mr. C. Leudesdorf, "On certain extensions of FruUani's
Tlieorem." The object of the paper is to supplement two papers communicated
by Mr. E. B. Elliott, and printed in the Society's Proceedings,
Thursday, May 9th. — ^Lord Rayleigh, President, in the Qiair. Messrs. W. M.
Hicks and T. R. Terry were elect^ Members, and Prof. G-. M. Minchin was
admitted into the Society. Professors Brioschi, Darbouz, Gordan, Sophus Lie,
and Mannheim were elected Honorary Foreign Members. Prof. Henrici, F.R.S.,
communicated a paper bv Dr. Klein, of Munich, " Ueber die Transformation der
Elliptischen Functionen;" Prof. Cayley, F.R.S., gave an account of a paper on
the " Theory of Groups j" Prof. Kennedy read a paper " On some new solutions
for certain problems in the Statics of Mechanisms.'' The special object of the
paper was to give an elementary solution of the problem : Given a linkwork or
plane mechanism of any number of links, with any force acting on any one of
them, to find the magnitude of the force necessary to balance the mechanisms if
acting in any direction on any other link. The method employed was the replace-
ment of the two links on which the forces acted by two others, which IukI the
same instantaneous centres and the same angular- velocity-ratio, but which were
80 chosen that they could be directly connect^ together by a third link. In this
way a simple combination of three links was used as a "critic^ mechanism" to
replace the original complex linkwork, and the solution became extremely simple.
Incidentally the author took occasion to insist on the advantages of the consistent
use of the action of the instantaneous centre even in the most elementary
treatments of mechanical problems. Mr. J. W. L. Glaisher, F.R.S., briefly gave
the principal results contained in his paper " GisneraUzed Form of Certain Series,"
and Mr. Kempe spoke "On Conjugate Four-piece Linkages."
R. Tuck BR, M.A., Hon. Sec,
( 28 )
AN ENUMERATION OF PRIME-PAlRS.
By /. JF. L. Qlaiiher.
§ 1. The present paper contains the tables of the number
of prime-pairs alluded to in the last paragraph of my paper
"On Long Successions of Composite Numbers'* (vol. Vii.,
p. 176), I call a pair of primes separated by only one number
a, prime-pair 'y thus 11 and 13, 17 and 19, 29 and 31, &c., are
prime-pairs. It is clear that, as the number of primes
decreases, as we ascend higher in the series of numerals,
the number of prime-pairs must decrease also, and I was
desirous of examining the rapidity of this decrease. I,
accordingly, had the number of prime-pairs in each thousand
or chiliad, in each of the first hundred thousand numbers
in each of the six millions over which Burckhardt's and
Dase's tables extend, counted: the results are arranged in
the following tables.
The explanation of the tables is as follows : The number
of prime-pairs between and 1,000 is 36, between 1,000 and
2,000 is 26, between 2,000 and 3,000 is 21, ... between 10,000
and 11,000 is 16, and so on. Thus, for example, the number
of prime-pairs between 83,000 and 84,000 is 8, between
1,064,000 and 1,065,000 is 7, between 8,041,000 and 8,042,000
is 3, &c. When a prime-pair extends over two chiliads, viz.
when it is ...999 and ...001, it is counted as belonging to
the lower of the two chiliads, and the number of prime-pairs
in that chiliad is marked with an asterisk. Thus the prime-
pairs between 1,028,000 and 1,029,000 are
1,028,099) 189) 327) 471) 579) 681) 747)
101), 191), 329), 473), 581), 683), 749),
939) 1,028,9991
941), 1,029,001/,
and are 9 in number, the 9 being marked by an asterisk;
and the last prime-pair is not counted in the 4 prime-pairs
between 1,029,000 and 1,030,000, which are
1,029,359) 407) 839) 881)
361), 409), 841), 883).
It should be stated that in the first chiliad of the first million
1 and 3, 3 and 5, 5 and 7 are counted as prime-pairs, the
number 2 being ignored.
The enumeration was made in duplicate from Burckhardt's
and Base's tables, and the two calculations were compared and
brought into agreement, so that the tables ought to be free
.r
HB. GLAISHEB, AN ENUMERATION OF PRIME-PAIRS. 29
from error. One enumeration of the portion from to
100,000 was made from Burckbardt and the - other from
Cbernac.
First Million.
0123466789
6
7
8
9
36 26 21* 21 23 17 19 13 15* 15
16 14 11 15 11 12 13 18 12 15
15 15 16 14 -6 12 11 15 12 9
11 13 19 13 16 11 9 12 11 9
8 12 12 11 11 7 12 11 13 17
9 14 8 11 8 11 14 9 11 11
10 6 8 10 9 14 & 10 9 12
16 10 11 6 10 9 11 9 9 11
13 12 13 8 12 11 8 14 10 8
10 10 12 12 13 8 7 15 13 8
Number of prime-pairs between and 100,000 = 1,225.
Second Million,
1
2 3
4
5 ,
6
7
8
9
11
6
9 9
4
9
ii
7
10
8
1
7
5
8 9
6
6
10
8
4
6
2
8
10
5 9
4
7
10
7
9*
4
3
7
7
7 7
13
4
9
3
7
5
4
9
9
7 8
6
9
7
11
6
7
5
7
6
7 6
6
6
6
6
8*
4
6
6
4
9 8
7
6
5
5
11
8
7
6
8
5 9
11
9
5
6
6
6
8
9
9
6 8
2
4
9
6
8
4
9
12
12
8 10
8
8
6
7
5
8
Number
of prime-pairs between 1,000,000 and 1,100,00 =
Third Million.
1
2 3
4
5
6
6
6
7
16
8
4
9
6
6
7 6*
5
5
1
3
6
8 6
8
8
9
4
7
6
2
7
6
10 6
9
6
5
8
8
11
3
8
2
11 8
3
6
7
8
5
5
4
8
1
3 6
7
8
6
7
1
8
5
6
10
5 7
7
5
7
8
6
6
6
9
4
8 9
6
5
8
5
7
5
7
8
5
5 7
7
8*
3
5
5
11
8
4
4
5 7
7
6
11
7
12
6
9
5
5
8 3
5
3
8
7
6
3
Number of prime-pairs between 2,000,000 and 2,100,000 = 644,
,-<^
y
30 MR. GLAISHEB, AX ENUMERATION OF PRIME-PAlRS.
&
iventi
I Milium.
1
2
3
4 5 .6
7
8 9
•
6
2
6
7
10 8 5
8
5 7
1
8
6
6
6
5 7 6
6
4 8
2
5
5
6
6
8 4 6
6
4 4
3
6
4
2
6
4 . 8* 1
7
3* 4
4
5
9
7
7
6 5 2
6
5 4
5
5
10
4
4
5 5 6
4
8 6
6
4
2
5
5
14 9
4
4 7
7
3
7
7
7
4 7 7
3
7 8
8
8
6
6
6
7 3 4
8
8 6*
9
6
4
2
3
5 5 8
5
5» 4
Number
ofprin
ne-pa
ITS between 6,000,000 and 6,100,000 = 6
Eighth Miliion.
23456789
1
2
3
4
5
6
7
8
9
Number
3
4
6
7
5
5
3
4
7
6
6
2
7
3
6
3
6
6
9
6
5
9*
3
5
5
5
5
7
3
5 .
8
5
5
7*
6
7
3
6
5
7
6
4
5
7
8
7
9
6
4
2
6
5
3
4
3
7
5
5
6
3
6
4
2
5
5
4
3
6
3
4
9
5
5
4
4
3
8
4
8
3
•9
5
3
4
3
3
6
6
9
6
6
5
6
6
5
8
9
4
2
of prime-pairs between 7,000,000 and 7,100,000 = 525.
Ninth Milium,
1
2
3
4
5
6
7
8
9
13
4
4
6
7
3
9
6
4
11
4
3
6
6
6
4
7
4
5
4
7
2
4
4
2
9
7
4
4
4
8
5
6
1
4
1
6
4
4
6
3
4
4
7
8
6
6
8
6
4
2
3
6
7
8
7
5
4
18
5
7
5
4
5
6
5
7
6
5
9
6
5
1
5
7
5
5
11
7
3
4
8
4
3
6
2
5
5
5
4
1
4
5
4
6
4
2
5
8
1
2
3
4
5
6
7
8
9
Number of prime-pairs between 8,000,000 and 8,100,000 s 518.
»i .
N,
§2. In the following table is shown the number of
MR. GLAISHER, AN ENUMERATION OF PRIME-PAIRS. 31
chiliads, in the first hundred chiliads of each million, each of
which contains no prime-pair, 1 prime-pair, 2 prime-pairs, &c.
Thus, for example, of the hundred chiliads between
6,000,000 and 6,100,000 two chiliads contain each 1 prime-
pairs, twenty contains each 6 prime-pairs, while only two
contain 10 prime-pairs.
Number of chiliads^ each of which contains
the first hundred chiliads of each
1,000,000 2,000,000 6,000,000
to to to to
100,000 1,100,000 2,100,000 6,100,000
n prtme-patrsj %n
million.
7,000,000 8,000,000
to to
7,100,000 8,100,000
1
2
3
4
5
6
4
7
2
8
9
9
10
10
8
11
17
12
13
13
9
14
6
16
8
16
4
17
2
18
1
19
2
21
2
23
1
26
1
36
1
1
1
9
8
21
16
15
16
5
5
2
1
2
1
7
5
19
20
16
18
4
2
4
1
2
5
6
19
19
20
14
11
2
2
4
17
13
23
21
10
5
7
1
4
5
6
26
17
17
11
7
a
2
1
The nnmbers of prime-pairs, as given by the aboye tables,
are rather less than one-tenth of the numbers of primes in the
same intervals. For the numbers of primes between and
100,000, 1,000,000 and 1,100,000, ... 8,000,000 and 8,100,000,
are respectively 9,593, 7,216, 6,874, 6,397, 6,369, 6,250. The
number of primes between and 10,000 (including both 1
and 2 as primes) is 1230, and it is a curious coincidence that
this is so nearly equal to the number of prime-pairs between
and 100,000, viz. 1,325.
32 MR. GLAISHERy AN ENUMERATION OF PRIME-PAIRS.
There can be little or no doubt that the number of prime-
E airs is unlimited ; but it would be interesting, though pro -
ably not easy, to prove this.
§3. On p. 106, vol VII., a proof is given that any
sequence of composite numbers, however long, must occur
at a definite place in the series of numerals, for if p^ q be
two consecutive primes, the y — 2 numbers immediately
following (2.3.5 7...^)+! are composite. This is at once
seen to be true, but the reason for it may be more fully
explained as follows :
Consider the process of formation of a factor table, which
shall give the least-factor of every number, but such least-
factor not exceeding a given prime o. The entries in such
a table will be periodic, the period oeing 2.3.5.7...^. For,
to fix the ideas, imagine the table to consist of an infinite
strip, the natural numbers being written in order at equal
distances from another, and the least-factor of each being
placed opposite to it, as below :
rH©qC0'^OCDI>a005OrHC^C0'^»CC0t^Q00iQ
To construct such a table we place a 2 opposite every
second number, starting from the beginning ; we may then
place a 3 opposite every third number, but we must then
erase the 3 from every sixth number which will have 2 as
its least-factor. Thus, as far as the factors 2 and 3 ai^e
concerned, the table is periodic, .the period being 6, or 2.3.
Similarly, if we place a 5 opposite to every fifth number,
we must erase it from every tenth and fifteenth number, and
we see that, as far as the least-factors 2, 3, 5 are concerned,
the table repeats itself after every 2.3.5 numbers. This
reasoning is clearly general, and if we enter all the least-
factors not exceeding p, the table repeats itself after 2.3.5.7...^
numbers. But observe that the first time any least^factor
appears in the table, it is equal to the number opposite to
which it is written, that number being a prime, viz. the first
time any least-factor appears in the table, it indicates a prime,
and whenever it appears subsequently it indicates a composite
number.
Now if we form a factor table, entering all least factors not
exceeding p, then the numbers 2, 3, 4, 5,...y — 1, where q is the
next prime to py will have a least-factor opposite to each of
them, and the table repeats itself with a period of (2.3.5.7...^),
v^
MR. SCOTT, ON SOME THEOREMS IN DETERMINANTS. 33
or say, for brevity, P; it follows that the numbers P+2,
P+3, P+4, ... P+q— 1 will have a least factor opposite
to each of them. The fact that the numbers 2, 3, 4, ...y- 1
have a factor opposite to each of them implies nothing in
regard to a sequence of composite numbers, as this will be
the first appearance of many of the least-factors; but when
they occur again opposit^e to P+2, P+ 3, ... P+y- 1, each
least factor indicates a composite number, and we have a
sequence o( q- 1 composite numbers. As the number P+ 1
corresponds to 1^, it may or may not be really prime, but if
only least-factors not exceeding p be entered, it will have a
blank opposite to it.
Practically, of course, in a factor table when the least-
factor is equal to the number itself, the latter is prime and no
least-factor is entered ; but in considering such a table
theoretically, it is convenient to suppose the least-factor
always entered, in which case the primes correspond to first
appearances of least factors, and the composite numbers to
subsequent appearances.
A glance at the above specimen table shows that if we
suppose only 2's and 3's entered as least-factors, then, as the
table repeats itself with period 6, only places opposite numbers
of the form 6w ± 1 will be blank, i.e. that, for numbers above 6,
only numbers of this form can be primes. This is of course
proved at once, and is well known, but it is noticed here as
the most simple consequence of the periodicity^ which is a
fundamental property of factor-tables.
ON SOME THEOREMS IN DETERMINANTS.
By R, F. Scott, M.A., St. John's College.
1. If when x=-a the elements of p columns of a de-
terminant become identical, the determinant is divisible by
{x - ar\
For subtract one of the columns from each of the
remaining p—l others, then when a? = a the elements of
these columns vanish, and hence each is divisible by x — a.
The value of the determinant not having been altered by
this operation the truth of the theorem is at once manifest.
The theorem is also clearly true when the elements of the
j> columns do not become identical but are only proportional.
VOL. VIII. D
34 ME. SCOTT, ON SOME THEOREMS IN DETERMINANTS.
Ex. If a,, at**-a» are any n quantities whose som is S^
and if A^=- o- a^, then
i) =
«-A) ««
...d.
a.
m X ^ ^^mt •••0^—
a.
. «,
= a!(a;-iS)"-.
For, adding all the columns to the first, the elements of
that column are all equal to x^ thus a; is a factor of the
determinant. Putting x=8j all the elements of the i^
column are a,, whence, by the above theorem, the determinant
divides by (a; -;SP.
Thus D^hx[x^Sr",
where Z; is a factor, which we see is unity by comparing the
coefficients of a;".
This theorem was given by M. Lucas, Comptes BenduSj
1870, p. 1167, without proof, and has been proved less
briefly by M. Albeggiani, Giomale^ xiii. p. 107, and by
M. Guenther, Zeithschrift^ XXI. p. 178, and Erlangen Sitzungs-
herichte^ 23 June, 1873.
In the same way it follows that
, A
• • • aX ~~ Cw
= {aj-.(n-2)fif}(aj-fif)-*.
2. Determinants of the form
JD =
x^j a?^, ajg, .,.x^
^«> ^1) ^s> •••^it-i
^a? ^8? ^4 J •'•^1
have been frequently studied and possess many interesting
properties.
Zehfuss, Zeitschriji VII., p. 439, states that they were
discussed by Bessel in a correspondence with Hesse, about
the year 1830. They have also been considered by Stern,
Crelley LXXIII. p. 374.
The peculiarity of the determinant D is that it *admits of
resolution into linear factors^ viz. it is the product of the
n factors
a?, + oaj, + aV, +...+ a^'V,
CO-
MR. S*(X)TT, ON SOME THEOREM^ IN DETERMINANTS. 35
when a IS a root of the equation «"- 1=0. See Spottiswoode,
Crelle^ Li., 375, or Baltzer, p. 100.
We have then
jD = (iCj 4 a?2 +...+ a;J IT (a?, + otr, +...+ a""*irj...(ii),
where the product FI includes all factors of the form (i).
I propose in the present note to point out one or two
properties of this determinant and to consider one or two
special cases. The factor corresponding to a = 1 is kept
outside the product as being more convenient in applications
of the formula.
If u be the logarithm of the determinant, we have
du
1
dxi a;,-f cCj-f ...+ a?,
Thus we have
du
dx^
-1 du
dx_
+ S
du ' du
dx^ dx^
du du «
-f a -7—+...+ a
. —
a-^
**
^1
^ + aa;5j+...+ a""V^
n
^,
n
•+ a"X
.(iii).
dx
dx.
...(iv),
where ^ is that root of 2" - 1 = such that ayS = 1.
Multiplying the equations (iv) together we see at once
by aid of (ii), that if i)' be the determinant formed from i>
by substituting -j-^^^-f^ ^^r a;,...a7^, then
If, for brevity, we denote the factors of D by X^...X^^
we see that
c?w
dx.dxt
r.-2
a'-^a*-^
d logX^ d logX,
dxi dxj^
Hence, if H{u) be the Hessian of w,
d logXr *
or
H[u) = -^
dxi
dxi
Thus i>'£^(u) = n* (- 1)*^'*("^^>^ ;
to which various other forms may be given.
0^
36 MR. SCOTT, ON SOME THEOREMS IN DETERMINANTS.
Many interesting results can be obtained by giving special
values to the quantities a? , a?^, ...a?^.
For brevity, let us write
(I) 2)= I a, a + J, a + 25, ... a + (« - 1) J |
which since a* = 1 gives
i? = in{2a + (n-l)J}n(^)
A special case of this is
I 1, 2...W I = i (n + 1) (- w)"-'.
(II) 1 1«, 2«, ...n' 1 = (-ip ("+I)f2^+1)>»'^ {(n-f2)--«-).
(III) 2>=Ic.,c.,...cJ,
where c„ c„ ...c^ are the coefficients of the binomial expansion
(l + ojp. Then
i> = 2"-»,ifnisodd,
= 0, if n is even.
(IV) 2)= I cosa, cos(a + J)...cos{«+ (n- 1) b] \ .
To effect the evaluation of this determinant, substitute for
the cosines their exponential values
cosa = i (w+-)) cos(a4 b)=^i^[uv-]- — J, &c.
Then D is the product of such factors as
i [(« + !) + («v4-i;)«+...+ («.- + ^)a-}
= iiw- +- :i — =i-r
cosg — cos (g 4- ?^i) — a [cos (a—b)- cos {q + (n — 1) &}] ^
afa +- — 2 cos J]
M. LUCAS, ON THE ANGLES OP FIVE CIRCLES, &C. 37
therefore
__ [cosa — cos (a + wJ)]* — [cos (a — J) — cos {a -h (n — 1) J}]*
2> =
2^1- cos nb)
(V). In like manner
I sina, sin(a + i)...8in {a+ (n— 1) J} |
_ [sing- sin(q-h?2&)]"— [sin (a — &)— sin (g-h (n— 1) b}]*
"" 2 (1 — Gosnb)
3. We are also able to solve the partial differential
equation
A, A, -A
where
i)r =
Cfer*
Viz. the solution of this equation is the sum of the solutions
of the different equations
du du ,_j du
-J — ha^} — h...+ a -J —
fltej dx^ dx^
= 0,
where a" = l. This is of Lagrange's form, and
Thus the solution of the given equation is
u = 2/*(a7j — oa?,, x^ — ' a*a?j, . . .x^ — oT^x^^
when the summation extends to all values of a being roots
of the equation «" — 1 = 0.
ON THE EELATIONS BETWEEN THE ANGLES
OF FIVE CIRCLES IN A PLANE OR OF
SIX SPHERES IN SPACE
By Edouard Lucas.
The definition of the potency of a point with respect
to a circle or a sphere, its expression in the system of
Cartesian coordinates, and the most elementary properties of
determinants, enable us to arrive immediately at a knowledge
of the most general relations concerning the distances of
points, angles, straight lines, and circles in a plane, and the
angles of planes and spheres in space. The relations of the
38 M. LUCAS, ON THE ANGLES OF FIVE CIRCLES, &C.
distances and angles are included, to a great extent, in
the following fundamental proposition.
The potencies of any point with respect to five circles in
a plane or with respect to six spheres in space are connected
by a linear and homogeneous equation in which the sum of
the coefficients is zero.
Let a?,- = a;* 4-y — 2a,-a5 — 2hiff + c,-,
the first member of the equation of a circle in rectangular
coordinates; we know that c,- and a?,- represent respectively
the potency of the origin and of any point J/, coordinates
ar, y, with respect to this circle. Eliminating a?, y and'a:* +.y'
between the four equations obtained with t= 1, 2, 3, 4, we
have the identity*
a-,1
o.>
K
1
c,.
''n
K
1
^,.
«.'
K
1
«,»
«..
K
1
a'.j
«..
K
1
"..
«..
K
1
a;.,
«4>
K
1
c«
««>
K
1
Denote by Z, ^, w, p the coefficients of a:,, a:,, a?,, a;^ in the
development of the determinant forming the first member, and
by q the value of the second member, we have then
Ix^ + mx^ + nx^+px^- j = (1);
but if we replace ajj, x^^ a?,, x^ i)y unity or by the same
number the first determinant vanishes, and we have
Z-fm-f 71+^ = (2).
We thus obtain the following principle :
The potencies of a point with respect to four circles in a
plane are connected by a linear equation, not homogeneous,
m which the sum of the coefficients of the powers is zero.
If the four circles are orthogonal to a fifth, the centre of
the latter has the same potency with respect to the four
circles, whence y = 0, and consequently :
The potencies of a point with respect to four circles,
orthogonal to the same circle, are connected by a linear and
homogeneous equation in which the sum of the coefficients
IS zero.
To prove the fundamental theorem it suffices to consider
a fifth circle, and we have, for example,
l'x^ + m'x^-\-n\ + p\-'q' = (3),
I', +7n' +n' +/ =0 (4).
Nouvelles Annates de Mathematiqnes, Second Series, t. XV. p. 205.
M. LUCAS, ON THE ANGLES OP FIVE CIRCLES, &C. 39
Multiplying the two members of the equalities (I) and (2)
by j', those of the equalities (3) and (4) by j, and subtracting,
we have the demonstration of the theorem.
We define the muttml potency of two circles radii r,- and r^-,
centres 0,. and 0,-, distance between the centres d^j^ to be -4 j,
where
2r.rjA^=^r; + r/-di/ (5);
when the circles cut, A^j represents the cosine of the angle
of the two circles ; we have also
^.. = 0, u4..,. = l, Aij = Aji.
Now suppose the point M to be at the centre Oj of a
.circle of radius rj ; the potency of this point with respect to
the circle of centre 0,. is
Xf = dj - ri",
viz. Xi = r/'-2rirjA^ (6);
but, by the fundamental theorem,
Zajj + mx^ + ncCj +px^ + qx^ = 0,
I +w +n -^p +j =0;
and consequently comparing the three preceding formulas,
Ir^rjAj -f mrjTjA^ + nr^rjAj -^pr^rjAj + qrjr^A^ = 0.
Making^ equal to 1, 2, 3, 4, 5, in the preceding equation we
obtain five equations, which give, by the elimination of
Z, m, n, p^ J, the determinant
'•.'•.^.1)
v,^..>
'•/.^..j
'•i'"«^u>
»'/.^..
v.^...
v,^«.
V.^M»
V«^M.
V.^M
Vl-^M)
WK^
V.^M.
v«^««.
v.^»
'^i'xK^
V,-^4..
v.^«>
V«^i4»
7* *• /I
' 4,' 6'^*46
V.^.o
vA»
Vs^5S»
V^^MJ
I* 7* /I
= 0,
(7).
K no one of the radii is zero, we can suppress all the radii ;
if one of them r,. is zero, we can replace r^vjAij by d^tj — r^*,
and r^r-Ai^ by 0; if two radii r. and Vj are zero, we replace
r-r^ by d'^^j. Thus, when the system of two circles is formed
of a circle and a point, we replace in the relation (7) the
mutual potency of the two circles by the potency of the point
with regard to the circle; and if the system is formed of
two points, we replace the mutual potency by the square of
their distance.
If the radius r,- of one of the circles increases indefinitely,
and if the circle becomes a straight line, divide by ri and
- ^
1
\
40 M. LUCAS, ON THE ANGLES OF FIVE CIRCLES, &C.
replace Aij by ~ or by the cosine of the angle between the
straight line and the circle, when the straight line and circle
cut one another. Thus, when the system of two circles is
formed of a circle and a straight line, we replace in the
relation (7) the mutual potency by the quotient of the dis-
tance of the centre to the straight line by the radius ; we see
also that when the system of two circles is formed of a point
and a straight line, we replace the mutual potency by the
distance of the point from the straight line.
In the system formed by a circle and a straight line which
passes to an indefinite distance, we replace the mutual potency
Dy the reciprocal of the radius.
In the system formed by a point and the straight line at
infinity, we replace the mutual potency by unity; in the
systeuL formed by two straight lines we replace the mutual
potency by the cosine of their angle ; if one of the straight
lines passes to an infinite distance, we replace the mutual
potency by zero.
Consequently, applying, as has been said, the notion of the
mutual potenjcy of the points, of the straight lines, of the
circles and of the straight lines at infinity, we have the follow-
ing theorem ;
The determinant formed by the mutual potencies of any
points, straight lines or circles in a plane, whose number is
at least equal to five^ is identically zero.
We have evidently the same theorem in space for any six
points, planes or spheres, or for any one or a greater number
of elements.
The following are some corollaries from this important
theorem :
1. If the circle 0^ is a point on the circle (?^, we have
the formula
AAA
^wt -^nJ -^ta
^ll> -^M) -^88
A A A
-^8t) -^8«» -^88
-^4l5 -^48? -^48
X.
x^
X,
8
' \ 1 8
> ^u>
A
^
^
> -^M'
1 -^441
, 0,
1
= 0.
K. LUCAS, ON THE ANGLES OF FIVE CIKCLES, &C. 41
This is the equation of the system of two circles cutting
ihree given circles x^ = 0, a;, = 0, ajg = at given angles.
2. We obtain, besides, the four groups of these circles
conformably to the rule of signs given in the Nouvelle Corre^
^pondance MatMmatique,
3. If -4,^ = Jj^ = ^3^ = ±l, we have the equation of the
► system of two circles touching internally or externally three
given circles.
4. If ^^^ = ^^ = -^3^ = 0, we obtain the equation of the
circle orthogonal to three given circles.
5. If the given circles reduce to three points, we obtain
the equation of the circumscribed circle.
6. If the circles become three straight lines, we obtain the
equation of the inscribed and escribed circles.
7. If, besides, A^^ = A^ = A^ = 0, we obtain the equation
of the double straiglit line at infinity.
8. Suppose now that the circle 0^ becomes the straight
line at infinity, we then have the formula
A A A A t'
•*3.,,. .^3.«.a XX... ^X.^a
11 J
12)
1S9
14?
^^.« ^«>« ^0>^% ■^m.M% ""
ai» «J
S89
^tC
^ai) -^89) ^M» -^84) r
-"...% -"..«* ^>..« -0.,,% -^
'■4l>
42^ '"48?
449
= 0;
-,-,—,-,
this is the equation giving the radii of the three circles which
cut three given straight lines or circles at given angles.
9. If u4,^ = ^^ = ^3^ = ±l, we find again the formula
giving the radii of the system of circles touching three given
circles. (Bauer, Journat de Schlomilch^ t. V.).
10. If -4,^ = ^2^ = ^3^ = 0, we find the radius of the circle
orthogonal to three given circles.
11. If rj = r,=:r3=l, we obtain the radius of the circle
circumscribing the triangle, given by the three points 0^^
0,, 0,, and consequently the area of the triangle.
42 MATHEMATICAL NOTES.
12. If r, = r, = rg = r = 1 , we obtain the relation given by
Cayley between tbe distances of four points in a plane
{Cambridge Journal^ t. n.).
13. If the elements 0^ and 0^ become two straight lines
at infinity, we obtain the relation between the cosines of the
angles of three directions in a plane.
14. If the elements 0^, 0^^ 0^ represent the origin of the
coordinates and the two axes, any circle whatever, and 0^ a
point on this circle, we obtain the equation of the circle 0^ in
oblique coordinates.
15. If the element 0^ is replaced by the straight line at
infinity, we obtain the radius of a circle with given centre,
which cuts at given angles the two axes of coordinates, &c.
These considerations are also applicable very easily to
space.
Paris, December t 1877.
MATHEMATICAL NOTES.
Proof of the Theorem in Kinematics^ vol. Vll. p. 190.
Let P, P be two points on the moving plane, and (P), {F)
the areas described by them.
Let PP==r^ and let PP' make -AT revolutions.
Let n be the total movement of P' perpendicular to PP.
Then (P) - (F) ^nr + N7rr\^
Take P as origin and the direction of FP in which n is
a maximum (= n) as initial line.
Then n = n' cos 0.
Thus (P) - (P) = nr cos + Nirr^
the equation to a family of concentric circles.
Transform to centre, then
(P) = JV7r(r«-a«)
where a = radius of circle corresponding to (P) = 0.
A. B. Kempe.
Fluid Motion in a Rotating Semicircular Cylinder,
It may be useful to put on record the solution of this
simple case of fluid motion. Take the centre of the semi-
circle as origin and one of the bounding radii as initial line.
MATHEMATICAL NOTES. 43
Let a be the radius and a> the angular velocity. Expand ^,
the Telocity potential in the series of harmonics
* = S {J.g)' + 5. (?)"} cos («<? + «),
The conditions to be satisfied by ^ are (1) it must be
finite everywhere, (2) —-rr: = tor when 5 = or = 7r and (3)
-^ = when r = a for all values of from 5 = to 5 = 7r.
ar
Condition (1) is satified by putting B=sO,
(2) a = 0, except when n=2,
when there must be a term ^oir^ sin 20. Thus
fj) = ^ft,r* sin 20 + 2 * ^^ (-\ cos 7i5.
To satisfy the third condition
^a sin 25 + 2 * - A^ cos n5 = from 5 = to 5 = 7r.
Therefore « is integral and by applying Fourier's theorem
we shall find that A^ = 0,
-^am+l- ^ (27W-1 2W + 1 "^2^1+3] *
Write 2 cos n0 = e"*' + e""", and put - 6*' = a?. We shall
require to sum the series ^
■^V2«i-1 2m+l"^2m+3>/
Now log(l + a;) - log (1 - aj) = 2 jaj + - 4...+
«?«"+*
2m+l
therefore 2,^ ^-^ a^'^" = Alog J-±^ - -
° 2/n + 3 2a; ° 1 - a? a:
therefore
+...
2r
^ ^ + ;r-^-^U''"^'
2w-l 27/H-l 2w
1-1
1 + 3)
-i(«..|.-.)^gl±|-(,.!).
44 MATHEMATICAL NOTES.
Hence
= W8.n2^+ _ |_|(^ + J,) cos2^- 2| log -,-^^^^3^^.
-2 (;;5 - -5 8'n2^ tan , .
This result was set to be verified in a St. John's Colleffe
examination paper in June, 1876 ; but was first obtained lU
the above manner. It is interesting as a solution obtained
by a series which sums to a finite expression.
W. M. Hicks.
On the Caustic hy Refraction of a Circle for Parallel Rays.
If the radius of the circle be taken as unity, and if ^, <f>
be the angles of incidence and refraction, the incident rays
being parallel to the axis to x^ then the equation of the
refracted ray is
(y COS0 — 0? sin^) cos ^' + (y sin^ + a: cos^ — 1) sin^ = 0...(i),
where sin<^' = A sin0 and cos^' = \/(l--^'* sin'<^), h being the
reciprocal of /tA, the index of refraction. Now put ^ = amu,
and (i) becomes
(y cnw — oj snM)dnM + (y snw + oj cntt-l)A snu = 0,
viz. snu(A;cnt« — dnu)a;4- (A;sn*M + cnw dnu)y = A;8nw.
MATHEMATICAL NOTES. 46
Multiply throughout by kcuu + dnuy and this equation
becomes
— k'* &nux + {cnu -^kAnu)y = k snw {k cut* + dnw),
and we have to find the envelope of this line.
Divide by snw, giving
— A'^j^H i/ = k{kcnu + inu) (ii);
difierentiate with regard to w, and we have
inu-^kcnu
&n*u
y = — Zs' SUM (dnw + k cnu),
whence y^k^nn^Uj
and substituting in (ii)
— k'^x = A: (A en w + dn tt) -^ i' sn*M (en w + A dn u)
= A* cn'^M + k dn^u.
whence the equation to the caustic is
that is (1 - Ai") a; = (1 - fJy^f + /*(!- M V)*'
The above investigation of the caustic is the same as that
given bv Professor Cayley in his "Memoir on Caustics,"
(Phil. Trans., 1857, pp. 281, 282) except that the envelope
of (i) is there found by differentiation with regard to ^, and
the elimination of <f> is difficult, occupying a page of work.
Now that elliptic functions are coming into general use, it
seems as proper in a piece of aleebra to replace sin <^, cos ^,
sin \/(l — k* sin*0) by sn m, en u, dnu (or, as is most convenient,
in actual work by s, c^ d) as to replace a, \/(l — «*) by sin a,
cos a ; and it is clear that in optical questions, where we have
sin 0, cos <f}y sin 0', cos 0' involved, the change may often be
convenient. Prof. Cayley himself puts k for /t"\ so that in the
memoir \/( I — ^''sin''^) is written for cos0', which indicates
that the elliptic function analogy was remarked by him.
J. W. L. Glaisheb.
-471 Algebraxcal Identity.
Let a, J, c, /, g^ h be the differences of four quantities
a, ^, 7, S, say
^j^^j/jS'? * = /3-7, 7-a, a-i8, a-S, ^-S, 7-8,
46 MATHEMATICAL NOTES.
then . A - ^r H- a = 0,
g-f . + c=0,
— a — h'- c .=0.
Now Cauchy's identity
patting therein a-^rh^ — c^ becomes
a' + J' + c' = laic i^c-^ca-^ ahf ;
hence we have
h^ — g' + a' = " lagh (- ga-\-ah — hgf^
g^^f . +c'^^7cfg{-fc+cg-gfr,
— a' - i* — c' = — lahc ( Jc + ca + aJ)* ;
whence, adding,
agk (- ga + ah' hgY + M/(- A6 + bf-fhy
or, as this m^y also be written,
agk [g' + A« + a'f H- bhf{h' ^f + bj
-^-cfgif'-^-g'-ircy + abcia'^b' + cy^O,
an identity if a, J, -c, /, ^, h denote their values in terms of
a, )8, 7, S. A. Caylet.
[l^ The identity may also be obtained by equating to
zero the terms of the seventh order in the trigonometrical
identity
Bin(^— 7) 8in(/8— 8)sin(7-S)+Bin(7— a) 8in(7— S) sin(a— 8)
+sin(a-^) sin(a— S) sin (^- 8) + sin (^-7) sin (7- a) sin(a-^)= ;
that is, in
sina sin^ smA + smb sinA siny+sinc sin/sin^ + sina sini sinc=0.
For the terms of the seventh order in the expansion of
sina sin 5 sine are
yj^c (3a* + 3J* + Be* + lOiV + lOcV + 10aV)...(l),
and if a + J H- c = 0, we have
a*-h6* + c* = 2JV + 2cVH-2aV,
MATHEMATICAL MOTES. 47
and also
Thus (1)
«= ^aic ( Jo + ca + aJ)%
80 that the identity is
+ ?/y(/*+/ + ^*)+«*^(«' + ^* + O-=0 (2),
or <igh (— ^a + aA — A^)* + &c. = 0,
or agh (^' + A* + ay + &c. = 0.
2\ It may be remarked that, by a known theorem,
whence, if a + i + c = 0, and n be uneven
a* + J"+c" = watc{a"""-i(w-3}ica"-* + &c.},
writing the two similar expressions obtained by interchanging
a, J, c, we have by addition
ct '\' V -^ c" =^ \nabc
x{a"-» + i»-» + c"^-i(n-3)aJc(a-«+i"-« + 0+&c.},
whence we can write down an identity of the r^ order, that
includes (2).
3". By Cauchy's theorem if n be of the form 6wi + 1|
then (a? + i/f --^ — y" is divisible by (aj* + a:^ + y*)'*, so that
By actual division, I find that
X (ar* -f ^x^y + 8a?y + WxY + 8a:y + 3ajy' +/),
and the last factor
= (oj* + ary + y')' + 2ajy (a^+ y)'.
1 find also that
(aj + y)" - a:"- y" = Wxy (a: +y) (aj' + a?y + y*)
X {(a?" + ajj^ + y")' + a?y (a? + y)*}.
J. W. L. Glaishej^].
48 MATHEMATICAL NOTES.
On Certain Sums of Squares.
The^n(n-l) squares 2(a, — Oy)* formed by adding the
squares of every pair of the quantities a^, a^, .,,a^ ^^Ji ^^
n be uneven, be expressed as the sum of n squares in the form
where c stands for cos , and s^ for sm .
For, in this expression, the coefficient of a* is i(n -1)
and the coefficient of a^j is equal to
2coB— ^^ — *^-^— +2C0S— ^ — *^^^...+ 2cos^^ ^-^ — *^-^—
n n 71
, 27r . . 2'7r
where cd = cos h * sin — .
n n
Thus the whole expression
= i (n - 1) Sa,.* - 2a,ay = ^2 (a, - a^)*.
If ri be even, then the expression
= 2 (a,- - a,-)*,
where i — y must be even. For the coefficient of a,a/ is equal
to the sum of the {i—jY^ powers of all the imaginary roots
of the equations a?"- 1=0, so that if i-J be uneven, the
coefficient is zero and if i —J be even it = — 2.
The simplest examples are {n = 3)
= 2 {(aj + a, cosfTr + a, cos|7r)" + [a^ sin |7r + «, sin |^)*},
and (w = 4)
=(a,4a,cosi7r+a,cos7r+a^cosf^)''4-(a,sini7r+a38in7r+a^sin|^)'.
J. W. L. Glaisher.
«
<.
»^*
MATHEMATICAL NOTES. 49
Centres qf Pressure.
The cientres of pressure of (1) a parallelogram with side in
surface, (2) a triangle with an angular point in the surface
and the opposite side horizontal may be deduced from that of
a triangle with one side in the surface by two methods. To
make the note complete we will find : —
(i). Centre of pressure of triangle with one side iii the
surface. Divide the triangle up into indefinitely thin
horizontal strips of equal breadth. Let DE^ FO (fig. 1) be
two such strips at equal distances from AB and C respectively.
Bisect AB in J2, join Ci? meeting DE in Jf and FG in N.
The pressure at any point of DE : that at any point of
FGwADxAFw GF\ CD :: FO i DE] therefore pressure
on strip Z>-E= pressure on strip FO^ and centres of pressure
of these strips are at Jf and N respectively; therefore centre
of pressure of these, or of any two such strips, is at middle
point of CH, But such strips comprise the whole triangle,
therefore centre of pressure of the triangle is at middle
point of CH.
(ii). To deduce the centre of pressure of (1) the parallelo-
gram and (2) the triangle. ABCD (fig. 2) is a parallelogram
with ABm surface, E and F are middle points of -4JS, CD
respectively, join BD^ DE^ BF. Centre of pressure of ABD
is in DE'^ centre of pressure of BDC is in BF\ pressure on
£Z>6^ = twice that on ABD^ because the areas of the 2
triangles are equal, but the depth of centre of gravity
of one is twice that of the other. Therefore centre of
pressure of ABCD is in a line parallel to DE and BF^ and
one third of the interval distant from BF^ but it is also in EF\
it is therefore at (?, such \!cizS, EG^^GF. (This does not
depend upon the actual position of centre of pressure
o(ABD).
Now centre of pressure of ABD is at S, the middle point
of DE. Join DG and produce to meet BFm K\ then -K'is
the centre of pressure of BDC^ and
FKiHEv.FGi GE:: 1:2;
therefore FK^^HE^IBF.
(iii) We may deduce these results by another method
from (i). Let ABC (fig 3) be a triangle, A being in the
surface ; produce BC to meet the surface in D. Bisect AD
in E. Then centres of pressure of triangles ABD^ A CD lie
on BE^ CE respectively, and at their middle points. The
line joining these will therefore pass through centre of pressure
VOL. VIll. E
50 MATHEMATICAL NOTES.
of the triangle ABGj which is their diflFerence. This line is
easily seen to be at one quarter of the distance from BD
towards A. Let BC become horizontal, D therefore goes off to
infinity. Then centre of pressure of each strip parallel to BG
in triangle BBC will be at the middle point of it ; therefore
centre of pressure is three quarters of distance from A to
middle point of BC,
(iv). Through G draw HOK parallel to AC^ meeting
AD^ CD respectively in H, K. Then HO : GK :: 2 : 1, and
when D becomes infinitely distant this will give the distance
below the surface of the centre of pressure of any parallelogram
having a side in the surface to be two-thirds of the distance
between the two horizontal sides.
(v). If the lines we have supposed in the surface be kept
horizontal but covered with liquid to a depth A, we must find
the centre of pressure by uniting with the pressure already
found (acting at its centre of pressure) a uniform pressure due
to depth h (therefore acting at centre of gravity of area
considered).
(vi). For example : to find the centre of gravity of a
narrow straight strip immersed in liquid. If one end of the
strip were in the surface, the pressure would be in magnitude
the same as that due to a depth ^ {d — d^) over the area, and
acts at two-thirds of the distance down the strip from the
highest point of it [dy d^ are the real depths of the two endsl.
"With this then we must combine a pressure due to depth a,
over the surface of the strip and acting at its middle point.
Let AB be the strip (fig. 4), trisect it in C and i>, and
bisect it in 0.
The force at may be divided into two equal parts and
supposed to act at C and 2>. We have, therefore, at CandD
forces proportional to ■^[d—d^) -hirf, and ^d^ respectively, or
proportional to d and d^ the centre of pressure is therefore at
a point P in CD, such that CP : PD : : depth of B : depth
of ^.
(vii). To find the centre of pressure of any triangle any-
where in the fluid.
Firstly, let the point A of the triangle ABC be in the
surface. Then, combining results of (iii) and (vi), we see
that the centre of pressure is three-quarters of distance from A
to a point P in Bu^ such that if J, c, be points of trisection of
BCj bP : Pc : : depth of C : depth of B.
MATHEMATICAL NOTES. 51
Secondly, if ^ be not in the surface we must find the
Centre of two parallel forces, one at Pand the other at the
centre of gravity of the triangle and respectively proportional
to the depth of that centre of gravity below A and the
depth of A below the surface.
T. C. Lewis.
June, 1878.
On the Deformation of a Model of a Hyperholotd.
The following is a solution of Mr. Greenhill's problem
set in the Senate-house Examination, January 14, 1878.
" Prove that, if a model of a hyperboloid of one sheet be
constructed of rods representing the generating lines, jointed
at the points of crossing ; then if the model be deformed it
will assume the form of a confocal hyperboloid, and prove that
the trajectory of a point on the model will be orthogonal to
the system of confocal hyperboloids."
Let.(ar„ y^, isj, (a?,, y,, z^ be points on the generating line
of "8 + ^- -i=l, then
a o c '
«i,
= 1,
a
a' ' b*
= j,
or what is
the same
thing, if
•
' 5
a
' c"
' b
1 5 =-?«'?«''*«'
then
P' + 2i' -
p: + 2," -
T
%
T T
=1,
=1,
=1.
Similarly if
line of ^^ + ^g
7"
V^, 0, (f., Vti
= 1, and if
(;)
he points on
generating
^ 5j 21 — -a fi r-^ ^ ^ — n n r
then p/ + q," - r/ = 1,
p.p.+qtq,'-r,r,=i.
E2
52 MATHEMATICAL NOTES.
Hence if (a?,, y,, «,), (f^, 17,, fj be corresponding points on
the two surfaces, tnat is if
a' 6' c a* i8' 7' "i'l' 2i» ^'
and similarly (a?^, y,, «J, (f„ 17,, 5^) are corresponding points,
that is if
then we have, as before, the system of three equations
Then if the two surfaces are confocal, that is if
a*, )8", -7* = a' + A, J' + A, - c* + A,
we shall have
(x. - *.)* + (y. -y.)*+ («. - «.)»= (?. - f .)* + (1,. - 1,.)* + (5;-?/.
For this equation is
tliatis (/'.-i>/ + (j.-2,)'-(r.-r/ = 0,
an equation which is obviously true in virtue of the above
system of three equations.
Hence, if on confocal surfaces
we take on the first two points P,, P^and on the second the
corresponding points ^„ Q^ ; then P , JP^ being on a generating
line of the first surface, Q„ Q^ will be on a generating line of
the second surface, and P^P will be = QxQt'i *°^ ^^^ ^sjhq is
evidently true for the quadrilaterals P^P^P^P^ and QiQ^Q^Q^
where jr,P,, P^Pg, Ps-f^i -P^;?, are generating lines on the nrst
surface, and therefore Q^Q^^ Q^Q^^ Q3Q41 Q^Qi are generating
lines on the second surface, which proves the theorem.
A. Catley.
( 53 )
ON A CLASS OF ALGEBRAICAL IDENTITIES.
By J. W. i. Qlaisher.
§ 1. Caucht's theorem referred to on pp. 46, 47 is that
(a; + y)*- a;* — y* is divisible by a? + xy-{']f if n be of the
form 6m ± 1, and by [x* + ary + j^)* if n be of the form 6m + 1.
The expression is also divisible by xy[x'{-y) when n is
uneven. The complete resolutions for n = 1 1 and ?? = 13 are
fiven on p. 47, and working out the resolution for w = 9, we
ave the remarkable series of identities
{x+yY -a:' -y» = 3a;y(a; + y),
(a; + 3^)» - a;' - y» = 3a;y (a; + j^) {3 (a:*+ icy +y')'+a;y (x+y)'},
(i» + 3^r-as"-y"=lla?y(a; + y)(a;* + a3( + y*)
X {(x" + a^^ + y)' + a;y (a: + y)'},
(a; + y)" - a:'» - y'' = 13ay (a; + y) (a^ + a^^ + y')*
X {(aj" + a;y + yj + 2ajy (a; + y)*},
§2. These formulae may be put in a symmetrical form
by taking x=^C''b^ y = «-"C, whence a? + y = a — 6, and
n being uneven
(a:+yr-a:''-.y'*=(i-.cr + (c-ar+a-5r.
Also xy {x •{- y) ^ {b " c) [c — a) (a — J)
= *{(i"Or + (c-a)Xa-Jn,
and a;* + a:y + y* = i{(i-c)* + (c-a)* + (a-in.
Now, let I {(J - c/ + (c - a)* + (a - 5)"}
be denoted by P^ ; then
ary(aj+y) = P3, x* ^^ xy -{- y* =^ P^^
and it is readily proved that P^ — \P^ (1).
Cauchy's theorem then shows that, ignoring numerical
factors and considering only algebraical divisibility, P is
divisible by P,, by P^ and P,, or by P,, P^ and P according
as n is of the form 6m + 3, 6m + 5 or 6m + 1.* In the last
case P^ is divisible by Pgi^*, and therefore by PP by (1),
but of course P^ is not divisible by the product PJP^^»
* I have ennndatod in this form Oauchy'B theoiem in the Quarterly Journal of
JfathmaticSf vol. XT. p. 366 (1878).
54 MB. aLAISHEB, ON ALGEBRAICAL IDENTITIES.
The formulae of the last section, when transformed, as
above, become
p. = -PA
p»= pap:+ip.\
p..= PAiP:+p,*h
P.. = 2P.P,(P,'+2P,«).
- The last two equations may also be written
P,, = P,(P/ + 2P3«),
p p p
so that -p } -pS ~^ ^® connected by a linear equation,
which is easily found to be
P P P
2^-5^ + 3^ = 0.
■*^7 "^6 •'■3
§3. Taking as in Prof. Cayley's identity (pp. 45, 46),
a, J, c, /, fl', A to be the differences of four quantities
a> /3j 7j S, via. a, i, c,/, g^ A=^-7, 7-a, a-^; a-S, ^-S, 7-S,
then
A— ;9' + a = 0,
.^-/. +c = 0,
— ^«5 — c, =0,
and taking the resolution of {x + i/f-x^^y^j and proceeding,
as on p. 46, we have
h^^g^ + a^ = Zagh {3 [-ga -{-ah- hgY - ayA'},
-A' . +/'4 i'= 3JA/{3(-Ai+ bf ^fhf •- h'KT\,
/ -/' . + c» = 30/^ {3 (-/o -^cg- gff - c'/y},
-a"-j«-c' =3aJc{3( Jc + ca + ai)' - a'JV} ;
whence
3a^A («^a + aA - A^)'+ 35A/(- hb+b/-'/hy'\-3cfg {^/c+cg^gff
. *
■^^
[ MB. GLAISHES, ON ALGEBRAICAL IDENTITIES. 55
Treating the other equations in § 1 in a similar manner,
and writing
^ = — ^a -f aA — Jig^
P= he -{• ca-i- aby
we have the series of identities,
affk + bhf + cfg + abc =0...(2),
agh A ■]- bhf B + cfg C -k- abc P =0...(3),
agh A^-^ bhf S*+ cfg C^ ahc P» = 0...(4),
^agh A' + 3bhf B' + Scfg C + Sabc P"
- a^W - b%Y - c'/V' - «'iV =0...(5),
agh J.*+ bhf P*+ cfg 0*+ abc P*
- a^h'A - VhYB - cYg'G - a'JVP =0...(6),
a^A ul*+ JA/ 5*+ c/^ C^+ abc r
- 2ayA'^" - ^b'hyff - 2c!r(7» 0* - 2a«6VP* = 0. . . (7),
Since
Jc + ca + oi = - 4 (a* + 6* + c*)
- ^a + aA ~ A^ = - i(a' +/ + A"), &c.,
if we put
i=a* + /+ A*,
if = 6« + A« +/«,
these identities may be written
agh + bhf + cfg + aJc =0.... (8),
agh' i + 6A/ Jf + cfg N -^ abc X =0.... (9),
agh r+ bhf iP+ cfg i\P+ abc Z''= 0...(10),
Sagh i'+ 36A/ iTH- 3c/^ iV^+ 3a5c Z'
+ 8ayA' + Sb'h'f + Scy*/ + Sa'bV =0...(11),
agh i*+ JA/ M*+ cfg N*+ abc X\
+ Sayh'L ■]■ WhyM-^ ScTff'N+ Sa'JVJT =0...(12),
agh L'+ bhf M'+ cfg N'+ abc X'
+ Ua'gVr + m'hyiP + UcyyiP + 16a»iVZ« = 0...(13),
of these (4) and (10) are Prof. Cayley's identities (p. 46).
%
I
\
56 PEOF. TANNEE, A THEOEEM EELATINQ TO PFAFFIANS.
I have not worked out any resolutions besides those given
in § 1 ; but it would be interesting to obtain the resolutions
for some values of n greater than n = 13, and if possible to
obtain the law of formation of the residual factor; i,e. the
remaining factor when the factors xy (a* + y), and x^ + xi/-\- y^^
or (a:' + ary + y^)', have been thrown out.
§ 4. I here add, though not connected with the subject of
the paper, the identity
(/8 - 7) (^ + 7 - 2a)'+ (7 - a) (7+a-2)8/+ (a-^) (a+^-27)»=0,
or, if a = /8 — 7, J = 7 — a, c = a - /8 so that a + J + c = 0,
a(6-c)'*H-J(c-a)» + c(a-J)' = 0.
A THEOREM EELATING TO PFAFFIANS.
By Prof. JSr. W. Lhyd Tanner, M.A.
The functions to which Prof. Cayley has given the name
" pfaffian " are of fundamental importance in the theoiy of the
transformation of the expression
In workinff at this theory I have been led to a theorem
concerning pfaflSans which seems of sufficient interest for
publication, if, indeed, it is new.
Definitions^ Notation^ &c. " The definition of a pfaffian
is as follows : taking the symbol 12, 13, &c. to be such that
12 = - 21, &c., then we have the pfaffians (1234), (123456), &c.,
viz.,
(1234) = 12.34 + 13.42+14.23 (3 terms),
(123456)= 12(3456), where (3456) = 34.56 + 35.64 + 36.45,
+ 13(4562) &c. = &c.,
+ 14 (5623)
+ 1 5 (6234)
+ 16 (2345),
15 terms, each with the sign + ; and so on."*
I have to than^c Prof. Cayley for this definition.
PROF. TANNER, A THEOREM RELATING TO PFAFPIANS. 57
An obvious induction shows that the transposition cf a
single pair of the symbols 1, 2... changes the aim of the
pfaffian ; whence, as In the case of determinants, it follows
that a pfaffian with two identical symbols vanishes. More-
over, we may change the order of the symbols 1, 2, &c., in
the above expansions ; if we prefix the sign + or — to a term
according as the order of the symbols 1, 2... in that term is
reducible to the order 12... by an even or an odd number of
transpositions*
By the symbols ({12}), ({13}),.. .({1234}),... will be indicated
the coefficients of (12), (13),.. .(1234)... in the pfaffian (12. ..2r)
which may itself be denoted by {1}. The symbol ({12})
represents a pfaffian which is derived from (12... 2r) by
omitting 1, 2, arid so of the others; ({12... 2r}) is unity. In
virtue of the above we may write
(I2...2r)({12...2r}) = {l}.l
=(12).({12})+(13)({l3})+...+(l,2r)({l,2r})
= &c.
In view of these relations, and of others which will be
discussed in the sequel, we may call such pairs of symbols as
(12), ({12}); (1234), ({1234}) complementary; (12...2r) being
the principal pfaffian.
It is easy to verify that the transposition of a pair of the
numbers in ({12}), ({1234}), &c., changes the sign.
The theorem to be jjroved is this : Any relation between
pfaffians is true of their complementaries, provided the
relations be made homogeneous {i.e. having the same number
of pfaffians in each term) ; and this condition can always be
satisfied by introducing {12... 2r} in the original, or {1} in the
transformed relations where required.
Thus having
(12). (34) + (13) (42) + (14) (23) = (1234). 1 = (1234). ({12.. .2r}),
there is another relation, or set of relations,
({12})({34}) + ({l3}).({42}) + ({14})a23})
. = ({1234}). {]} = ({1234}) (12.. .2r).
Again, we should have
{1} ({123456})= ({12}) ({3456}) + ({13}) ({4562})+ ({14}) ({5623})
+ ({15}) ({6234}) + ({16}) ({2345}).
58 PROF. TANNEB, A THEOREM RELATING TO PFAFFIANS.
Consider, first, the equation
({12}).({34})+({13}).({42}H({14}).({23})-({1234}).{1}=0...(1).
If this be expanded in terms of (12), (13)...(23)...&c., each
term will contam a factor (12), (13)... or (1, 2r) involving the
symbol 1. If, then, it be shown that the coefficients of
(12), (13)...(l,2r) all vanish the truth of (1) will be
established.
Take then the coefficient o^ (12). In the first term
({12}). ({34}) it is only in the second factor that (12) can
occur [for ({12}) is a pfaffian formed from (12... 2r) by
omitting 1, 2 J. In this term the coefficient of (12) is
({12}). ({1234}). In the second and third terms of (1), (12)
does not occur, for in each case 1 is left out of the first factor
and 2 out of the second. In the last term the coefficient is
— ({1234}) ({12}), which annuls the portion derived from the
first term. Hence in (1) the coefficient of (12) is zero. In
the same manner it may be shown that the coefficients of
(13), (14) also vanish.
It is otherwise with the coefficients of (15), (16), &c.
That of (15), for example, is
({12}) ({1534}) +({13}) ({1542})+({14})({1523}) - ({1234}) ({15}),
or
({12}) ({1345}) - ({13}) ({1245}) -t- ({14}) ({1235})
-({15}) ({1234})... (2),
which does not obviously vanish. Herein the symbol 1 does
not enter, but every term of the expansion will contain 2.
The coefficients of (23), (24), (25) identically vanish.
If, then, the coefficients of (26), (27), &c., also vanish, (2)
will vanish; in other words, the coefficient of (15) in (1) will
vanish, and, since (16), (17), &c., may be dealt with in just
the same way as (15), the truth of (1) will be proved. Now
the coefficient of (26), say, in (2), is after re-arranging
({1234}) ({1256}) - ({1235}) ({1246}) + ({1236}) (1245})
- ({1^} ({123456}).
But ({1234}) is the conjugate of (34) with respect to ({12})
or (34... 2r), and if we take this as a new principal pfaffian,
the identity we have to prove may be written
({34}) ({56}) - ({35}) ({46}) + ({36}) ({45}) - {1} . ({3456}) =
(3).
M
PBOF. TANNER, A THEOREM RELATING TO PPAPFIANS. 59
This relation (3) diflFers from (1) only in that the new
principal pfaflSan {1} contains two symbols less than the
original {1}. But (1) is true if (3) be true; that is, (1) is true
when the principal pfaffian involves 2r numbers, if it be true
when there are only 2 (r- 1) numbers. Now (1) is true when
2r = 4, for it then becomes
(34) . (12) + (42) (13) + (23) (14) - (1234) = ;
hence it is true when 2r = 6, and so generally.
In precisely the same manner the second identity may bo
proved. This may be written
({12}) ({3456}) - ({13}) ({2456}) + ({14}) ({2356})-({l5})({2346})
4 ({16}) ({2345}) - {!}. ({123456}) =0 (4).
The coefficients of (12), (13), (14), (15), (16) all obviously
vanish; the coefficients of (17), (18), &c., do not so. But,
taking as example (17), the coefficients of (17) (23), (17) (24),
(17) (25), (17) (26), vanish ; and it only remains to prove that
the same is true of the coefficients of (17) (28), &c Now the
coefficient of (17) (28) is
^({12}) ({12345678}) + ({1238})({124567})-({1248})({123467})
+ ({1258}) ({123467})- ({1268}) ({123457})+({1278})({123456}),
which if we take ({12}) as a new principal pfaffian differs
from (4) only by the new {1} containing two numbers less
than the original {1}. Hence (4), being true by definition
when 2r = 6, may be proved by induction to be generally
true.
The results obtained amount to this; that the symbols
({12}) V {1}, ({13}): {1},...({1234}) : {1}, &c., satisfy the relations
by which (12), (13),. ..(1234), &c., were defined. Hence, any
theorems true of the latter are also true of the former symbols.
And this is the theorem to be proved.
July, 1878.
Concerning the above paper Prof. Cayley writes : —
" I think the true form is not exactly as you put it : ' An
identity subsists when the pfaffians are replaced by comple-
mentary pfaffians,' but ' It the elements of the principal
pfaffian are replaced by their, complementarles, then each
minor pfaffian is converted into the complementary minor
pfaffian'; your form being, in fact, an immediate corollary
of this.
60 PROF. CAYLEY, NEW FORMULA FOR
^^Thus taking tbe principal pfaffian to be 128456, the
complementaries of 12, 13, &c., are 3456, 4562, &c., and the
theorem is, that substituting in any sub~pfaffian, such as 1234,
the foregoing values in place of 12, 13, &c., respectively,
this is changed into the complementary sub-pfaffian 56.
" Of course an identity, 1234 = 12.34 + &c., subsists when
the elements are changed into any other elements, viz. we
have l'2'3'4' = l'2'.3'4'-f &c., and the theorem is, that writing
herein 1'2' = 3456, &c., it becomes
56 = 3456.1256 + &C....
(where, on the left-hand, there is a factor 123456 to be
inserted). •
''The like theorem holds, as regards determinants, viz.
for a determinant (11, 22, 33, 44, 55, 66), if each element
11, 12, &c., is replaced by the complementary first minor
(22, 33, 44, 55, 66), (21, 33, 44, 55, 66), &c., then every other
minor (12, 33, 54), &c., is converted into the complementary
minor (21, 45, 66), &c., and the theorem for pfaffians follows at
once (though the working-out would, perhaps, be rather
difficult) from that for determinants, by taking a symmetrical
skew determinant which is the square of a pfaffian."
NEW FOEMULJS FOR THE INTEGRATION OF
By Professor Cayley,
I HAVE found in regard to tbe differential equation
^ + ^
is/(a — x,h''X,c—x.d''X) 'sl^fl — y'h—y.c—yA—y) *
a system of formulae analogous to those given, p. 63, of
my Treatise on Elliptic Functions^ for the values of sn (w + v),
en (m -h t?), dn (m + 1?) ; viz. writing for shortness
a,b,c,d =(Z--a:, J — a?, c — a;, cZ-a?,
»!) K ^ij ^i=«-y» ^-yj c-yj ^-y>
and (&o, ad) to denote the determinant
l^x-^yyxy
1, J -f c, be
1, (Z-f e?, orf
I
THE INTEGRATION OP -7^ + Tv= = 0. 61
and {cdy db)j {bd^ ac) to denote the like determinants. Then the
formulae are
// g — z \ _ V(a — ft ,a — c) {V(adb^cJ 4- V(a^d,bc)}
V \d^) " (^ ad) '
_ iJ{a-~h.a''C) (a? — y)
" V(adb,cJ - V(aAbc) ^
_ \/(<* — J.a — c) {\/(abc,d,) + \/(a,h,cd)}
(a-c)V(bdb,dJ-.(6-rf)V(aca,cj *
_ V(a — ft.g - c) {V(acb|dJ + V(a,c,bd)}
" (a-6)V(cdc,d,)-(c-£Z)V(aba,bJ '
//j«,x _ \/(!r-5) {(^ - ^) V(Mb A) + (ft ^ d) V(aca,c,)}
V U~i
^){V(abc,dJ-V(a,b,cd) }
V(adbjC,) - V(aid,bc)
(a - c) V(bdb,d,) - (6 - d) V(aca,c.) '
(a - 6) V(cdc,d,) - (c - rf) V(aba,b,) '
V(adb,c,) - V(a.d.bc) '
"" (a - c) V(bdbjdj) - (6 — rf) VCaca^cJ
(a - 6) V(cdc,d,) -{c-d) v/(aba,b,) '
62 TRANSACTIONS OF SOCIETIES.
The twelve equations are equivalent to each other, each
giving z as one and the same function of a;, y ; and regarding
2; as a constant of integration, any one of the equations is a
form of the integral of the proposed differential equation.
Writing in the formulae a; = a, J, c, d successively, the
formulae become
x — a x=^b aj = c x=^d
a -z _ a^ c — a b, h — a c, a-h.a-c d,
c? — ^:~d' d—hc,^ d—c K^ d—b.d-c a,,^
J-«__b, c-Ja, J — a.J-c d, a — bc
1
d—z dj' fl?— aCj' d—a.d-ch^^ d — c b.^^
c — z_^ c, c — a.c- b dj b- c sl^ a — c b,
flf— « "" d/ rf— a.fZ— J Cj ' rf— abj' d — b s.^^
viz. in the first case we have«=^, and in each of the other
cases z equal to a linear function — — = of y.
Cambridge, July 3, 1878.
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, Jtrne, 18th, 1878.— Prof. H. J. S. Smith, F.R.S., Vice-PresidefU,
in the chair. Mr. T. R. Terry was admitted into the Society and Mr. J. D. H.
Dickson, M.A., Fellow and Tutor of St. Peter's CJollege, Cambridge, was proposed
for election. Dr. Hirst, F.R.S.. communicated a paper bv M. HaJphen " On the
characteristics of systems of Conies." Mr. J. J. Walker read a paper " On
a method in the Analysis of Plane Curves." He discussed, by means of the
method, the problem of the Inflexion-tangential curve for the ^uartic. Mr. Tucker
communicated the following papers : " On the Calculus of Eqmvalent statements,"
Mr. Hugh McCoU; "On the Flexure of Spaces," Mr. C. J. Monro j "On the
Decomposition of certain numbers into sums of two square Integers by Continued
Fractions," Mr. S. Roberts, F.R.S. ; " On a new method of finding Differential Re-
solvents of Algebraical Equations," Mr. R. Rawson. Questions were asked by Prof,
Cayley, F.R.S., (" Has a proof been given of the statement that in colouring the
map of a country, divided into counties, only four distinct colours are required,
80 that no two adjacent counties should be painted in the same colour ? *') ; by
Mr. Merrifield, F.R.S., on the uniform distribution of points in space; by
Mr. Tucker, with reference to exceptions to Fermat's statement that 2* + I is a
prime (see Nature^ No. 447, for the recent discovery that in addition to the well-
known exception of 2 + 1, 2 + 1 is also ^n exception).
R. TuOKBR, M.A., Ron. Sec,
( 68 )
c
/
NOTE ON CERTAIN THEOREMS IN DEFINITE
INTEGRATION.
By J, W, L. Olaisher,
§ 1. This note refers to Boole's Memoir " On the com-
parison of Transcendents, with certain applications to the
theory of definite integrals" {Phil. Trans.j vol* CXLVII.,
pp. 745-803, 1857), and two papers of my own "On a
theorem in definite integration, Quarterly Journal^ vol. x.,
pp. 347—356, 1870, and " On the reduction of functional
transcendents" {Messenger^ vol I., pp. 153—162, 1872) ; it con-
tains transformations and examples relating to the contents of
these papers.
§2. The theorem*
(\ real, a real and positive) may be put in a slightly different
form ; for c being real, but otherwise unrestricted, the left-
hand side may be written
Now a; - - —
i»'+(A. + r 1«?
\ x + \ x-{-\ '
so that / fix r- jdx= I /{x) dx^
(X real, n real, positive, and not less than unity).
Boole's general theorem when transformed as above, is
that
♦ Phil Trans.i vol. CXLVii. p. 780.
64 MB. GLAISHEB, OX DEFDCITE ISTBGRALS.
(X,, X^..A^ real; ji,, « ...», real pontiye, and Dot leas than
nnitj; A^^ A^ ...A^ w of the same sign}. Whether A^^
A^..„A' are pontiTe or n^^tive they are to be taken as
pomtiTc on the right-hand side of the equation, Ce, the factor
IS really V(-4, + -4^...+ -4j'*.
$3. We know that
. jiSC ZaC ^aB p / « \
**"*"" ar'-(iT)' ~ar'-(|ir)'~ar'-(i7r)* " ^"^ ^'^»
, «» 1 1 1 .
whence
tanx j 1 1 1
+ aJ- - (iTT/ + x' - (1^)' + x* - (fir)' + '^"i
- ^^ 2x* 2j* „
- (iTT/ {x'-Ci^} + (f,r/ {ar'-d^y} + (|^/{xMSt/} +*''•
(12),
and
1 /^^tan^\ ^ J_ /_!_ + _l_^
a? \ X } {^trf \x-\ir x-^^ir)
-f 7i-w f i- + — -^-"j +&c....(3),
when by the theorem {^Quarterly Journal^ vol* x. p. 349),
This example is similar to, but somewhat simpler than, that
contained in Example I {Quarterly Journal^ vol. X. p. 350).
Boole's theorem gives that
MB. GLAISHER, ON DEFIKITE nfTfi0BALS< 65
We have from (2)
i f tanv^N 2 2 j.
wbence
and we may ootice that from the original series (1), Boole's
theorem gives that
/l-^(^+^)^='/y(^)^^ (^>
§4. From(l)
1 2x 2x 2x •
**°« " (Wx*- 1 ■•■ (iTr/sB*- 1 ■*■ {^irfa? - 1 ^ **'*
whence
and, therefore,
TO. fjyxM^x)^=fJ!^d, (7),
§5. Boole's theorem, applied to the equation
cotg 1 . , _ 1 2g* , 1 2a^ .
gives ^/(x+i- 2^) dx^fj{v)dv (8),
and from the Quarterly Journal theorem, we have
§ 6. It is to be remarked, as in Messenger^ vol. i. p. 158,
that when the expression unaer the integral sign changes its
form within the limits of integration {as ex. gr. in
VOL. VIII, P
66 MB. QL^USHEBi ON DEFINITE INTEaBALB.
(4), (6], ...(9)}, the equation is really a theorem in the
companson of transcendents ; thus ex. gr. (7) is that
j"/(Va>taiiV«) ^ + fj{^x\m^x) § = [j^dvy
and the second integral
*j /(tVi» tan «Vaj) — « ,
80 that the theorem is
r /(V»tanVa) ^ + J fi-^/ximh^x) -^^ j -^ dvy
For example, taking /(v) = vV**
I tan'ya5exp(— ojtanVic) —
tanh* V^ cxp (— aj tanh' V^) — 5= ^ir,
where exp u is printed for e*.
§ 7. With regard to Boole's theorem and that in vol. x. of
the Qv^arterly Journal^ there is one point that deserves notice.
Boole obtains his theorem bj means of the transforming
equation
a, a a
» — X, aj — X, *""\
and, supposing X,, X,,...X, to be in ascending order of
magnitude, as v passes from — 00 to 00 the least root passes
from — 00 to X, the next from X, to X •.., and the ^eatest
from X^ to 00. Now when we take the transforming
equation to be
^J 1_ ]_ 1_ 1 1
jc — TT aj + TT aj-27r a? + 2^'" x—nir x-^-nir *
the least root passes from — oc to — t»r, the next from — tiTr ta
» (n — \)ir and so on. In the limit we take n infinite, and
since the least root has to pass from — 00 (the limit of the
integral) to - nv (» infinite), it is clear that we, in effect,
assume the infinity of the limit of the integral to exceed
the (puQiericall^) greatest multiple of ir for which cot x is
infinite. This 13 pmeetly allowable whenever the function /
is such that f{x) becomeo zero when x is infinite, for the
^ ^
KBU OULISHEK, ON DEFINITB INTEGRALS. 67
I auantity under the Integral sign yanishing^ near the limits of
I tne integral, the nnmerical value of the integral cannot be
effected by an assumption of this kind with regard to the
infinities of the limits.
The Quarterly Journal theorem is obtained by means of
\ . the transforming equation
f
a. a. a.
— V + — V — "*• — V "^^^
and the greatest root passes firom X to oo , changes ugn and
passes from - x> to X^ as t? changes from oot to — oo .
Thus, on the whole, it appears that in an^ one of the
equations, when both the integrals are finite m value, and
independent of the particular infinities which form the oo 's of
the limits, i.e. such that the values of the integrals are not
altered bj changing the oo of any of the limits into oo + A,
or 00 * &c, then the equation will certainly be true ; but that
when these conditions are not satisfied, a special examination
of the case is requisite. K the integrals oecome infinite for
Particular values of ;v or v their principal values are to
e taken*
It is to be observed that all the results in §§ 1-6 are
derived from the theorems
and jj{x^)dx-fj^dv,
where, for brevity,
• •• T
fic — Xj a; — Xj a? — X^
is denoted by x^i ^^^ ^^^^ these can be proved directly
without the use of Boole's symbol 0, as in Prof. Cayley's
note, (Quarterly Journal^ vol. X. p. 356. The other theorems
in Boole's paper and in the Quarterly Journal^ in which the
\ ^ integrals are of the form
I <^ {x)f[ai - xx) dx and | ^ {x)f(x^) dx
were obtained bv means of 0, one of the operations of which
is to subtract the coefficients of x'^ in the expansion of a
certain expression in descending powers of x. When x^
f2
M
68 MS. GLAISHEfi, ON DEFINITE INTEGRALS.
contains an infinite number of tenns, the expansion in
descending powers of x is not always possible, and before
replacing x^ ^7 cot a?, or -r- tan-p , or other such functions,
in these theorems, it is necessary to examine each case
separately, and determine whether the theorem is applicable.
For an example in which the substitution is not permissible,
see Messenger^ vol. i. p. 153 (1872). I do not here consider
any of the equations, similar to those obtained in §§ 1-6, that
can be obtained by these theorems.
§ 8. Boole has shown (p. 787 of his memoir) that
where x^ has the same meaning as in the last section, and by
means of this theorem, by putting h = bi. and A = - bi. and
subtracting and adding the results, we obtain reductions for
the integrals
Taking the simplest case of x^^a* i Xj we find
whence
These formulas can also be deduced from
{Quarterly Journal^ vol. X. p. 351); for, taking
/Nil 1 1 1
^^ '^ ^ x-a ^ x+a d^
x — \
te '
t f
i.
^
Mfi. GLAISHEB, ON DEFINITE INTEGRALS* 69
we have
agreeing with (10).
§9. As the remaining portion of this paper will be
chiefly devoted to evaluations derived from (11)* I here give
another investigation of this equation derived from the
formula
which is proved by Boole on p. 783 of of his memoir. If wo
put a=l this becomes the formula given by Caucby in his
Mixercices de MatMmatiques^ 1. 1, p. 54 (1826); and (13) can be
readily deduced from Cauchy's theorem. Another proof
of (13) is also given in the Messenger^ vol. il. pp. 78, 79 (1872);
it is there remarked that (13) does not follow from Cauchy's
result, but this statement is corrected in vol. ill. p. 9 (1876),
of the Proceedings of the Cambridge Philosophical Society.*
Putting in (13) w= 1, 2, 3... we have
r s>?f(X) dx=j' dvfiv) ja* + »» I ,
C x*f{J.)dx=r dvf{v)[a*+ 3aV+ v* |,
r x*fiX) dx^f dvf(v) L' + ^ oV + 5a»»« + v'l ,
" On a formula of GekochyB for the Eyalaatioh of a class of Definite Xntegrah,"
Camb, Phil, 8oc,y Proe, vol. in. pp. 6-12. The paper relates to Cauchy's own
proof of (18), and to a sinular theorem.
70 MR. GLAISHER, OM DEFINITE INTEORALS.
where X-x .
X
Multipljing by 6 ", - &'*, J"*... and adding
X
whence
{H'4T-iHT^4{^-T^''^'-^]^
and, therefore, '
agreeing with (11). Thus, if we denote by ^(J)the value
of the integral
r fix) dx
tbea
§ 10. In the investigation in the last section two points
may be noticed. First, although the series
l-fci:» + A;V-&c.-(l+fcB«r,
only when Jca? is less than unity, still we have
x*/{x)dx- &c.,
<<
MB. aULIBHEB, ON DBFINITS INTSaAALS. 71
if A; is less than unity and i sc^f{x) dx does not become
infinite for any positive even value of n, for we may regard
the right-hand side of the equation as the expansion of the
integral on the left-hand side, given by Taylor's Theorem.
It has been assumed that b is greater than unity (otherwise
the series on the left-hand side might be divergent), but this
restriction is easily removed by transforming the integrals
by putting a? = /acc , v = /av', ana taking a' = a : fi^b' = b : fi.
Secondly, the theorem has been proved for values of /(a?)
such that I xy{X) dxj and / v"/ W ^ ^^ never infinite,
viz. ex. gr. we have
where X^x , J5=J + -? .
'' x^ b
Now regarding this merely as a result, without reference
to the manner in which it has been obtained, if we gradually
diminish k indefinitely, the limiting equation is
r F{X)dx B r F{v)dv . .
and, if both these integrals are finite^ there is no reason to
suppose a discontinuity takes place when A; = in (14); and
thus we obtain (15), or at all events (15) is true in the sense
of being the limiting form of every equation, similar to (14),
in which /is any function such that / aj"'^(X) F[X] dxj and
I v*-^ (v) F (v) dx are never infinite.
§ 11. If/ (a?) is an even function of a?, then
r /{x)dx^2 f f[x)dx\
and in the equations we may take the limits to be oo and 0.
From the integrals
f* cos COS , ^ -^ f^oj sin CO?, ^ .
9
»>
72 MR. GLAISHEB, ON DEFINITE INTEORAliS.
we deduce by (II) and (12)
,» aj Bin c I « - - 1 - ,.
and, putting*
erfca? = I exp (- a?) dx^
we^have
whence
To verify this result, put a* = c"* and replace c* by c ; the
equation thus becomes
£• a^^£) ^= ^ „, (^.^ J.) erf„ (j ^., J) ,
differentiating with regard to c, changing the signs and
addingjthe original equation, we obtain
r.^(-^-^)^=^.-
which is a well-known evaluation.
§ 12. In the Messenger t. i. p. 156, it b shown that
/(t?) do
coffi*
* Jn the Messenger, roh I. p. 157, edx is defined as j ^ (-a;^ (^ ; but it is
more conyenient to take
edx = I exp (- a*) <fo, erfco? = f exp (- «*) <fe j
the two functions are connected by the equation erf « + ezfc x = i^.
*» «
MR. GLAISHEB, ON DEFINITE INTEQ&AX8. 73
whence, from (11),
and writing as before
we haye, bj applying (16) to the examples on p. 157 of the
paper rderred to at the beginnmg of this section,
rCOS(cCOtZ) , W / ^1 T5\
r "P^-^^**-^ <& = 5^ exp (c* coth'5) erfc (o coth 5),
r exp(-^coy|^8(2ycotJ0^^^ exp:(c«coth«J)
X jexp (- 2^ cothJ?) erfc Tc coth -B-^j
4 exp ( ^g coth-B) erfc fc coth JJ+^U ,
13. While referring to the paper in voL i. of the
/ ^ (aj)/(cot o^Ax^x y^ (»)/(cot x) db,
J -00 Jo
where
'^[x)^4> {x) + if} {x — ir) + 4> {x + v) +&C.,
and (ii) in equation (12), p. 157, coth a should be substituted
for a in the expressions subject to the functional sign erf.
74 MS. GLAISHER, OK DEFINITE IKTEdAAtS.
§14. It is eas^ to deduce from (II) a similar formtila on
which the denommator iBx^ + b*: for replacing 6* by 6% and
asmg the formate
1 1 2a!'
_j L___H*!*_
we find
These results can be verified by putting a = &, for the
second equation then becomes
07
VIZ,
which ia a case of the fundamental theorem. The first
equation becomes
r /{X) dx )_ r /{v)dv
which is true, since
£/(-!)?=«' £^«*- .
Similarly from (12) we have
f.
00
X
OB
/
A
( 75 )
ON A CASE OF WAVE MOTION.
By Professor CI Niven,
Wave motion is essentially a case of continnal transfor-
mations of kinetic and potential energy^ the kinetic energy
which any set of particles acquired as the wave passes over
them bein^ due to changes either in the amount of potential
energy which they possess or of the kinetic and potential
energy of the system with which they are connected, or it
may oe due to both of these causes. This view of wave
motion is, in general, masked by the ordinary method of
solving questions of this sort by means of differential
equations and arbitrary functions. It is interesting, therefore^
to find a case in which we can dispense with this method
and exhibit the successive transformations which take place
at each element of matter as the wave passes over it.
We can do so in the following problem, due to Mr.
C. H. Prior, of Pembroke College, Cambridge: An elastic
filament is hung from one end and cut at any point ; it is
required to determine the subsequent motion of each part
The initial distribution ot each being of displacements,
we shall assume that any solution which satisfies the principle
of conservation of energy and the initial conditions b the
solution of the problem.
2. Jjet AB (fig. 5) be the filament, and let the part
BC{= I) be cut off J we shall first discuss the motion oi AC.
If any element dx of the unstretched string be stretched
under the influence of a stress T^ which may be either
tensile or contractile, the extension or contraction will be
T
— dxj where X is Young's modulus, and the potential energy
At •
stored up in doing so will be ^ — db. The weight of the
At
element will be mgdx ; but it will simplify somewhat the
symbols to assume that the unit of mass is the unit of length
of the string. We may, therefore, put 7/i = J , but the results
obtained for the velocities of the points of the string and of
the propagation of the waves may be reduced to ordinary
units by writing — instead of X.
76 PROP. NIVEN, ON WAVE MOTION,
Consider, first, the lowest element of the string, AO
whose unstretched length is dx ; before the instant of cutting
it b under a tension = gL while after cutting its tension is
zero. The amount of energy set free is, therefore, ^^ ^j
and this is converted into kinetic energy ^ Vdx^ where
The subsequent motion of the string AC will be of the
following character : each element will, m succession, assume
the velocity F, all those below it retaining this velocity and
the distribution of tensions in the part over which the wave
has passed will be the same as if the string A G were hanging
freely. To verify this, consider what takes place in the
element PQ (fiff. 6), as the wave passes over it, the
mistretched lengths of CP, PQ being a:, dx.
Before the wave reaches P the tension in PQ iag{x + T}:
after the wave reaches it the tension is gx. The potential
energy given up by the element is, therefore.
If d^^ be the extension of the element in the first state and
d^^ in the second, the work done against gravity is
gx{dS^^d^;j
which is equal to ^-^- dxj
since ^fi='^'^ — ' ^% and rffj=^ dx.
The kinetic energy acquired by the element may be put
equal to \ifdx^ we have, therefore, the equation
\v^dx^ii (P+ 2&)di»-^ dx\
therefore v = ^ = Fl
The motion until the wave reaches A will therefore be of
the character stated above.
3. With regard to the velocity of tranamission of the wave^
we may investigate it in the following way.
«►,
'^
PKOF. NIVEN, ON WAVE MOTION. 77
While the wave is passing over PQj the upper part of it
is under a tension g {l^ x)^ while the tension at Pis gx; the
resultant force on FQ is therefore glj and in the time dt this
face generates momentum gldt. But the momentum gene-
rated in this time is equal to Vdxj hence we have
dxidt^'^ -^\.
This is, therefore, the rate at which the wave passes along
the particles of the string ; and we see that, considered wim
reference to the unstretched string, it is uniform.
This result will apply to all the waves which we shall
have to consider, and we need not, therefore, repeat it.
4. We have now to enquire what takes place when the
wave reaches A. The highest element may be supposed
to receive for a moment the velocity F, but to have its
velocity instantly destroyed by impact at the fixed obstacle to
which the string is attached. The other parts of the string
will continue to move upwards until their velocity is destroyed
by the pressure which is produced by the contraction of
the string. What takes place is this : each particle of the
string is reduced to rest in succession till it is all brought
to rest, and, if we suppose BC>AC^ the strain in the
string will be the same as if GA were produced to ff
(where CB* =• CB), and the contraction proceeded uniformly
m)m Jffj the stress at any point P being ^g.FJff.
[I{AC (fig. 7) were greater than BCwe should still have
to find a point B by the same construction, and AB would
remain stretched during the return wave, while B'PC would
be contracted].
To verify the above statement, let the unstretched length
of PC be Xj and consider an element dx. Before and after the
passage of the wave the stresses in dx are^x. and a (Z — a?).
The kinetic energy taken up is therefore ^ .
The work done against gravity is
^:^{x+l-x)dx^^dxj
and the kinetic energy given up by the reduction of the
element to rest is ^ P(kc. These parts exactly counterbalance
each other, and therefore the motion is of the kind described
above.
78 PBOP, KIVEN, ON WAVE MOTION.
5. For a moment the lowest element is in a state of
contraction : but, the end bein^ free, the tension is instantly
relaxed, ana it commences to shoot downwards, each particle
receiving in succession the velocity V till the upgoing wave
reaches Aj and leaving the part of the string below it in
the same state, as regards tension, as if it were hanging
freely. The reader may readily verify this statement.
6. After the npgoing wave reaches A^ the highest
element, though momentarily in motion, is instantly reduced
to rest, and a wave of rest runs' down the string, which
meanwhile continues to stretch until, when the wave reaches
Cj it is in exactly the same condition as when it was
newly cut.
If (fig. 8) the unstretched lengths of CP, PQ he x^ dx
the tensions in FQ, before and after the passage of this wave,
are gx and g{l + x)] and the potential energy acquired by it
wUlbe £(PH-2te).
Gravity does an amount of wOTk=^ (Z + a — a?) <£», and
A
the kinetic energy destroyed is ^ V^dx = i ^ dx.
The principle of conservation of energy is therefore
verified.
The string is now as it was when it was cut, and will
continue to wriggle up and down through the same changes.
Motion of the falling part.
7. The motion of the falling part is even more interest-
ing than that which we have iust been considering, and may
be investigated in somewhat the same way.
When first cut the tension of the element at C (fig. 9) is
reduced from gl to zero, and therefore it acquires the velocity
F, as we have seen already, for the other part A C,
A wave of F-velocitjr runs down the string, the lower
part remaining, meanwhde, suspended at rest in space till the
wave reaches it, and the stress in the upper part being that due
to a contraction of the same absolute amount as if the string
were inverted and hung freely from the point over which the
wave is passing.
\^
i PBOP. NIVBN, ON WAVE MOTION. 79
To Terify these Btatements let the imstretcbed lengths of
OP, f^Q be as before a?, dx^ the energy surrendered by FQ
In addition, gravity does an amount of work upon OP in
pulling it down which is equal to^{l^x + x)dx =^^ dx.
The kinetic energy acquired by PQ is therefore that duo
to a velocity F, where F* ^ , the same as before.
8. When this wave of contraction reaches B^ the con-
traction of the lowest element will be immediately relaxed,
and it will commence to shoot out with a velocity 2 F, which
will be communicated in succession to every element of the
string, and the stress in the lowest part over which this
upgomg wave has passed will be a tension equal to that due
to the weight of the string below each point.
To show this, let the unstretched lengths of BPj PQ be
ar, dxj and let the contraction and extension of PQ before and
after the |)assage of the wave be d^^ and d^^.
Assuming the distribution of stress to be that given above
the energy surrendered by PQ during the passage of
the wave will be
The work done by gravity on the string is of two parts,
(1) that done on the whole string, considered as rigidly
moving with the original velocity F, during the passage of
the wave over PQ^ (2) that due to the separation oi the parts
CQ^ SP from each other. The velocity of CQ is F, and
tibe time oociipied by the wave in passing over PQ = -yr , and
hence the distance through which CQ falls in this time
Vdx
= -rrTr-r 5 the part (1) of the work due to gravity is thus equal
to ^dx^Tdx.
Part (2)
as before.
80 PBOF. NIVEN, ON WAVE MOTION,
If, therefore, F, be the new velocity acquired by PQ^ th.e
only change of kinetic energy experienced by the whole
string daring the passage of the wave over it will be
whence we have at once
verifying the statement made above.
9. When this wave oi extension reaches it will be
succeeded by a wave of contraction down the string
corresponding to veloci^ 3F, and thereafter by a series of
wormlike movements wnose velocities, beginning with the
first, are F, 27, 3F, 47, &c.
Suppose, for instance, that a wave of extension has just
reachetd (7, the velocity corresponding to it being nVj it will
be followed by a wave of contraction corresponding
to(n + l)F.
For let (fig. 11) the unstretched lengths of CP, PQ be
Xj dx^ the stresses in PQ before and after the downward wave
being numerically gil-^x) and ^x, the energy surrendered
hyPQia^iP-2lx)dx.
The work done by gravity on the whole string is
(1) That on the string as a rigid body supposed to move
fi Vdoij
with the velocity wF, and therefore ^gl. — jr- .
(2) That due to the collapse of OP on JSP, which is
equal to ^ (Z— a;4 x)dx^^ dx.
The only change of kinetic energy which takes place in
the string is that acquired by PQ, whose velocity changes
from n Fto P^j, suppose we have, from the equation of energy^
^(P^^^-n"P)c& = ^(P-2&)db + fi^[F&:+^£fe,
whence - P,^, = n'P + 2wP+ P,
or F.,,=:(w+1)F.
The time which each of these separate movements requires
for its accomplishment Is -^ .
'^ -.
\
( 81 )
t
MATHEMATICAL NOTES.
On Long Successions of Composite Numbers.
Mr. Glaisher has given in the Messenger for November,
1877, and March, 1878, interesting results upon the sequences
of composite numbers in the natural series. To the explana-
tions given by the author, I add the following reflections
upon the appearance of sequences much earlier than the
theory, indicates.
Denote by -AT", =2w + 1, any uneven number, by P{q) the
product of all the primes 2.8.5... 9, q being the greatest prime
inferior to n^ and by S (x) the series of the N consecutive
tiumbers xP{q) — w, ...aj-r (j), ...xP{q) + n.
If we suppose l<a<n+l, the number xP(q)±a is
evidently composite, whatever integer value x may have.
Therefore, i£xP{q) — 1 and xP{q) + 1 are composite num-
bers, the series 8^ (x) will be formed of N composite numbers.
We have, for example, for a; = 1,
P(17) - 1 = 61 X 8369, P(17) + 1 = 19 x 97 x 277,
P(19) - 1 = 53 X 197 X 929, P(19) + 1 = 347 x 27953,
i^hence the series
510492,... 510510,,.. 510528
contains 37 composite numbers, and the series
9699668,... 9699690,... 9699712
contains 45 consecutive composite numbers.
Thus, again, the series 8^^^{x) contains 21 composite
numbers for the values of a;,
8, 15, 25, 26, 31, 33, 34, 37, 38, 45, 52, 54, 56, 58, 62, 71, 77,
79,80, 82,84,91,98, ....
In the general case, where q is given, we determine x by
solving the two 'simultaneous congruences
xP{q) = l (mod^r) and xP{q) = '' 1 (mod^),
g and h denoting the two primes that follow q.
Edouaed Lucas.
Farii, Attgtut, 1878.
• ■ ■
G
82 MATHEMATICAL NOTES.
The Intrinsic Equation of the Elastic Curve.
The intrinsic equation of the elastic curve or lintearia
(yp = c')i8
t = 2am(A,A:),
2c
where A = — , a being the greatest distance of the curve from
a
the axis.
For 1 = ^ = i dn -
p ds he kc^
and sin-^ = - ^ = sin2am t-
= 2sn -5- en T ;
kc kc
therefore ^ ~ T ^^ F 5
and therefore yp = c*.
If i < 1, i|r can increase indefinitely, and the curve is of
the form given in fig. 34, Thomson and Tait, Natural
Philosophy^ §6^11-
If A; = 1, we have fig. 33.
If A>1, -^ oscillates between the values ±sin r; and
1 ^
putting A? = T- , we have
whence i, = ^ , and the curve is of the form given in figures
*iC
28 to 32. A. G. Greenhill.
A Theorem in Combinations.
The number of combinations ofn elements r at a time when
every element may be repeated any number of ti^es in the same
combination^ is
n{n + l){n + 2)...(w + r - 1)
I have given in " Choice and Chance^^ Prop. X. p. 67, (3rd
edition), an algebraical proof of this proposition, dependent
MATHEMATICAL NOTES. 83
only on first principles, and at p. 45 I have cast the same
proof into the form of a proof of a rule of arithmetic.
But though that proof is very short, it is not very easy ;
and the following, though not so concise, may be preferred.
To fix the ideas, think of the n elements as being an
alphabet of n letters ; and observe that the number of per-
mutations of any letters r at a time in a particular order,
say alphabetical order, is the same as the number of com-
binations of the same letters r at a time. The problem is
therefore equivalent to finding the number of r-lettered
words that can be formed with an alphabet of n letters,
each letter appearing as often as we please, but all the
letters of the word being arranged in alphabetical order.
At equal distances draw n parallel lines a-4, ^JB, ...f^
(fig. 12) to represent the n letters of the alphabet, and at the
same distances draw r + 1 lines at right angles to them, so as
to form a series of squares (for convenience, we will call then
square inches), baving w — 1 squares in each horizontal row,
and r squares in each vertical column. Then any r lettered
word subject to the required conditions may be represented
by proceeding vertically from a for as many inches as the
number of times a is to occur in the word, thence proceeding
horizontally to the line representing the next letter which
comes into the word, and going down vertically as many
inches as that letter is to occur, and so on, until we finish at the
point Z. The number of words is therefore the same as the
number of ways of proceeding from a to ^ by a zigzag course,
Le. it is the number of orders in which w— 1 horizontal steps
and r vertical steps can be taken, or the number of ways m
which w + r — 1 steps can be taken, so that n — 1 of them are
horizontal and r are vertical, which is known to be
L^ + ^-1 n(n4-l)(w-f 2)...(w + r-l)
Therefore, &c. q. e. d.
W. Allen Whitworth.
Generalisation of Prof. Gayley^s theorem on partitions
(vol. V. p. 188).
Prof. Cayley's theorem is, that if u^ denotes the number
of partitions of w, no part less than 2 and order attended to
(so that e.g, forn = 7 the partitions are 7, 52, 25, 43, 34, 322,
232, 223),'then w^^l, 1*3 = !, t^^ = 2, 1/^=3, w^=5, w,=8, u^^lZ^
&c., where each term is the sum of the preceding two terms.
02
84 MR. LEWIS, ON ampere's THEORY.
The generalisation Is, that If n^ denotes the number of parti-
tions of «, no part less than r and order attended to, then
«^ = (n — 1), -f (n — r), i,e. each term = the next preceding term
+ the r^ preceding term. When r = 1, so that all the numbers
may occur in the partitions, we have w =2(w — 1), whence
n, = 2**'*, as is well known. The general formula is obtained
by considering the coefficient of a?" in the expansion of
1 4- {1 - (aj'+ aj*^^+&c.)}, = (1 -u:) ^ (1 - a?-.af).
J. W. L. Glaisher.
•^x
ON AMPJfeRE'S ELECTRODYNAMIC THEORY.
By T. C. Lewis, M,A,, Fellow of Trinity College, Cambridge.
Some of the results of Ampfere's Theory, which are usually
deduced by integration from his celebrated formula, seem to
be capable of a much more elementary proof. The methods
usually employed are all, except that by Quaternions, very
long, and without exception they are complicated, and the
meanmg of the processes is masked under the difficulties
of the analysis. It seemed likely that those results which
admit of a simple verbal statement, might possibly admit of a
correspondingly simple proof, based immediately on the ex-
5erimental laws which are at the foundation of his analysis,
'he proofs given in this paper are (a) of the existence at
every point of space of the "directrice de Taction ^lectrody^
uamique;'' {b) that the action on an element of current at
any point is perpendicular to that element and to the * direo-
trice or action at the point; (c) that the action on an
element is proportional to the magnitude of the directrice,
and to the sine of the angle included between it and the
element. The formula itself cannot of course be proved
without analysis, but, as it gives the eflfect of an isolated
element of a circuit upon another element, it can only be
practically useful when mtegrated, and some of the integrated
results are what are here demonstrated without help from the
formula.
Ampfere's experimental laws are —
I. Equal and opposite currents in the same conductors
produce equal and opposite eflFects on other conductors;
whence it follows that an element of one current has no effect
on an element of another which lies in a plane bisecting the
former at right angles (and therefore is not affected by it).
II. The effect of a conductor bent or twisted in any
MR. LEWIS, ON ampere's THEORY, 85
manner is equivalent to that of a straight one, provided that
the two are traversed by equal currents, and the former nearly
coincides with the latter.
III. No closed circuit can set in motion an element of
a circular conductor about an axis through the centre of the
circle and perpendicular to its plane.
IV. In similar systems traversed by equal currents the
forces are equal.
To these we add the assumption, that the eflfect of any
element of a current on another varies directly as the product
of the strengths of the currents and of the lengths of the
elements. It will not be necessary to assume that the action
between two elements of currents is in the straight line joining
them.
1. Since in nature electricity must flow in closed circuits,
and therefore the results we wish to obtain are due to the
whole circuit or circuits (and not to their isolated elements),
we need not consider more than that part of the action of one
element which does not necessarily vanish when summed up
for all the elements of the circuits. Hence, in considering
the action of one element on another, we may make use of
the results observed when a whole circuit is acting instead.
Now, in the latter case, we know by Ampfere's second law
that for any single element of a second circuit, we may sub-
stitute two others which do not deviate far from it, and
which provide a road for the electricity to go from one end
to the other of the element. Hence, in considering the
eficct of one element of the first circuit, we may also also re-
solve the given element of the second circuit into two parts,
and consider the effect on both together to be the same as on
that for which they were substituted ; there may be some
difference in these effects, but, at any rate, if this difference
be summed up for the whole of the first circuit, it vanishes,
and therefore need not be considered in this investigation.
From Law III. it is inferred, that the effect of a closed circuit
upon an element of a current, resolved in the direction of that
element, is zero ; therefore we need, as in the case of Law II.,
only consider that part of the effect, even of an element of
the circuit, which is perpendicular to the element acted upon.
2. By Law I. the action of one element upon another
vanishes, when the plane bisecting the latter at right angles
contains the former. There is therefore one direction of the
86 MR. LEWIS, ON ampere's THEORT.
element a^ which is being acted upon, such that the action of
any particular element of the system of current electricity
upon it is zero, and this direction is perpendicular to the
plane containing the influencing element and the middle
point of a^
Let An be this neutral position of a^ for the action upon
it of any chosen element of the influencing system. Take
any other point G near AB^ and join -4(7, BG. Then the
actions on AG and G5, so far as they affect our results, are
together equal to that on AB^ which is zero ; therefore the
actions on BG and AG are in same straight line, but they
are also perpendicular to BG and AG respectively. The
only line which Is perpendicular to both is the normal to the
plane BAGj and is therefore perpendicular to AB. Thus
the action on the element for any direction of the element in
the plane ABG is in a constant direction ; and for any direc-<
tion whatsoever of the element, the action on it is perpen^
dicular to AB^ and therefore lies in a fixed plane ; and is
such, that it is perpendicular both to the actusd and to the
neutrsd position of a^.
3. Let ADy AE be two positions of a in the plane per-
pendicular to the neutral position. Let AD equal AE. Join
DE, Then since the action on AD is the resultant of those
on AEy ED by Law II. (if the same current flows by either
path from A to D) resolve parallel to DE] then since all the
actions we consider are perpendiculars to the elements on
which they act, we see that the action on AD is equal to that
on AE. Therefore if the element a, be rotated round its
neutral position as axis, the action on it due to the correspond-
ing element of the system of current electricity is un2dt^:«d«
4. It is now necessary to prove that there is a neutral
position of a^ under the action, not merely of an element of
another current, but of any complete system of current
electricity ; and then the proofs of §§ 2 and 3 will become appli-
cable in the genersd case. We have seen that, if the effect
of one element only of the system of currents is considered,
the neutral direction at any point is found by taking the
5 lane containing the element and the point considered, and
rawing a line perpendicular to the plane. Now by Law IL
we may replace a^, so far as it is necessary to consider the
action of this one element of the closed circuit, by two pieces,
one perpendicular to the above plane — on this the action is
zero — and one parallel to the plane— on this the action is
«"^
MB. LEWIS, ON ampere's THEORY, 87
Eroportlonal to Its length, {.e. to a, And, where is the angle
etween the actual and neutral positions of a, ; it is also pro-
Sortional to its intensity t, and to some ftinction F^ which
epends upon the particular element of the system of current
electricity whose effect we are considering, and upon the
point from which a^ is drawn, but (as we have provea in §§ 2
and 3) is independent of the direction in which a, is drawn,
^is in fact the action of the element of the system in question
per unit length of unit current placed along the direction of
that part of a^ which is alone affected by the element, i.e.
the action per unit length of unit current in any direction
perpendicular to the neutral direction, as proved in §3.
Now the action of the whole system upon a, is the sum of the
actions of its parts. Divide the system up into very small
elements, which may be supposed ultimately straight. Uence,
if we suppose a number of forces F (corresponding to the
various elements considered) to act along the corresponding
neutral directions at a^, F sin will be the resolved part of
any one of them perpendicular to a,, and if we find the
resultant of these, and turn it about the element a^ through a
right angle (which is of course equivalent to rotating the
components, and then finding their resultant), we shall get
the action upon a unit length of unit current placed along a^
due to the whole system under consideration. We may dis-
tinguish any force F as the action at a, due to an element,
and the resultant of the forces F as the action at a^ due to the
system of circuits. If the system of forces jPsin^ be in
equilibrium, the resultant action on a^ is zero; i,e, if a, be
directed along the resultant of the system of forces Fj the
action on it is zero. Hence^ there is always a neutral direction
for the element a^. The resultant of the system of forces F is
the same as Ampfere's *'^ Directrice de Faction ilectrodynam'
5. Hence, as in §§2 and 3, we may prove that for any
direction of the element a^ the action of any system of electric
currents on it is perpendicular to its actual and neutral
directions, and hence it is always in a fixed plane, viz. that
to which the neutral direction is normal. Further, if F^ be
the resultant of the system of forces F^ 0^ the angle between
that resultant and a,, we see, by means of § 4, that the action
on ttj is equal in magnitude to ioL^F^ sin 0.
Auffuit, 1878.
( 88 )
TEANSACTIONS OF SOCIETIES.
7%6 Meeting of the British Association at Dublin.
The forty-eighth meeting of the British Association for the Advancement of
Science was opened it Dublin on Wednesday, August 14, 1878, under the
Presidency of Mr. W. Spottiswoode, r.R.S. The greater part of the address of
the President related to mathematics, his chief purpose* being * to inquire whether
there be not points of contact in method or in subject-matter between mathematics
and the outer world.* Three methods peculiar to modern mathematics, viz., 'first,
that of Imaginary Quantities; secondly, that of Manifold Space; and, thirdly,
that of Oximetry not according to Euclid,' were explained and examined. The
Sections assembled on Thursday morning, Angust l!6, the following being the list
of officers of Section A (Mathematical and Physical Science) :
President.— Prof. G. Salmon, F.R.S.
VicerPresidents.—PTqt R. S. Ball, F.R.S. ; Prof. S. Haughton, F.R.S. j Prof,
Henry Btennessey, F.R.S. ; Dr. T. A. Hirst, F.R.S. ; General Menabrea ; Rev. Dr.
Molloyj Prof. S. J. Perry, F.R.S. ; Prof. John Purser ; Prof. R. Townsend, F.R.S. j
Prof. H. J S. Smith, F.R.S. ; Mr. G. Johnstone Stoney, F.R.S. j Prof. J. J.
Sylvester, F.R.S. ; Sir William Thomson, F.R.S.
Seeretaries.'-FTof. John Casey, F.R.S. ; Mr. G. F. Fitzgerald ; Mr. J. W. L.
Glaisher, F.R.S. ; Mr. OJiver J. Lodge.
Among the mathematicians present at the meeting, besides those mentioned
among the officers, were Prof. J. D. Everett ; Mr. R. Harley, F.R.S. ; Mr. W. M.
Hicks J Mr. H. M. JefEery ; Prof, 0. Niven ; Mr. F. Purser; Prof. G. G. Stokes.
F.R.S. ; Mr. Benjamin Williamson.
Prof. Salmon was unable to be present at the meeting, having met with an
accident, and a short inaugural address was given by Prof. Haughton. The
f o4owing is the list of papers read in the department of mathematics on Monday,
August 19, Prof. H. J. S. Smith in the chair :
C. W. Merrifield.— Report of the Committee on Babbage's Analytical Engine.
Janjes Glaisher. — Report of the Committee on Mathemq-tical Tables, with an
Explanation of the Mode of Formation of the Factor Table for the Fourth Million.
J. W. L. Glaisher.— On certain special Enumerations of Primes.
J. W. L. Glaisher.— rNotes on Circulatiijg Decimals.
Prof. J. J. Sylvester.— On the Irreducible Derivatives of certain Binary Fonng
and Form-systems.
Prof. H. J. S. Smith. — On Qua^ric Transformation.
Prof. M. Falk.— Elementary Demonstration of the Theorem for the Multipli-
cation of Determinants.
W. Spottiswoode.— On the " 18 Coordinates" of a Conic in Space.
T. Archer Hirst.— On Halphen's new form of Chasles's Theorem on Systems of
Oonics satisfying four conditions.
Pfof . Casey. — On Curves of the Third Order.
H. M. Jeffejy.— On Cubics of the Third Cl^ with three single Foci, both
Plane and Spherical.
H. M. Jeffery.— On Cubic Surfaces referred to a Pentad of Cotangential Points.
Frederick Purser.— Note on the Geometrical treatment of Bicircular Quartics.
Prof. H. J. S. Smith.^-On the Modular Ciirves.
Prof. Casey.— On a New Form of Tangential Equation.
Robert Harley. — Oij certain Linear Differential Equations.
J. W. L. Glaisher.— On the Solution of a Differential Equation connected with
Biccati's.
Frederick Purser.— On the occurrence of Equal Roots in the Determinantal
Equation of Small Oscillations.
J. W. L. Glaisher.— On the Law of Force to any point whea the Orbit is a
Conic.
Prof. R. S. Ball.— On |;he Principal Screws of Inertia of a free or constrained
Rigid Body.
Prof. J. Purser.— On the applicability of Lagrange's Equations to certaiQ
Problems of Fluid Motion.
W. M. Hicks. — On the Motion of two Cylinders in a Fluid.
Besides these the following papers relating to mathematical subjects were rea4
during the meeting :
PROP. GREENHILL, ON FLUID MOTION. 89
ffir W. Thomson and Admiral Evana. — On the Tides of the Sonthem Hemi-
aphere and Mediterranean.
Prof. S. Haughton.— On the Snn-heat received at the several Latitudes of the
Earth, taking account of the Absorption of Heat b^ the Atmosphere, with con-
jclnsions as to the absolute Radiation of Earth-heat mto Space, and the Mimmnm
Duration of Greological Time.
G-. H. Darwin. — On the Precession of a Viscous Spheroid.
G. F. Fitzgerald. — Note on Surface Tension.
Prof. J. Thomson. — On * Dimensional Equations' and on some Terbal expressions
in Numerical Science.
The Babbage Committee reported that in their opinion the cost of completing
Mr. Babbage's analytical engme would be expressed in tens of thousands of
pounds at least, and they did not recommend tne Association to take any steps
to procure its construction. In the report of the Tables Comtaittee it was stated
that the factor table for the fourth miUion was completed and reader for press, and
that the factor tables for the fifth and sixth millions were being actively proceeded
with. A new Committee, consisting of Prof. Sylvester and Prof. Cayley, was
.appointed for the calculation of tables of the fundamental invariants of algebraic
forms, and the sum of £50 was placed at their disposal for the purpose. The
whole sum granted on the recommendation of Section A was £430.
The Association adjourned on August 2]L till Wednesday, August 6, 1879, at
Sheffield (not Nottingham, as decided at the Plymouth Meeting). Mr. G. J.
Allman is the President elect. In 1880 the Meeting will be held at Swansea.
J. W. L. G.
FLUID MOTION IN A ROTATING QUADRANTAL
CYLINDER.
By Professor A» Q, Oreenhill, M.A.
Mr. Hicks has found by summation [ante^ pp. 42-44) the
expression for 0, the velocity function of fluid motion in a
rotating semicircular cylinder, bounded by r=^a^ and ^ = 0,
6 = IT J rotating about its axis with angular velocity o) ; ana
the current function ^jr^ being the conjugate function to 0,
can of course be immediately written down.
Generally, for liquid contained in a sectorial cylinder
bounded by r = a and 5 = ±a, rotating about its axis with
angular velocity ca, it is well known that the current function
, - » cos2d
cos 2a
2a
©
cos(2n+ 1)-^-
^ 2a
+ 32flia'a'S f— 1^"** , .
T cuuu-^^ V ) i (2n + 1) 7r- 4a}(2n+ 1) tt {(2w+l)7r+4a} '
and therefore the velocity function
V-*
90 PROP. GREENHILL, FLUID MOTION IW A
sin 2d
^=5 Jwr*
COS 2a
Va) ^'^(2^ + ^)2^
4- 32tt)fflVS*°* f — 1 r** ^^^ '^^
- ^ ^ {(2n+l)Tr-4a}(2n+l)7r{(2n+I)7r+4a}'
being the conjugate function to ^.
For we must have
(1) V V = ^) *^^ V^ finite, within the sector.
(2) '^ = JoDT*, when 5 = ± a.
(3) -^ = ^ ©r', when r = a.
Conditions (1) and (2) ^e satisfied by assuming
and In order that condition (3) may be satisfied, namely
'^ = ^©r* when r = a, we must have
«"^ A f^ .v*^^ 1 «/^ C082d\
between the limits d = ±a, and therefore by Fourier's
theorem
- ©a' r» /, cos2d\ ,^ ,x T^ 7/1
_ ©a" r 2 8ln(2n4 l)^'rr ^ sln{(2n + l)^7r-2a}
"" 2a L (2n+l)- cos2a {(2/i 4 1) Jtt - 2a}
^ ^2a
8m{(2n4-l)^?r4-2a} "I
cos2a{(2n + l)i7r + 2a}J
= ©a (- 1) |(2n 4 1) TT - 4a "" (2n 4 1) tt "^ (2n + 1) "w + 4a j
= 32©aV (- ir* {(2n + l)7r-4a}(2n+l)7r{(2nH-l)7r4.4a} '
BOTATIKG QUADRANTAL CTLINDEB. 91
Therefore
cos 2a
> .A^^^'S:
.2-.2 -"'•
+ 32wo a 2„ |(2n+ 1) w-4a}(2n + 1) ir {(2n+ 1) ir + 4a}
. , as* aa* —. ,,^, (Z'^Ds
x{ 1- i- + _i 1
a?
cos 2a
l-^, . JL . I*'
- -^7- I a? I tail ^ +aj I 1 f
+ 0? 1 +a?
putting x^- ef\
IT
The integrations can be performed when — is an integer,
iSa
that is, when the angle of the sector is a submultiple of two
right angles.
In a semi-circular cylinder ^ = 1) and we obtain the
expression given by Mr. Hicks on p. 44.
In a quadrantal cylinder r- = 2, and the first two terms
in the series for yjr -f t<^ assume the indeterminate from
00 — GO , and must be evaluated.
Putting in these terms — = 2 + 5?, and neglecting «",
, .^ 2©a' for' x^ x -i^ f * ^\ f ^^)
2a)a
TT
s
|- »' logo; + tan-'as' + i(x'- ^) log (1 + x*)\
92 PROP. GREENHILL, FLUID MOTION IN A
' f 2-.cos2d 1 + 2 -j8m25+-;
+ — Uan'^ 4 + JVlog
I 1 i 1-2 -5 8in2d+-i.
a a o
X U log (l + 2 ^ sin4e + ^1) + ; tan-^ -^
"4 Sin 45
. 7
cos 40'
a"
and therefore
^=x-— r'cos251og- + — r'sm25.£?
7*
, 2 -5 cos25
+ — tan^ —
IT _ 7^
'-a*
«o' '-»
4ir
g-j')cos251og(l+2^,cos4i? + ^)
/
©a /r' a \ . ^^ , _, a
sin2a tan —
27r W • rV , r* Tl'
1 + -i C0845
a
6 = 7^00825.5 7^8^2^102:-
^ IT IT ^ a
T . _ ^ T
. l + 2-j8in25 +
ft)a , a a
1-2 •^8in2i?+ -4
a a
"#N
BOTATINa QUADKANTAL CTLINDEB. 93
#
1 + -T cos45
a
It is easily seen that v'*^ = and v^(f> = from the con-
sideration that if x^ and y^, and also x^ and y^ are conjugate
functions, then x^x — y y, and x^y^ + a^y^ are conjugate func-
tions (Maxwell, Electrtcityj § 187), ana we may put
x^^^r^ cos2d, y^ = r' sin 25,
also a:^ = (^,-p)cos25, y, = (^, + ^J8in25,
7*
a?,
4 8 -4 sin 4:0
, = 41og(l + 2~cos*5+J), y^=tan-^-^-s ,
^ * ^ l+-iCOs45
a
and then oi^^^—y^y^ and ^{y^-^-^Jj^ form component parts of
'^ and ^.
Also when r^a^
'^= sm2i?.5 8m25 tan* tan25+ — tan^oo
and when 5 = ± Jtt, -^ = Jcar*,
and therefore '^ and ^ satisfy the required conditions.
If we put '^ = X 4- ^6)r', then within the sector v'x ~ ~ ^®)
and Y = round the boundary ; % is the current function of the
liquid relative to the cylinder, and ;^ = (7 is the equation of
the curve described by a particle of liquid relative to the
cylinder.
94 PROF. OREENHILL, FLUID MOTION IN A
If a sectorial cylinder filled with liquid perform finite
oscillations under gravity about its axis which is held fixed,
it will oscillate like a simple circular pendulum, and to find
the length of the simple equivalent pendulum, we require to
know the kinetic energy of the liquid due to a given angular
velocity of the cylinder about the axis.
If T denote the kinetic energy per unit length of the
cylinder due to the angular velocity «, and if ^ denote the
velocity function, p the density of the liquid,
by Green's theorem.
For a sector bounded by the arc r^a, and the radii
= a and 5 = - a, ^ = when r=^a, and — =s±{or when
^ an ^ an
5=±a; also ^. = — ^^.
Therefore r= \wp f (^. - (f>^ rdr
cop
For a semi-circular sector bounded by the radii i? = and
^ = J(»r'8in2i?
1 + 2 ^cosfl + -,
©a" \ \[f a'\ ^>» ^1 , a a
47r
•[{(^^-^'-4'°^79
cos5+ -%
a a
and, therefore,
Therefore r= ©p / ^^rd/r
4
A
BOTATINa QUADRANTAL CTLINDEB. 95
T
and putting - =«,
Denoting the mass of the liquid per unit length of the
cylinder hj Jf, and the effective radius of gyration of the
liquid in the semi-circular sector by k^ we have
{Mlf^' = r= ip«» ^ (4 - i^,
So*
and therefore ^ ** ^ — ^^'
If the liquid be solidified, the effective radius of gyration
of the liquid will be given by
and therefore y^ = -5-1=5?,
the ratio of the lengths of the equivalent pendulums, when
the semi-circular sector is filled with liquid, and when the
liquid is solidified.
In a quadrantal sector, bounded by the radii i? = ±^7r,
A = - — r* cos 2^.5- — r»sin2d log -
r* sin 4:0
+ i — -1--, cos2i?tan*-r— I
cos4d
, l + 2-2sin25+-r
+ i — log 4 ;
1-2-. sin25 + -7
a a
>
96 PROP. GREEKHlLL, FLUID MOTiaN IN A
and therefore
r*
and T=^p(o l ib rdr
if «= -i.
a
Performing the integration^ we shall find
T=pa.««^(log2-^^),
and therefore if k denote the effective radius of gyration of
the liquid in a quadrantal sector about the axis,
T
pa>' I* (log2«g
and
^ 16, „ , I
the ratio of the lengths of the simple equivalent pendulums,
when the quadrantal sector is filled with liquid, and when
the liquid is solidified.
IT
For a sector of 60*. r- = 3, and therefore
-^ + 16 = tt>aV + tan aj
Zone? f ^ t dx _« r x^dx
TT
( ^[ dx _j r icWa; \
?
ROTATING QUADRANTAL CYLINDER.
oa'aj* + tan" V
97
TT
:(.'+i.)(tan-^-^ + 2tan-«.)
TT
= ft)r* cos 2^ + t(ar' 8in25
2.-11 cos 3^
6}(Z
TT
_..„„„. 1 + 2^81113^ + 5,
- ^tan J + it log
1--
1-2^81113^
a
-7)
a
V3 coa'
TT
X
ilog
1+ 2V3- co8^ + -2(2co82^ + 3) + 2 V3-8C085 + ~.
a Grj cr a*
1 - 2 V3 " COS ^ + -7; (2 COS 2^ + 3) - 2 V3 -. cos ^ + -.
a a a a
+ ttan'
2V3^8b^(l-;)
1 + -i (2 COS 2^
a
-v{(^?)-»^^'-e-S''-4
2 - cos ^
X h tan-* 2-
^ 1-(1 +
(-5)
7~7
(l + 2cos2^)^ + -4
Ji
+ i » log
1 +2 - sin ^ + -^1 -2 cos2^) + 2-,sm ^+ ^
a a_ ' a a
l-2-sin^ + -Jl-2cos2^)-2-3Sin^ + -^
a a a a
2 - cos
.8
l + 2^sin^+-.
+ tan
1--T, l-2-8intf + V
a . a (A
98
PROP. aBEEKHILL, FLUID MOTION
•••
using the theorem that
log(a + t)8)=ilog(a' + ^) + it^
and tan ^x = \\ log - — r ;
and therefore
tan-(a + iV3)=iflogJ±|^
1^-1 2a ,., (14/9)'+*
Therefore, equating the real and imagit
a 2 -8 cos 3^
» » /I ^^ —1 ^
Yr = fi)r cos 2^ H tan ^
^-?
_ V3 wa' /r^ __ a*\
4 7r Va' W
cos 25
X log
I 4- 2 V3 - cos 5 + ~ (2 cos 2 5 + 3) + •
a <r_" '
l-2V3-cos5 + -„(2cos25 + 3)-t
+ -V - + - siu25tan
l + jPc
2 - COS tf
f tan- _^_
\ 1-^,(14
i^ + 2 tan"
(1 + 2 cos 25) + -4
a
L
:. ctl;nd£B.
r r
1- - + -.,,
a cc \
r r'/
-- + -
a a
101
tf-.
.*.
4*
a
1--
a a
an
-1
V3-
1-^
u>
]2
Tra^o) •'^
us of gyration of the liquid,
S^4>ardr.
i' 2^ = w, an integer
• a? CDC? _
ja«-:i-+2 — tan"*a;»
COS-
n
_ wW / J raj*"»daj
(-/s
a"
+«■
Jl + a^j*
100 * PfiOF. GREENHILL} FLUID MOTION IK A
r . ^ r*
( — ?)•
a a
Obviously V V = 0, and y V = 0, and -^ and ^ are finite
inside the sector ; also
(1) when r = a,
«»«' . -t ^ ©a'
-Jr = ©a* cos 2tf H tan"* oo - 2 — cos 2^ tan"^ oo
(2) when ^ = Jtt,
7* .T ^T T
V3 wa /r' o \ , ( a a a a \
a a a a
a a a a a
T T ' T
» /^ 2. / V3- V3- ' V3-V
+ 1 — -5+ -) 3 tan^— -5-tan*— -y-.2tan*i— b)
a" a' a*
= i©*-'
and so also when 5 = — Jtt, -^ = ^©r* ; therefore -^ satisfies
the required conditions.
Also
i^
, VS , ao", a*
1+3- +4 --3 -, + -.,
^i^(t: + %) logf 2 ^ 4-4)
«*
^
a a a a
BOTATINa QUADRANTAL CTLJNDER. IQl
A* fj^ At 01^ tj^
a a a a a
r
• a/3-
V3 coa* /»•* a'\ , -, a
-TV b-?j*"^ —7
a-
-^^ .•^^ a. ^^' l^«. 5^
2 TT f
a
1 1 H h -i
a a a
a
and A; denoting the effective radius of gyration of the liquid,
j^,_ 2T 2a>pf:<l>rdr
!o4>ardr.
7ra*a>
• IT
Generally, if — = n, an integer
.»
'^ + t^=:^ft)a' — + 2 — tan'^aj*
C08-
n
ntoc? ( « {Q^'^dx - (x^^dx\
102 PROP. GEEBNHILL, FLUID MOTXW IH A ^
Now f^T^» h ^ suppose)
= «l2co8(n-2)glog(x«^2xco8g+l)
sir •
X - cos —
I . / ^v «^ X -1 2n
+ -S8m(n-2)-tan ^ ,
sinr-
2n
and f fq:^ (= -S suppose)
= - 1 Sco8(n + 2) glog(t«'-2xco8g + l)
Sir
X — cos -—
1 87r 2w
+ - 2 sinfw +2) — tan'' ,
n 2w . sir '
sin— -
271
wliere S means a sum to be* formed by giving to 8 all the
odd integral values from 1 to 2/i - 1 inclusive.
Therefore,
V 9
^ + 1^ = jo,a' — + 2 — tan"'aj"
cos -
n
|(,._|)(^-5)+(.-+^)(^+5)}
-- iftja' — + 2 — tan 'a?
cos-
n
:-(^-i)s.»=»a'^l.is(.^-!.»<+t)
CC — COS —
ril (a»«4,^) s sm -rr cos — tan^ ■■ i
^ ^ ^ sin —
2n
, r'cos2^ . . r'sin2g
cos - cos --
f
KOTATING QUADBANTAli CYLINDEB.
103
. 2 -i» COS nO
+ — tan ^ ^
TT 1 ^
, 1 + 2 -li Bin w5 +
- . ©a'
+ 4« — log
TT
a a
l-2--Binn^+-=
a a
©a
27r
{S-pj-^'+'-S+p)''-^''}-
2 Sin — sm —
2 n
[4 log
r >, tfTTr*/ .^_ , «7r\
1 - 4 - cos ^ cos
a
87r r (
cos 2^-2 cos*— J
— 4 -- cos cos
a'
sir r*|
2^"*"?J
— ttan
2 - sm ^ cos i sm 2^
a 2n a
TT
S Sin -- cos
2
1 - 2 - cos ^ cos 7— + -5, cos 26^-"
a 2n a
, Sir f Sir r ^\
Bin -- cos- cos^
_i 2/1 V 2n a /
" i
sir ^r ^ sir r
cos 2 - cos ^ cos —- -t- -5
n a 2n a
?[i
1-2- cos .
and, therefore,
+ i;iog— :£32i—L4i
- „ COS 20 . ©a ^ -
•^ =Ao>r — — + — tan
^ ^ IT IT
cos —
2 -= cos ntf
a
3r
1-
a
"»
ma
47r
1 — 8 cos 2^5 sm—- sin —
xloff|l-4-costf COB ^ + 2-]fcos2tf-2co8*^'\
^ \ a 2n a \ 2n/
104 PROF. GREENHILI^ ON FLtilD MOTION.
— 4 -5 co» ^ COS ~ +
5}
+ -— -^ + -i 1 sm 2^2 Bin — sin —
27r \a* r*y 2 n
2 - sin i? cos -- - -r sin 2d
, _, a 2n a
X tan .^
1 — 2- cos ^ cos -— + -5 COS 20
a 2n a
^ . sir f sir r A
2 sm ;r- ( COS cos a
^ -1 2n \ 2n a J
^•T ^ r ^ STT r
cos 2 - COS a cos h -i
n a n a
1
-2- COS —-^ + -=
- a \2n J or
a \2n J a
• o/i « l+2--sin«^+-5ii
- , sin 26 ma , a*> a
0=:^a,r' —+ ^log ;» -sir
cos- 1 — 2-iiSinna4--5s
n a a
cos2a2 sin-r- sin —
2'jr W ry 2 " n
■ r . ^ / STT r A
2 - sin ^ cos t; cos ^ )
X -1 ^ \ 2n a /
X tan^ i
1 — 2 - cos tf COS-— + —5 COS 2^
a 2n or
. mc? /r* . a*\ . ^^^ . «7r . «7r
+ -r- I -i + -3 1 sm 2^S sin -r sin —
4ir \a* rV 2 n
X log i 1 - 4 - costf COS ^ +2-3 (cos 2^-2 cos* — J
^ ( a 2n a*\ 2n/
' II
>
V
/
MB. WHITWORTH, ON ARRANaEMENTS. 105
t! — 4 -i COS COB
2n ^ a*j
COS 2a2 sin -— cos —
2 9t
cog* fr^ _ a^\
1-2-C08 ---5 +-,
- a \2n / a
xlog-
1 + 2- cos --.+ ^ + -
a \2n / a
s
1-^ — -^j 8m2^Ssin-^ cos —
^,f lir W r 2 w
xtan'
. sir f sir r ^\
sin ^r- cos r cos cr )
2n V 2n g /
sir ^r ri sir 1^ *
cos 2 - cosa cos -- + — «
n a 2n a
ARRANGEMENTS OF m THINGS OF ONE SORT
AND w THINGS OF ANOTHER SORT, UNDER
CERTAIN CONDITIONS OF PRIORITY.
By W. AUen Whitworth, M.A.
Op the two integers m and w, let n (throughout the
paper) be one which is not greater than the other.
The conditions of priority which I propose to consider
are such as place limits on the proportion of the two classes
of things in any part of the arrangement reckoned from its
initial extremity.
For example, if an urn contain black balls and white
balls which are to be drawn out in succession, the order in
which they are drawn out may be limited by the condition
that the number of white balls drawn must never exceed
the number of black balls ; or, again, by the condition that
the excess of black balls over white must never be more
than a given number.
I use the symbol c^ ^^in preference to c^ to denote the
number of combinations of n things taken r at a time, and
it is convenient to notice that
_ n
106 M«, WHITWOBTH, ABRANGEMENTS
and, therefore,
_7n — n + 1 __m — n+1
and, as a particular case,
Also that c . = Jc .
ft) ft— 1 4 ft} ft
1. From any origin (fig. 13) draw OX horizontally and
Y vertically, and let P be the point which would be reached
by starting from and taking m paces horizontally in the
direction OX and n paces vertically parallel to OY.
Let A be the point reached by taking one pace hori-
zontally and one pace vertically, £ the point reached by
taking 4wo paces in each direction, G by taking three paces
in each direction, and so on; so that ABC... lie in the
straight line bisecting the right angle XO Y.
Let 0, a, J, c... be the points reached by taking one
horizontal pace from each of the points 0,-4, 5, G... and
0), a, ^,7... the points reached by taking one pace vertically
from Oj A J -B, G... respectively.
In considering the number of routes from one point to
another we shall proceed on the understanding that each
route is to be traversed by paces taken only . horizontally
and vertically, and without retrogression.
2. The total number of routes from to P must be the
same as the number of different orders in which m horizontal
paces and n vertical paces can be arranged. For in any
order m horizontal paces and n vertical paces will take us
from to P and every different order will give a different
route. Hence, the number of routes from to P is
_L^*+^_(w + w)(7n + n-l)(w + w-2)...(n + l)
C = T — i ^- ' — '
**» * j m\ n I ra
__ (m + w) (m + n — 1) (m + w — 2)...[m + 1)
3. Let f^ ^ denote the number of routes from to P
which do not cross the diagonal line OAB... , and let x^^ ^
denote the number of routes which do cross this diagonal;
so that
f ■X.x =c
\1
OF m AND n THIKG8. 107
4. All the routes included in the number x^ ^ must Bass
along one or other of the paces Oo), -4a, J5/8.,. ; some of the
routes may pass along more than one, but we will classify
all the routes according to the point at which they first
cross the diagonal whether they recross more than on^
or not.
Let us consider the routes which cross the diagonal at K
(after making say r horizontal and r vertical paces) wilJK)ut
having crossed before, but independently of any consiideri^tion
of the number of times they may cross afterwards.
Any such route OKkF may be divided into three portions
OiT, Kk^ kP, of which
OJTcan be made In^ ^ ways (Art. 3),
Kk in one way,
^^"i<'m-r,n--iWay$(Art. 2).
Hence, the whole number of ways in which the route
can be made, or the number of routes through K^ is
Giving /fall values from to w - 1 we shall get all the
routes comprehended in the number x^ .
XT *^ •*»•
Jtlence,
or. In virtue of the relation,
5. Considering the particular case when m = n we have
pr, in virtue of the identity
^r, r-j = i<^r^rj
•^*ij * ~ ^H, » "" 2 l/o, o^«, »'^/l, l^n-1, n-l ^'^, /^»-8, «-8+' • •'h^i-i, n-l^i' J*
PKOPOSITION I.
6. To prove that
«, n __
•'"i •• n+ l"^ *» " ^^«+»» *+>•
. 108 MR. WHItWOBTH, ABRANaEMENTS
Observing that c,, , = 1, c,, , - 2, c,, = 6, c,, „ = 20, c,^ , = 70,
and actually counting the routes for tne first tive values of w,
/o,o = ^3/i,i'=^)/2,s = 2, /3^3 = 5, /^^ = 14,
we see that the proposition is true as long as n does not
exceed 4.
To prove it generally, we shall show that if it be true
when w = 0, 1, 2, ... or a? — 1, It will also be true when n = x.
By Art. 5 we have
But, since the theorem is by hypothesis true when w = 0,
1, 2, ... or a; — 1, we have
/o, ~ 2^0, o"" 2^1,1)
/,,t = 2c,,i-K«»
&c. = &c.,
Hence, substituting, we get
But we know that
(or we may obtain it by expanding (1 - 4aj)"* by Bin. Theor.;
squaring, and equating coeffs. of a;**), therefore
or X, « ~ 2c^ ^ — Jc^j^ ^^,
which shows that if the theorem is true for the first' a; values
of w, it is true for the next value.
But we have shown that it was true for initial yalueS|
therefore, universally,
fuf n ~ 2C^^ ^ — a^n+1, «+l»
or (which is the same thing), v
f ^
ft
or again (which is the same thing).
^
OP m AND n THINGS. 109
7. Since all the routes from to P commence at the
point on the diagonal and terminate at the point P to the
right of the diagonal, they may be classified according to the
points at which they first pass to the right of the diagonal :
%.e, according as they first pass along Oo or Aa or Bb^ &c.
Consider those which pass along Kk^ any such route consists
of three portions OK^ kK^ kP, of which OK lies altogether to
t^e right of the diagonal, and can therefore be described in
7C^ r ways ; Kk can be described in one way, and kP can
be described in c^.^,, ,^ Ways. Hence there are
1
such routes, and giving r all values from to n inclusive we
must obtain all tne c^, ^ routes from to P. Therefore we
must have identically
1
This is true for all values of 7/1 and n ; therefore, writing
m + 1 and n — 1 for w and n respectively, we obtain
1
PEOPOSITION II.
8. To prove ihatf ^ = c,^ , - c^+,, ^.,.
From Art. 4 we have
Substituting for /^ q, /, „ /,, , their values as found in
Art 6, in the form
' * n+ 1 *' *'
we obtain
. «.
_ 1
»
110 MB. TfHITWORTH, ARRANQBMENTS
or in virtue of the identity proved In Art. 7
N.B. We may also write the result in the form
^ m — n + l ^ m — 71+ 1
9. If, In Art. 7, the routes Instead of starting from
had started from the point / situated A paces horizontally to
the left of 0, the whole number of routes would have been
c„^.,. And classifying them as in the case of Art. 7, any
route passing along Kk would have consisted of three parts^
JK, Kk, kP, of which
A+ 1
JK could be described in 7—- c^^f^^ , ways (Art. 8),
Kk in one way,
*-P in c^^,, ^^ ways.
Hence there are ^^^^^ c,^,, ^ c,^, , such routes. And
giving r all values from to n we obtain the identity
A+1 A+1
A+w + l ^*-^'*' ^«»— »*o»
or
A + l~ A + 1 ^ A + 2 "^ A+3 ^'"^ A + n + 1 '
Write A for A + 1 and »» for m — 1, and we have
"T"-"~"A""^ A + 1 "^••'■^ A + n •
Now write n — A for w, and we have
A "" A A + 1 *** n
PROPOSITION III.
10. To find the number of routes from to P which touch
or cross a line SB parallel to the diagonal OK at a vertical
distance ofh paces above it.
OF m AND n THINGS. Ill
The routes may be classified according to the points at
which thej first touch the line HR,
Let 0\'LP be any route which first reaches this line
at a point Z, distant r horizontal paces and r + ^ vertical
paces from 0, and let \'L be the pace by which the point
L is approached.
The route may be divided into three parts OX', \'X, and
LP] of which
Ox' can be made in v c^^,_j» »• ways (Art. 7),
\'L in one way,
LP in c^^^^^ws.jb;
therefore the number of routes first touching at L is
•s
-. I I H-*-l» r^mr^t n^-kJ
and the whole number of routes will be got by giving r
all values from to n — A inclusive, and addiuj^.
The summation is that of the final series of Art. 9.
Hence, the whole number of routes required is c^^^j, ^^.
11. COEOLLAET. It foUows that the numher of routes
from to P which do not cross or touch the diagonal h
paces above the diagonal through 0, is
m» n »i<-A, N-A9
and writing A f 1 for A, the number of routes from to P
which do not cross (but may touch) the diagonal h paces above
the diagonal through 0, is
for If they do not cross this diagonal, they cannot touch the
diagonal next bevond it.
Observe that if A = this reduces to the case of Prop. II.
(Art. 8).
12. Examples.
(I) A man drinks in random order n glasses of urine
and n glasses of water (all equal) ; shew that the odds are n
to i against his never having drunk throughout the process
more ^ wine than water [J&lucational TimeSy June, 1878,
question 5669).
112 MR. WHITWORTH, ARRANGEMENTS
Let glasses of water be represented bj horizontal paces
and the glasses of wine bj vertical paces, then the chance
required must be the same as the chance of keeping always
to the right of the diagonal OV in passing at random from
to V.
The total number of routes (by Art. 2) is c ^.
The number which keep to the right of the diagonal
(by Art. 6) is c„.„-(n+l).
Therefore the chance is l■^(r^+l), or the odds are n
to 1. Q. £. D.
(2) Ifn men and their wives go over a bridge in single
file^ in random order^ subject only to the condition that there
are to be never more men than women gone over^ prove that
the chance that no man goes over before his wife is (w + 1) 2~*
(Educational Times^ September 1878, Question 5744).
Eegarding the diflferent men as indiflferent and the
different women as indifferent, the number of orders in ^hich
they could go over would be (as in Art. 2) c^, ^ and the
number of orders subject to the condition that never more
men than women are gone over would be (as in Art. 6)
c^»„-f (n+1).
But the men may be arranged among themselves in
In ways, and the women in In ways.
Therefore the total number of possible orders, subject to
the given condition, is
[w[nc^ ^-^ (n + 1) or [2w -5- (w + 1).
These include amongst them all the orders in which each
man has gone over before his own wife. But amongst all
the possible \2n orders in which the 2n persons could cross,
any assigned man will be before his wife in (\) [2n orders
and all the n men will be before their wives in (i)**| 2n orders.
Therefore the required chance is
(4r|2n
^— or (n + 1) 2"**. Q. E. D.
L2n-j-(n+l)
(3) In how many orders ca/n m positive units and n
negative units be arranged so that the sum to any number of
terms may never be negative f
By taking horizontal paces to represent positive units and
vertical paces to represent negative units, this question is
OP VTL AND n THINGS, 113
seen to be equivalent to that of Art. 3. And, therefore,
(by Art. 8) we have the result
[ 7n + 1 1 n *
(4) In how many orders can m even poioers of x and n odd
powers ofxhe arranged^ so that when a? = — 1 the sum to any
number of terms may never he negative f
This Question diners from the preceding in that the terms
are all aifferent. The positive terms may therefore be
permuted amongst themselves in \m orders and the negative
terms in In orders. The required number of ways is therefore
(w — n + 1) f m + n
mTl '
(5) In how many orders can a man win m games and
lose n games so as at no period to have lost more than he
has won f
The question is equivalent to question (3) and the answer
is identically the same.
(6) A man possessed of a-^l pounds plays even wagers for
a stake of 1 pound. Find the chance that he is ruined at the
(a + 2aj + 1) wager and not before.
He necessarily loses the last wager.
Let wt = a + 05, the number of wagers he loses before the
last one.
n = Xj the number he wins.
The number of orders in which his gains and losses can
be arranged will be represented by the number of routes
from P to (fig. 13) without crossing the diagonal OV.
And this must be the same as the number of routes from
to P which is {by Art. 8)
m — w 4- 1
c ;
m+l •"•*'
and, therefore, the chance that m loses and n gains are
arranged in such order that the man is not ruined before
the {m + n)"* wager is
m — n + l
m + 1
VOL. Yin.
114 MB. LKWISy ON CENTRES OF
Bat the chance that his first m wagers should give m
losses and n gains is
and the chance that the final wager gives a loss is ^.
Therefore the chance that he is not mined before the
{a + 2X'\- 1)^ wager, and that he is rained then, is
j-n+1 i ^ + n
971
•n+1 Iw» + n (a+l)|a + 2a?
+ 1 \m\n ^^^ " (a + a; + l[a; ^^^
(7) In haw many different orders can a man possessed qfh
jpounds win m wagers and lose n toagers of 1 pound each
without being ruined during the process f
The number of ways must be the same as the number of
routes from to P without touching the diagonal JB. The
result is therefore (bj Art 11)
(8) If a man playing for a constant stakcj win 2n games
and lose n games ^ the chance that he is never worse offUuin at
the beginning and never better off than at ike end is
n* + n4-2
4^T6n + 2*
(9) If he win 2n + 1 games and lose n+1 games^ the
chance is
n
4n + 6
{Educational Times^ November, 1878, Question 5804).
ON CENTEES OF PEESSUEE, METACENTEES, &a
By T. a Lewis, Jt£.A., Fellow of Trinity College, Cambridge.
The centre of pressure of a plane area bounded by a closed
conic may be found by the following elementary geometrical
method.
(i) Through every point of the boundary of the given
area draw vertical lines to the surface ; then if the vertical
i
♦ 1
i
'. ♦
PRESSURE, METACEMTRES, &C. 115
line through the centre of gravity of the liquid enclosed by
these lines meet the plane area in* H^ H is the centre of
pressure required; for the centre of parallel forces is not
altered if* the forces are altered in any constant ratio, or if
their directions are changed, provided only their points of
application and their parallelism are preserved. The problem
of finding the centre of pressure for an area is therefore
identical with that of determining the vertical line through
the centre of gravity of the liquid which lies vertically
over it.
(ii) Let ABC be any closed cOnic section touching the sur-
face at A (fig. 14) ; let G be the other point of the conic at
which the tangent is parallel to the surface, i.e. the other end
of the diameter through A. Through G draw a vertical line,
and take in it some point above the surface of the liquid.
Construct the cone whose vertex is and base ABG^ let the
section made by the surface of the liquid be Ahc^ c being on
OG. Bisect -40, Ac in i>, JS, respectively; join i>0, mJO]
take jF, G^ in DO^ EO respectively, at one quarter of the
distance fi:om i>, E towards 0; join OF^ and produce to
intersect the given area ABG in H\ ^will be the required
centre of pressure.
For Ahc Is a conic section, and the tangent at c is parallel
to that at (7, i.e. to that at A ; therefore D, E are the centres
of the two sections ; therefore F^ O are the centres of gravity
of the two cones OABG^ OAbc] therefore the centre of
gravity of the liquid between the two sections must lie in GF,
Also &F is parallel to JDE^ and BE is parallel to Cfc, which
is vertical; therefore OF is vertical, and is in the plane
OGAj therefore ^is on AGj and
BHi.BG:: BF:BO:: 1:4.
These two results, that GF'm vertical, and that DHia one
quarter of DC, are independent of the position of on the
vertical through (7, and of the angle which the plane ABG
makes with the surface. They are therefore true whatever
this angle is, and whatever the distance GO is. Take at
infinity ; the wedge of liquid between the two sections then
becomes the same as the liquid vertically over the plane area
ABG; and H^ being determined as before, is seen to be the
centre of pressure required.
(iii) If a circle be immersed so that its highest point is at
a depth h below the surface, its centre, 0, at a depth A + jfc,
and if a be its radius, and OF^ ^a, measured along the radius
12
^ .
L
116 MR. LEWIS, ON CENTRES OF
to the lowest point, the centre of pressure is at Q in OP
(fig. 15), such that
OQ: QPiikih.
A similar construction holds for an ellipse.
Radius of gyration.
The radius of gyration, A;, of a plane figure, of area -4,
about anj axis of y lying in its plane, may be defined by
the equation
da being any small element of the area, and 2 denoting a
summation extending to all such elements.
Some useful results may be obtained without the aid of
Integral Calculus.
(i) Let the area be immersed in a liquid so as to touch the
surface ; let the tangent in the surface be taken as axis of ^ f -^
let x^^ x^ be the distances of the centre of gravity and centre
of pressure of the area respectively from the axis of y. The
pressure at a point of the area at a distance x from the axis
of y is proportional to a?, call it 6x ; therefore x^ and x^ are
determined by the equations
and x^A^-lxda]
therefore, by multiplying these two equations together, we get
x^x^ = A' ;
or, the radius of gyration of the plane .area^ Jnst immersed^
about the tangent in the surface of the liquid is a mean pro^
portional between the distances of the centre of gravity and,
centre of pressure of the area from that tangent.
(ii) Let a? = aj, +ajj,
and let \ be the radius of gyration about an axis through the
centre of gravity of the area, and parallel to the axis of y,
then _
A*^ = Sa;Vcr = 2 (oj, +a?JV(r
= x^A + Saj^Vcr = x^A + k^A
therefore
fC — K "" fiC- •
i
PBESSURE, MBTACENTRES, &C. 117
(ill) Applying this to the case of a triangle having one
Joint, Ay m the surface, and the side BG horizontal at a
istance p^ from -4, we have
therefore ^* = iPi*5
therefore A^" = (i - 1 ) Pi^ = i^«>i"-
(iv) If BC be in the surface,
therefore J(? = ^p*j
and A;^" = (i - i) P,' = i^Pi% as before.
(v) If we take a parallelogram with one side in the
surface, and the parallel side at a distance 2a from it, we have
therefore k* = fa*,
and ^0""^'
(vi) If we consider the case of a closed conic touching the
surface in a line whode perpendicular distance from the centre
is py we have, as shewn above,
therefore k* = ^p* ;
therefore k^ = ip\
Metacenires.
All the radii of gyration found in the previous note are
useful for finding metacentres.
(i) If V be the volume of liuuid displaced by a floating
solid, A the area of the section made by the plane of floatation,
k^ the radius of gyration of A about an axis through its
centre of gravity and perpendicular to a plane which divides
the solid symmetrically, H the centre of gravity of the
displaced liquid, and Jf the metacentre, then it may be shown
in an elementary manner (vid. Besant's Hydromechanics^
3rd edition, Art. 62), that
118 MB. LEWIS) ON CENTRES OF PfiESSURE, &C.
Let the immersed part of the solid be an inverted cone
with its vertex, 0, vertically below the centre of gravity of
the area -4, and at a depth A, then F=! ^Ah ; therefore
and Oif=|A + 2jL=|lJ^
(ii) If A be an isosceles triangle, and jp^ the perpendicular
from the vertex on the base,
by (iii) of preceding note.
If A be an equilateral triangle, and OA^ OB^ 0(7 make
an angle a with the vertical, f jp, = h tan a ; therefore
OJIf=P(l -fitan^a).
(iii] If ^ be a rectangle with sides 2a, 25,
for the two symmetrical displacements.
If ^ be a square, and the lines joining to its angular
points make an angle a with the vertical, *s/[2)a=^hi2i,TLa\
therefore OJf = | A (1 + 1 tan*a).
(iv) K ^ be an ellipse with semi-axes a and 5,
02f=f-^orf-^,
for the two symmetrical displacements.
If ^ be a circle, and the semi-vertical angle of the inverted
cone be a,
a = iltana;
therefore OJf = |A (1 + tan"a)
= f A sec*a.
StpUmber 2, 1878.
( 119 )
CAUCHT'S THEOREM REGARDING THE
DIVISIBILITY OF {x + y)' + {-x)' + {-yT.
By ThomaB Muir, M.A., F.R.8.E.
Caucht's theorem in regard to the divisibility of
(aj+y)"**'-a!'"*'-y"'*' to which Mr. Glaisher has recently
drawn attention may be established in a simple manner
quite distinct from that employed by its author.
Let -&, = (x + yf + (- »)• + (- y)*,
then, by ordinary moltiplication, we have
2/8Li8L + ZS^S^^ =
[x+yP +2(a! + y)'(-a;)" 4 2(a! + y)»(-y)"
+ 2(-xr' +2(-a;r(-yr
+ 2(-y)»(-x)" +2(-yr'
+ 3 (x + y)* (- »)-« + 3 (x + y)» (- y)'
+ 3 (-«)«(- yy
2
+ 2
+ 2
+ 3
+ 3
+ 3
-a!)'(a5 4yr' + 3(-a:)'
-3')'.(a>+yr+3(-yr(-xy
vn+1
V«+l
Now the sum of the 1st, 5th, 9th, 10th, 14th, 18th tenns
here clearly
= 55[
n+8 >
the sum of the 4th, 7th, 13th, 16th
= (a;+y)"*'{-2aj* + 2ajy-2y« + 3aj*+3y'}
the sum of the 2nd, 8th, 11th, 17th
= 2(-a:r{(aj + y)»-y»}+3(-a:r{(^+yr + y*}
= (- a:)*** (- 2aj' - 6a:y - 6y* + 3aj'' + 6a:y + Sy*)
and, similarly, the sum of the 3rd, 6th, 12th, 15th
= (- yi
k»H-8
120 MB. MUJR, CAUCHT'S THEOREM, &C.
Consequently we have the identity
or aSs)>». + (i^J^n.i= ^.« I ^ ^*
In this, putting n = l, 2, 3, and noting that /Si = 0, we
see that
S, is divisible by (JS;)^
/8g is divisible by ^8^ and ^/Sg,
S^ IS divisible by neither.
Then, as S^ is got by multiplying S^ by ^/S, and 8^ by
J 5, and adding, it follows that
8, is divisible by (^5,)' and Jfi',;
as 8^ is got in the same way from 8^ and 8^ it follows that
8^ like iSj is divisible by ^5,,;
and, as 8^ is got in the same way from 8^ and £^, it follows
that
8^ like S3 is divisible by J S3.
Consequently
8,, like S', is divisible by (^8/,
8^^ like Sg is divisible by i^8^ and J S3,
and so on in a cycle of six elements.
Now. ijSa=i»" + ajy + y',
and i^8 = ^(^ + y);
hence we have the following theorem which includes Cauchy's
ITie expression (a? + y)*+ (— a?)*4-(— y)" is divisible
hy neither xt/ {x-^-y) nor x^ •\- xy-k- y^ when n — 6m
hy xy{x-^ y) and {a?-\-xy-¥ yY when n = Qm-\- 1
by a?^'\-xy+y^ when n=^Qm + 2 ^
by a?y (a: + y) when w = 6w + 3
by {a? + xy-\-y^Y when n = 6m 4 4
and by xy [x + y) and x^ + xy + y^ when n = 6m + 5 ^
{B).
( 121 )
NOTE ON CAUCHY'S THEOREM.
By /. W. L. Glaisher.
The resolutions of [x-\-yY -x'—y'' forw = 9, 11, and
13, which I gaTe on p 53*, were, I have found, given bv
Cauchy himself in his "Note sur quelques th^orfemes d'algfebre"
(Exercises cC analyse et de physique mathSmatique^ t. ii, 1841,
pp. 137-144).
The general formula for {x + y)** + (- xf + [—yT in terms
oixy[xA-y) and x^^-xy^-y* is given .by Todhunter in the
second edition of his Theory of Equations [\^^1)^ p. 189. The
subject is also considered in two papers by Mr. Muir and
myself in the current number of the Quarterly Journal of
Mathematics,^
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, November 14, Lord Bayleigh, F.R.S., President^ in the Chair.
After the Treasurer's and Secretaries' Reports had been read and adopted, the
meeting proceeded to the election of the new ComidL The Scratators (Messrs.
0. Pendlebnry and R. F. Scott) declared the following eentlemen elected :
President, Mr. C. W. Merrifield, F.R.S. j Vice-Presidents, Prof. Cajley, F.R.S.
and Lord Rayleigh, F.R.S. ; Treasurer, Mr. S. Roberts, F.R.S. ; Bon. Sees., Messrs.
M. Jenkins and R. Tucker ; Other Members, Mr. J. W. L. Glaisher, F.R.S. ; Mr. H.
Hart; Dr. Henrici, F.R.S. j Dr. Hirst, F.R.S. j Dr. Hopkinson, F.R.S. ; Mr. A. B.
Kempe ; Dr. Spottiswoode, F.R.S. ; Prof. H. J. S. Smith, F.R.S. j Mr. H. M. Taylor ;
and Mr. J. J. Walker.
Mr. Merrifield having taken the chair, Mr. J. D. H. Dickson, M.A., was elected
a member, and the Rev. A. Freeman and Prof. Reinold were admitted into the
Society. Prof. W. S. Jevons, F.R.S.. was proposed for election. The Chairman
read and laid before the meeting a letter addressed to the Society by Mr. Warren
de la Rue, F.R.S., respecting a memorial to M. Leverrier.
The following communications were made to the Society : " On the instability
of Jets," Lord Rayleigh ; " On self -strained Frames of Six Joints^" Prof. Crofton,
F.R.S. (read by Mr. Hart, the two conclusions establiahed were (i) j in order that
a hexagonal fiame ABCJDEF, stiffened b^ means of diagonal bars AB, BE, OF,
shall be capable of being selfetrained, it is necessary and sufficient that the six
Joints lie on a conic, (ii) there is only one other case where a frame of 6 joints
and 9 bars can be self -strained, i.e, when the lines BA, DE, CF meet in one
point); *'0n the Calculus of Equivalent Statements, H. Mo Coll, BA.., (an
abstract, read by Mr. Tucker, stated that the paper contained the solution of a
test problem to show the power of the authors method of ^mination; then
an explanation with illustrations and applications of another allied method,
called ''The method of unit and zero substitution"; next, a brief indication of
the way in which the algebra of logic may render important B&mcQ to scientific
men in investigating t£e causes of natural phenomena; and, lastdy, a brief
criticism of Prof. Jevons's method of solving logical problems).
B. TuoKBR, M.A. Hon, See,
* Ona class of algebraical relations, § 1.
t On an expansion of {x'\- 1/)* + (— x)* + (— y)* Quarterly Journal, voL xvr,
p. 9, On Cattehy's theorem relatwg to the factors oj{^ + y)* - jc* — y*, Ibid, p. ^9.
( 122 )
MATHEMATICAL NOTES.
Geometrical demonstration of a known Theorem relating to
Bur/aces.
Take an infinitely small arc aa^ (figs. 16 and 17) of a
curve traced on a surface. Draw at the extremities of this
arc the normals an^^ aji^ to the surface. The angle which
a,n, makes with the plane naa^ is the angle of geodesic torsion.
Suppose that the curve (a) is the intersection of two surfaces
cutting one another at the same angle along this curve.
By drawing at a the normals aw, am to each of the surfaces,
and also the normals a^n^, a^m^ at a^ we have the angles man^
mflM which are equal to one another.
We can bring these angles into coincidence by means
of two rotations; one, which brings into coincidence the
planes of these angles, takes place round the characteristic
of the plane man ; the other takes place round a perpendi-
cular to this plane, drawn from its focus. The characteristic
of the plane man is the axis of curvature G of the curve (a) ;
the perpendicular to the plane man drawn from the focus of
this plane is the tangent aa^ to this curve.
Turning round 5, a becomes aj, and the angle man be-
comes ma^n\ then this angle turns round a, and becomes
coincident with m^a^n^.
It follows that the angle ma^m is equal to n a^n, and as
aa^ is normal to the plane of the moving angle, tnese angles
measure the angles which the normals a^m^^ a^n^ make with
the planes maa^^ naa^^ viz. these are the angles of geodesic
torsion of a relative to the surfaces of which this curve is the
intersection.
We thus obtain a geometrical demonstration of the known
theorem : The angles of geodesic torsion of the curve of inter--
section of two surfaces which cut one another at a constant
angle are equal.
A. Mannheim.
On Triangles self-conjugate vrith respect to a Parabola.
Observe the following construction : The diameter inter-
cepted between a point and its polar is bisected by the
parabola in a point at which the tangent is parallel to the
polar. In other words, the tangent to a parabola parallel
to a given ^straight line lies midway between that Ime and
MATHEMATICAL NOTES, 123
its pole the point of contact and the pole being on the
same diameter.
Now let ABC (fig. 18) be a triangle self-conjugate with
respect to a parabola ; D^ £j F the middle points of BGj
CAy AB] P, Qy R the points where the diameters through
-4, By G cut the curve ; then the above construction shows
that the sides of the triangle DEF will be the tangents at
P, Qy -B,* Also since the tangents at Q^ R intersect at D
a point on BG the polar of -4, the chord QR will pass
through A.
II. The perpendicular from A upon its polar being
parallel to the normal at P, the portions of both intercepted
by the axis, and their projections upon that axis will be
equal. This is an extension of the property that the sub-
normal is constant.
III. The area of the triangle ABG will be double of
that of PQR. Let Q lie between P and R and produce BQ
to meet -4C in Y^ Then since BY is parallel to PA^ and
BQ=- QY, BQP^BQA=^\BYA. Similarly BQR = BQG
= ^BYC. RtncQPQR^^ABG.
IV. The orthocentre of the triangle BEF is the centre
of the circle circumscribing ABG, and the circle circum-
scribing the former is the nine-point circle of the latter.
The directrix passes through the one, the focus is situate
on the other. An independent proof may be subjoined of
the former of these propositions.
Let the diameters AP, GR meet the directrix in Z, N^
respectively. The angle LEN is double of JDEF, the triangle
LEN is isosceles and therefore bisected by the perpendicular
from E upon the directrix. (This is the triangle used in
Drew's Geometrical Conies for drawing a pair of tangents).
Compare this with the triangle formed by joining A, G to
the centre of the circle circumscribing ABC. The vertical
angle of this isosceles triangle A DC is double of ABC, i.e. of
JDEFj and therefore the triangle is similar to LEN. And
the base LN is the projection of AC upon the directrix,
through an angle equal to that between OE which Is perpenr
dicular to ^u and the perpendicular from E upon the
directrix. This perpendicular must therefore equal the
-F^<W^>^*^^— -^a— — — -•^~-^->iT«»~>W>^>^r^»»^
* I find that this theorem is given in Wolstenholme's Problems, No. 442
(ed. 1, 1867). It was set bj me in an examination paper in 186'3. Bteiner's
theorem that the orthocentre of DEF (§ iv) lies on the directrix was originally
proved with the help of Pascal's hexagram. For anof^ eJementaxy proof, see
the Lady'i and GenUemwfi Diary, 186S.
124 MATHEMATICAL NOTES.
projection of OE through the angle between them, therefore
must be a point on the directrix.
The following properties are therefore established : tiz.
(1) If a triangle he self-conjugate with respect to a parabola
the straight lines joining the middle points of its sides are
tangents to the parabola ; and touch the curve in points the
diameter through one of which meets the chord joining the other
two in one of the angular points of the self-conjugate triangle.
(2) If a perpendicular be dravm from any point upon its
polar with respect to a parabola^ the difference of the abscisses
of the point where it cuts the axis and of the given point from
which it was drawn is constant,
(3) The area of a self conjugate triangle is double of that
formed by joining the points of contact of the three tangents
parallel, to its sides.
(4) The centre of the circle circumscribing a self-conjugate
triangle lies on the directrix of the parabola.
(5) The nine-point circle of such a triangle passes through
the focus.
A. F. TORRY.
Note on the Centre of Gravity of the Fru^strum of a
Pyramid.
Let ABCabc (fig. 19) be the frustrum; ABG, abc its
parallel faces ; F^ f their centres of gravity, and G that of
the frustrum.
Divide the frustrum into three pyramids by the planes
Abcy Bbc. Then clearly
vol. of cABC _ vol. of Aabc _ vol. of ABbc
a^ ¥ ^6 '
where a = AB^ b = ab.
Now the centre of gravity of a pyramid is that of four
lequal particles placed at the vertices ; hence, we may replace
the three pyramids first formed by three sets of four particles
proportional to the volumes of the pyramids, and so replace
the frustrum by the following system of particles
a', a* 4 aJ, cf-{-ab-{- i', at (7, jB, -4, respectively ;
&', &* + a&, J'* + oJ -f a', at a, i, c, respectively.
MATHEMATICAL NOTES.
125
But two other similar arrangements can be similarly
obtained. Superimposing these three systems, we find that
the centre of gravity of the frustrum must be that of
three equal particles at -4, B^ (7, proportional to 3a' + 2ah + J*,
and „ „ a, J, c, „ 36* + 2aJ+a*,
hence, we have at once
FO fO ^ Ff
36" + 2a6 + a' "" 3a* + 2ai + i' "" 4(a*i-aJ+ i') '
J. W. Sharpe.
A Formula hy Gauss for the Calculation of log 2 and
certain other Logarithms.
Gauss has given, Werke, t. Ii., p. 501, a formula which
is in effect as follows :
oi-e^lO" (^X (}^^^\' /g^GOy /15624y /9801Y
U024/ U048575/ V6561/ U5625; V9800y '
viz., this is
'5".41\Y 2*" ^ Y5.2^41\Y 2^3^7.31 \Y 3\11' \*
^WW.3.11.31.41JV 3' )\ 5* )[2\5\r)'
where on the right-hand side the several prime factors have
the indices following, viz.
=2~5«^(^
2, index is (59 + 160 + 15 + 24-50-12
(16+ 16-* 8-24
(59+ 10+ 3-16-48- 8]
( 8- 8
(8-8
( 8- 8
(5+ 3-8
or the right-hand side is = 2*^ as it should be. The value
of log2 calculated from 2'"* = 10'* is log2 = 3% = -301020,
viz. there is an error of a unit in fifth place of decimals. <
The actual value of 2^^ has been given by me Mr. Glaisher :
2*'«= 10043 36277 66186 89222 13726 30771
32266 26576 37687 11142 45522 06336.*
3
5
7
11
31
41
= 196,
= 0,
= 0,
= 0,
= 0,
= 0,
= 0,
* The value was deduced from Mr. Shanks's value of 2>** in his JUct\fication
of the Circhy (1868), p. 90. J. W. L. G,
126 MATHEMATICAL NOTES.
Supposing log 2 calculated by the form, we then have
41 = (HI t) 2" -MO', giving log41,
and 3» = 104m • 2*.41, giving log 3,
and formulaB may be obtained proper for the calculation of
the logarithms of y, 11.31, and 7.31.
A. Catlet.
}fbte on a Definite Integral.
The integral J= I -jj- ,- — 715-57 of Weierstrass is at
once seen to be =K—JEj but the proof that the other integral
/•> kVdK
J' = I *-7r-5 — 7—; — 7T-s^ is =^' is not so Immediate.
J^ v(^ — 1.1- Aia; J
We have
rf yV(l-y') ^ l-2y' + ^y ^
dy V(l - ky) (1 -.y«)i (1 _ ky)^ '
and thence
J. (l-/)l(l_r/)*'
A" 1
viz. replacing tbe nnmerator by ~ T? + B (^ + ^V)* ^'^^^
becomes
A:']. (i-/)i(i-Ay)i"^A''j, (1-/)* »
that is f -— ^(^ r=4-^;
Jo (1 - y)* (1 - Ay]» *" '
or writing A' for A:
J. (i-y«)i(i-r3^«)* A* • .
The integral .T writing therein x = -jt- — t^-jt becomes
viz. its valae is thus = E.
A. Catlet.
' \
MATHEMATICAL NOTES. 127
On a Formula in Elliptic Functions.
en u
Writing enw = -5 — , then the formulae p. 63 of my
Elliptic Functions give
8n(M + v)=-^— ^,, en(M + v) = .^-^,;
and, substituting for T, T^ 5, B\ and C, C their values
we obtain
, . snuent;4 snvenu
sn { M + 1?) = ' . ,, ,
^ 1 +A; snuenwsnvenu'
, . V enuenv — snusnt;
en (w+ t?) = - — T^i ,
^ ' 1 — A; snuenusnvenv'
fonnulse which, as regards their numerators, correspond
precisely with the formulae, sin (m + v) = sin u cos t; + sin v cos u
and cos(M+t;) = cosm cosv — sinw sin v, of the circular functions,
and which in fact reduce themselves to these on putting A = 0.
The foregoing formulae putting therein A;* = — 1, are the
formulae given, Uauss, Werke^ t. III. p. 404, for the lemniscate
functions sin lemn [a ± b) and cos lemu [a±b)'^ where it is to
be observed that these notations do not represent a sine and
a cosine, but they are related as the sn and en, viz. that
cos lemna = V(l^ - ^^^ lemn' a) -5- V(l^ + shi lemn' a).
A. Cayley,
Value of a Series.
The series is
1 1 1 1 p -r . ^
1.2.3.4 5.6.7.8 9.10.11.12 13.14.15.16
the value of which is
ilog2-^7r.
^^^^ ijHa " * In " 2:3;
5.6.7.8 'V5.8 6.7^
_1 l(l L^
9.10.11.12 '\9.12 10.11/ '
[^ &c. = &c.|
\
128 MATHEMATICAL NOTES.
80 that the serws
1 1 1
2.3 6.7 10.11
+ &C.
where
^. = i(i-i + i-i+ i - 1^ + iV - iV + &c.),
Now l-i + J-i + i-i + f--i + &c-=l«>g2,
and evidently
l + J-i-i + i + *-|-i+-&c. = iir+ilog2,
whence by addition and subtraction
3-S, = l-i + i-i+ i-3'5 + &c. = i7r + f log2,
therefore
i (^t - ^J = Hl^^ + i l0g2 - iTT + i l0g2}
= ilog2-^i^7r.
The above series was suggested by the formula
1 1 1 « TT
1.3.5.7 9.11.13.15 17.19.21.23 ' 96 (2 + V2) *
which was set in the Senate-House Examination for 1850
(Thursday morning, January 17, Question 6). This formula
can be proved by a method very similar to that employed
above by means of the equation
which may be obtained at once by putting n = 4 in
. TT f, 1 1 1 1 p )
= wsm-- iH 7 — + — &CX .
n{ n-1 w + 1 2n-l 2n+l J
A solution is given by Ferrers and Jackson in the Solu'
tions of the Cambridge Senate^Hoitse Problems^ 1848-1 851,
pp. 226, 227.
J. W. L. Glaisher.
IT
r
MATHEMATICAL NOTES. 129
Note on " Choice and Chance.^^
The proof of Prop. XLVI. in the 3rd Edition of my Choice
and Chance is needlessly complicated, and it labours under
the disadvantage of depending on Prop. XLV. which itself
depends on Prop, xxvii. The following independent proof
is much to be preferred.
PfiOPOSiTiON XLvi. If tJi*he the chance of an operation sue--
ceeding in any trial^ the chance that in n trials there shall not
he k consecutive successes is the coefficient ofx*^ in
1-aV
Let K^ denote the chance that there shall not be k consecu-
tive successes in n trials. Then, whatever be the result of
the first trial, K^_^ is the chance that there are not k con-
secutive successes in the remaining w — 1 trials ; and therefore
that there are not k consecutive successes in the n trials
unless the first k trials be successes, and the A; -f l"* a failure.
But the chance of this case is
Therefore we have
But it is plain that K^j K^,,.K^ ^ are each unity, and that
Therefore K is the coeflScient of x" in the series
>S= 1 + a; 4 a;* + ...+ a^"* + (1 - /Lt*) JB* +... .
But from this series we obtain in virtue of (1]
{1 _ a; 4 (1 -/*)/**»**•} -S= 1 - mV,
Ik h
-fiTx
or 8=
l-a; + (l-/iA)/iA*a;
fc^t+i •
Therefore the required chance is the coefficient of a;" in the
expansion of
Q. E. D.
W. Allen Whitworth.
VOL. VIII.
130 MATHEMATICAL NOTES.
Theorem relating to a system of conies.
It is a known theorem, and an old one, that if a system
of conies be inscribed on the same quadrilateral, i.e. have four
common tangents, the pencil of tangents drawn from an^
point forms a system in involution. The converse of this
theorem is also true, i e. that if a system of conies be such
that the pencil of tangents drawn from any point be in
involution, then the system of conies possesses four common
tangents.
As in the old theorem it is at once proved that the director
circles all pass through two fixed points, and that therefore
the centre of the conies all lie on one straight line. Hence
it is clear by projection that the locus of the pole of any
straight line with respect to the system of conies is a straight
line. Hence the coefficients in the tangential equation must
contain an arbitrary parameter in the first degree; or the
equation to any one of the system must be U-\- \ V= 0, which
^ proves the theorem.
R. Pendlebuby.
Note on the Theorem in Kinematics (vol. VII. p. 190).
Prof. Liguine, of Odessa, has pointed out, with reference
to the theorem given by me in the Messenger^ vol. Vll. p. 190
and vol. viil. p. 42,
(1) That the zero-circle may shrink to a point and become
imaginary, its radius being V{(-" <*")}) so that the formula of
p. 42 becomes (P) = Ntt (r" + a') ;
(2) That the centre of the concentric circles is always real
and the area described by it a minimum.
A. B. Kempe.
Note on an Inequality.
If a be greater or less than unity, n being a positive
integer, then
a'"^-l n + 1
a(a'^-l")^ n '
Demonstrations of this proposition have been given in the
first series of the Messenger by Dr. Ingleby, vol. Hi, p- 235 ;
by A. Q. G. C, vol. IV. p. 39 ; by a Student in rrofessor
Kelland's class, vol. IV, p. 109 ; and by Mr. Sharpe, vol. IV.
p. 123. The following proof may be added to the preceding,
MB. SCOTT, ON FORMS OP DETERMINANTS.
131
Let a be > 1 : then
a
.SfH-V
-1 >n + l
ndr*- w> (n + 1) a'"*'- (n + l)a,
(a-l)4w(a-l)^a*^*-a,
na
.»H-1
na
.»H-1
> ^«»
S»-l
+ n^a*"+a'^'+...+ a,
a«-(a-l) + a*^(a«-l)+...+a"**(a*-l)
^a"-l+a*-*-l+a"''-l+...+ a-l,
a"" + a'^^[a + 1) + a*""^ (a« + a + 1) -H...+ a"**(a*-*4 a"-*+...+ 1)
^a""' + a'^H-...+l + a"-"-fa''-»+...4l4...+ a+'l + l,
a*'-.l4(a«^*«l)(a4l)4(a"*^--l)(a«4a4l)4-...
4 (a"**- 1) (a"-*4a"^4...4l)^0.
Evidently the sign > is the correct one.
Let a<l and put a= -, where a> 1: then
a
,VHS
-1
.«»«
-1
a(a«"-l)"^a(a**-l)»
and is therefore, by the preceding proof, >
n4l
n
W. Walton
ON SOME STMMETEIOAL FORMS OF
, DETERMINANTS.
By R. F. Scott, M.A., St. John's College.
§1. If
A =
c, a, 0, 0, 0, ...
bf c, a, 0| 0, ...
0} bf Cj a, 0, ...
0, 0| bj c, a, ...
n being the order of the determinant^ we get^ on expanding
K2
132
MB. SCOTT, ON SOME SYMMETRICAL
the determinant according to the elements of the first row
and column, bj Cauchj's theorem
an equation of difierences for D . Solving this in the usual
way and determining the constants from the known values
of i), and i)„ we get
^•'"" 2"* Vic''- 4ai)
By giving special values to the quantities a, J, c we get
results of more or less interest —
(i) If c = a + J
D =^- T—
(ii) Ifcs2cosd| a = J=l
^ _sin(n4 1)
A result given by Prof. Wolstenholme, Beprtnts from
Ed. Timesj XXI. p. 83.
(iii) Ifc = 2tan5, a = -l, J = l,
J. _ (tan g + sec g)"^* - (tan - sec g)*^*
•" 2sectf
(iv) Ifa = 6=xc=l
^ 2 . 7r(n + l)
jj ss -7- sm — ^ .
" V3 3
(v) If c^ 8 4ai the solution in the general case becomes
A= 5 (« + !)•
§2. Let
^.=
U« A/* Kj Kj • • •
X* U« Kj Km •••
X 1 i 1 U| n/| • • •
1, 1, 1, 0,...
All the elements in the diagonal being zero, all above it
k% all below it unity.
.J
FORMS OP DETERMINANTS.
133
Subtract the third row from the second, the fourth from
the third, and so on, then
k
0, 1, 1, 1, ...1
0, -1, h^ 0, ...1
0, 0, -1, A, ...1
1,
= (-1)
1, 1, ...0
" 1. • K^ V| Vr, • • •
V, ~" M. • A/a V« • • •
n — 1,
or
A;
= (-ipU'^+
1,
1,
0, -
1,1,..
Km U, est
X • Km • • •
Pro-
The suffix denoting the order of the determinant,
ceeding in this way
J5:*^- 1
§3. In the determinant of the preceding paragraph
multiply every column by h and write Tcb = a. Then we get
0, a, a, a, ...a
5, 0, a, a, ...a
i, i, 0, a, ...a
i, i, 5, i, ...0
^ ' a- 6
This expresses a geometrical progression in the form of a
determinant.
§ 4. By means of § 3 and Prof. Cayley's theorem for the
expansion of a determinant according to products of elements
in the principal diagonal {Crelle^ xx^viii. p. 93) we can
evaluate determinants of this nature when the elements of
the principal diagonal do not vanish. We then arrive at
the followmg results :
134
MR. SCOTT, ON SOM£ SYMMETfilCAL
(i) If all the elements of the principal diagonal are equal
to c all above it a, all below it h
Cj ttj ttj (Z| •••
bj bj Cj Uj •••.
bf bj b^ Cj •••
a{c^bT-b{c^aY
a-'b
If c a a + & this becomes identical with § 1 (1).
(ii) If in the diagonal there are p elements equal to c,
q elements equal to d^ r elements equal to Cj &c, where
then
i> + S' + r+...= w,
D^
ajo-hYid'-bYie-bY. .-bjc-ayid'-ayie'-aY...
a — b
(iii) If the elements of the principal diagonal are c,, c,, ...c^
and /(a;) denotes the continued product
then
(c.-a:)(c,-a:)...(c^-a;),
a —
By putting a = 5 and evaluating the vanishing fractions
we are led to some known theorems, e.g. (iii) gives us the
result of Ex. 15, p. 16 of Salmon's Higher Algebra. A result
due to Mr. Michael Eoberts, Nouv, Ann,, 1864.
§5. In the same way in which the result of §2 was
obtained we may prove that
= (- Ip PA«a— «« + *A«a— «•
+ bfij>fi^...a^+...
+ *A «Ji
0, fl^, a„ a^, a^, ...
^) ^» «»5 «45 «6? —
*iJ Kj ^} ^i1 ^6) —
hi *«> *») ^) «6) •••
*1J *S) ^> *4) ^ •••
where the law of formation is obvious.
By giving to the a's and V& different values we get
numerous results.
By filling in the diagonal we get results analogous to § 4,
but of greater complexity*
FOBMS OP DETERMINANTS.
135
§ 6. From § 4 (ii), if we put all the elements of the diagonal
equal to c, except the last, which is put equal to zero, we get,
after dividing both sides by, ai.
c, a, ...a, 1
6, c, •••6f, t
&, &, ...c, 1
1, 1, ...1,
a result we shall find useful in what follows.
There is a corresponding result for § 4 (iii).
§ 7. It is, of course, clear that the reason why the deter-
minants in the preceding paragraphs reduce to simple forms
is the symmetry with respect to the principal diagonal. It
is a natural step then to inquire if there are any similar
results when the symmetry which exists is with respect to
both diagonals.
As a first step, in this direction, consider the determinant
a,
c,
c,
c,
c,
£
c.
a,
C)
«>
^
c
c.
c.
«>
i,
<?»
c
c,
c,
^
a.
c.
c
c,
i,
c.
0,
a,
c
h
c,
c,
c,
c»
a
of order 2n.
If a = i, the first column is the same as the last, the second
the same as the last but one, and so one ; hence the deter-
minant divides by [a — J)**.
Adding all the columns to the first, each element becomes
[a + i + 2c (n — 1)], which is therefore a factor of the deter-
minant.
Add the last column to the first, the last but one to the
second, and so on, then the first n columns would become
identical, viz. all their elements would be 2c, if a + J = 2c,
whence (a + J - 2c)*"* is a factor of the determinant. Thus
the determinant divides by the product
(a-J)"(a + J-2cP{a + 6 + 2c(n-l)}.
This being of the same order as the determinant, and the
coefficients of a^ being the same in both, the determinant
must be equal to the product.
136
MR. SCOTT, ON SOME SYMMETRICAL
§8. On looking at the determinant in §7, we see that
what we have now got is symmetry with regard to the centre
of the square, in which the elements of the determiDant are
written. Also that the form of the determinant is different
according as its order is 2n or 2n + 1 .
For the present let us confine ourselves to the determinants
formed as follows : Draw the two diagonals of a square and
two lines through its centre parallel to the sides, thus dividing
it into eight triangles. All the elements in each triangle are
to be the same, and pairs of triangles symmetrically situated
with regard to the centre are to have the same elements ; all
elements in the principal diagonal are to be x in the other y.
The other dividing Imes will or will not contain elements
according as the order of the determinant is 2n + 1 or 272.
§ 9. Evaluation of the determinants
Xy a, a, J, J, y
dy Xy a, bj y^ c
dj dj Xy y, c,
c, c, y, Xj t?, d
c, y, i, a, a;, d
y, bj bj a, a, x
...(1),
x, a, a, jp, J, J, y
J, X, a, p, J, y, c
rf, d, X, p, y, c, c
2» ?? ?» ^7 ?) ?? 2
^> Cj y^ P^ ^j ^> d
c, y, b, p, a, X, d
y, J, J, p, a, a, a;
...(2),
of order 2w and 2w + 1.
In (1) subtract the last column from the first, the last but
one from the second, and so on. Then in the determinant so
transformed add the first row to the last, the second to the
last but one, and so on. Then all the elements in the square
block, which in (1) contains the elements
c.
C)
y
c,
y>
b
y>
i,
b
vanish, and the determinant reduces to the product of the two
determinants
£C — y, a — by a —b
d -^ Cj x — y^ a — b
d — Cy d— Cj X -y
and
^ + y) c + rf, c-\-d
a + bj x-\-yj c-\- d
both of which are of the form (i) in § 4,
FORMS OF DETERMINANTS.
137
Thus the value of the determinant (1) is the product of
the two expressions
a — 6 + c- d
a + i — c— d
Pursuing the same process of reduction with the deter-
minant (2), we reduce it to the product
(3).
x — y^ a — bj a — b
d—Cj x-yy a — b
d — Cy d— Cj x—y
and
^j 2 J 2 J 1
2p, X'\-yy c -f rf, c-\-d
2p, a + J, x + y^ c + d
2p, a + bj a + i, x + y
the first of which is of the form § 4 (i) ; the second is the sum
of two determinants, one of the form §4 (i), the other of the
form § 6.
Thus (2) reduces to the product of the expressions
[•
a — 6 + c — d
{a + b){x-\-y-c-'dy -jc + d) {x-^y-a-by
a+b—c—d
[x'\-y-a-bY-{x'\-y-c-d)'
a-^b — c — d
+ 2^2
]•
§ 10. If the elements in the diagonals are not all equal
but only equal in pairs equidistant from the centre we can
evaluate the resulting determinants of § 9 by § 4 (iil).
The property possessed by the above of being resolvable
into the product of two determinants is common to all
determinants symmetrical about the centre.
By introducing special relations between the quantities
involved a great number of special cases may be deduced,
some of which are interesting, but it seems unnecessary to
enumerate them here.
§ 11. The determinants of § 9 may now be extended.
These determinants are formed from that in §4 (1) by
taking the two determinants
Xy a, a
h h y
a, x^ a
h y, <^
dj rf, X
?
y, cj c 1
138 MR. fiOBERTS, ON CERT AIN^DETERM IN ANTS.
and forming a new square array, by repeating each twice so
as to get §9 (1).
But we can take two determinants^of the form § 9 f 1) and
repeatmg each twice obtain a new determinant of oraer 4n.
This a^ain, by the same transformation, as above can be
resolved into the product of two determinants of the form
§9 (1), and hence its value is known, viz. if the elements
of the two determinants of the form §9 (1) are denoted by
suffixes, the value of the new determinant is the product of
four factors, the first and second of which are got from §9 (3j
by writing for a?, y, a, J, c, ^; a?, - a?,, ^i - y^, «i- ^jj &c., ana
the third and fourth by writing x^ + a;^, y, + y^, a, + «j, &c.
We get determinants of order 4n + l, 4n + 2, 4n + 3 by
taking a determinant of the form (1) with another of the
form (2), or two of the form (2), or two of the form (2) with
a dividing line, all of which can be evaluated in like manner.
We may then extend these new determinants, and so on
interminably.
§ 12. By interchanging the order of the rows and columns
we get from the determinants written down other forms of
sjjnnmetry. For example, from that in § 7 by removing the
nrst row to the last place continually n times we change
the determinant into one^where all the elements are the same
except those lying on the sides of a square inscribed in the
square of the original elements, the angular points of the
second square being at the middle points of the sides *of
the first. Similar remarks apply to the others.
NOTE ON CERTAIN DETERMINANTS CONNECTED
WITH ALGEBRAICAL EXPRESSIONS HAVING
THE SAME FORM AS THEIR COMPONENT
FACTORS.
By Samuel Boberts, M.A., F,RS.
If we assume Lagrange's equivalence
(aj« + ax,' H- hx^' 4 aJa?,«) (X* + aZ^« + JX,« 4 abX^")
= { xX 4 aa?,-X, 4 ^^^X^ 4 abx^X^Y
4 a {-a:Z,4 a:,X - Ja:,Z,4 te.ZJ*
4 J f- xZ, 4 oajjZg 4 x^X - ax^X^Y
4 oJ {^ ajZg - x^X^ 4 x^X^ 4 x^X }*,
MR. ROBERTS, ON CERTAIN DETERMINANTS.
it Is plain that the determinant
+ X, +aZ„ +5i,» +«*-^8
-X„ + X, - JX„ 4 JZ,
-X„ +aX„ 4 Z, -aX,
139
-^., -
-3;, 4
Z, 4 X
IJ
must be a power of {X* + aX* + hX^ + ahX^) to a constant
factor pr^. The power is the second and the constant is
unity, as appears by inspection.
In like manner, if, we consider the corresponding equi-
yalence for forms of 8 terms whose coefficients are the terms
of (1 -f a) (1 + b) (1 -f c), an analogous determinant of 64
elements is arrived at. It is not necessary to write down
the extended equivalence in full. We have
(a?* -h ax^^ H- abx* + bx^ + acx^ + cx^ 4 bcx^ + abcx^* )
X (Z^ + aX,' + abX^^ + bX^' 4 acX; 4 cZ/ 4 bcX; 4 abcX,')
= P" 4 aP^'' 4 abP^' 4 JPs^* 4 acP/ 4 cP/ 4 JcP.* 4 ahcP^,
where P, Pj, ...P^ are linear and homogeneous in a?, a?^, ...a?^
and X, X^^ •••'^7*
If, then, we form the condition in JT, X,, ...X^ that
P, P„ ...P^ may vanish, the function must be a power of the
second factor of the left-hand member of the above equation
to a constant factor pr^. The power is seen to be the fourth
and the constant is unity for the form of determinant here
given, namely :
+Xj +aX^^ 4aJXjj, 4J-X3, 4acX^, 4cXg, -{-bcX^^ -VahcX^
-Xj, 4 X, 4 JX3, -SX„ 4 cXg, -cX^, -JcX^, 4- bcX^
-X„ - X3, 4 X, 4 X„ 4 cX,, 4cX„ - cX„ - cX,
-X3, 4aX^, -a Xj, 4 X, 4acXy, -cX^, 4 cX^, -a cX^
-Z„ - J„ - JX„ -5Z„ 4- X, 4- Z„ 4* X„ + i X.
—X,, 4-aX^, -aJXj, 4JX,, -a X„ 4- X, -J X„ +a6 X,
-X„ 4-aX„ 4-0 X^, - Xj, -a X,, 4- X„ 4- X , -o X,
-^,.-
X., 4 X^, 4 X^, — X,, — X„, 4- X., 4- X
The forms P, P^, ...P^ are found by multiplying each
element of the first column by a?, of the second by ajj, and
so forth.*
♦ See another expression for the case of a = i = tf = l in Mr. J. J. Thomson's
paper, "On the resolution of a product of two sums of eight squares into the
sum of eight squares," vol. vii. pp. 73, 74.
140
MR. GLAISHER, THEOREMS IN ALGEBRA.
The determinant may, of comrse, be varied (1) by writing
negatively one or more of X, -X,, ...-X^; (2) by interchanging
columns with or without change of sign in all the terms;
(3) by similarly interchanging rows, operations which may
be effected simultaneously.
If in the determinant we change the signs of X„ X, •••X,,
leaving the diagonal unchanged, and then convert tne first
column into the top row, by turning it about X till X, takes
the place of ahcX^^ the second column becoming the second
row in a similar way, and so on. the determinant thus obtained
(having the same value) will, wnen multiplied into the original
determinant by the usual rule, give
4>, 0, 0, 0, 0, 0, 0, 0,
0, 4>, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, *,
where O is second component of the left-hand member of the
equivalence.
In fact, each of the principal minor determinants contains
4> as a factor, and the reciprocal determinant is of similar
form.
There are then analogous determinants of 4 elements,
16 elements, and 64 elements.
The continuance of the series depends on the extension of
Lagrange's equivalence to similar forms of 2* terms. But
the literature on this subject is singularly conflicting, and
seems to require critical examination.
THEOREMS IN ALGEBRA.
By J. W. L, Olaisher.
I.
§1. Ka* + c*-2Jrf=l and J" + eP --200 = 0, then
a, hj c, d
dj a, bj c
c, dy a, h
by Cj dj a
= t
MB. GLAISHER, THEOREMS IN ALGEBRA.
141
and in this determinant
a =
Of bj c
,j=
J, c, d
, c =
bj c, e2
,rf=
bj Cj d
rf, a, b
dy a, &
a, J, c
a, 5, c
Cj dj a
c, rfj a
c, (^, a
dj a, &
t.6. the r^ element in the first row = (-)*^^ the minor of the
r^ element in the first column.
This is easily proved for the determinant
and also
aa — db-^- CC" bd=lj
ca-'bb'\-aC'- dd^ 0,
the first and third of these being the given equations, and
the second and fourth identities. Writing down from these
equations the values of a, i, c, d as the quotients of two
determinants, and observing that the determinant which
forms the denominator is equal to unity we obtain the second
det of relations.
§2. If a'-<f+2ec-2i/=l,
e*-J' + 2ca-2/rf = 0,
c*-/"+2ac-2a» = 0,
then a, J, c, rf, 6, / =1,
/, a, &, c, (?, e
e, /, a, J, c, rf
^, e, /, ay bj c
Cj (?, 6, /, a, 6
J, c, (?, e, /, a
and if Aj Fj J?, D^ C^B denote the minors of a,/, 6, eZ, c^ i
in the first column, then
142
MR. GLAISUEK, THEOREMS IN ALGEBRA.
§3. In general, if the 2n quantities a^, a,, ...a^ be con-
nected by the n relations
^fii -«««» +«»<*»-i —-«»•«« =1)
afi^ -afi^ +a/$^ ...-ajx^ =0,
«i««,-i
a^a^^ + «a<3^»,.s •••- «»<*«» = ^j
then
^8» » ^t ) ^11 •••^211-1
«« J «a ) «4» — ^t
= 1
(i),
H **tn> •••^2
and if -4„ -4 , -4^^,, ...-4, denote the minors of a^, o,
in the first column, then
The law of formation of the given relations is, that the suf-
fixes of the first factors in the terms run forwards from 1 to
2n, and those of the second factors run backwards round the
cycle, beginning with 2r - 1 in the r^ relation.
It can be shown that if we denote the n expressions which
form the left-hand members of these equations by a^, a,, ...«,»
then the determinant in (i) is identically equal to the deter-
minant of n rows,
and when a^^lj a, = 0, a, = 0, ...a^siO, this determinant
obviously = 1.
For example, if n = 3,
a, S, c, rf, e, /
/, a, J, c, eZ, e
c, /, a, J, c, eZ
c?, c, /, a, J, c
c, eZ, c, /, a, J
i, c, <?, e, /, a
7, a, ^
fit 7) a
MR. QLAISHER, THEOREMS IN ALUEBEcA. 143
where a = a* - d* + 2ec - 25^,
;8=6»- J' + 2ca-2/c?,
7=c'-/' + 2a6-2d5*
The relations a^ = (— )'^^-^9n-r-i *re obtained, as in §1, by
introducing between the n given relations the n identical
relations
afy - a^a^ + a^a^^. . .- a^^j = 0,
The above properties occur in a paper [' On a special form
of determinant, and on functions of n variables analogous to
the sine and cosine" [Quarterly Journal^ t. XVI. pp. 15-33),
but they seem worth stating as mere algebraical results, apart
from the notations with which they are connected in that
paper.
II-
If a, hj Cj a\ h\ c' be any six quantities, then
( 2a+ i+ c+ . - l'+ c'Y
+ ( 0+2J+ c+ a'- . - c'Y
+ ( a 4- & + 2c- a'+ V . )*
+ ( .'+ 5- c+2a'- y- c7
+ (- « . + c- a' + 2y- c'j"
+ ( a- i . - a'- J'+2c7 (1),
= 8(a" +J' +c* +a''4i'*+c'»
+ 5c +ca +aJ +J'c + c'a + a'J
- iV - cV - a'i' - hd -cd- ab') (2)^
= 4{(i + c + 5'-c7+(c + a + c'-a7+(a + 5-fa'-J7}...(3>
In the expression (2), all the product terms appear except
aa , hh\ cc •
It follows from (2), that
a« + J» + c'4a'» + 5"4c'» + Jc + ca4aJ
4 b'c 4 c'a 4 ab > Vd 4- del 4 at' 4 bd 4 ca' 4- oJ',
* By means of this theorem, I have calculated the yalne of the detenmnant on
the left-hand sidei Quarterly Journal, t, XYI. p. BBi
•^^
144
MR. GLAISHEK, THEOREMS IN ALGEBRA.
By changing the signs of letters in this inequality, we may
reduce the number of product terms on the left-hand side to
four (but not to any four), e.g. changing the sign of a,
a« + J» + c» + a'* + J"' + c'« + Jc + i'c
+ ah '\-aV>ca'\-ab-^ Vd + cV + dV + hd 4 ca' -f da.
The equality of (1) and (3) is evident in virtue of the equation
(^ + 5/ + (^-5)» = 2(^» + 5»).
I met with the system of squares in (1), in the calculation of
the sum of the squares of the residuals in the solution of the
equations
y = *j
a; = a.
2J =
i/+'^=f\
(4),
by the method of least squares.*
The most probable values of a;, y, z are
a; = J( 2a- i + rf-64-/),
« = i(- i4 2c-rf-f e+/)y
and the residuals x- a, y-i, ...aj + y + «— /are
J (-2a- b . + d- e+ f)j
J(- a- 26- c+ (/+ e . ),
i( . - 6- 2c- c?+ 6+ /),
J( a+ i- c-2rf . + /),
J(- a+ i+ c . -2e+ /),
i( a . + c+ rf+ e-2/),
viz. the squares of these are the same as the squares in (1)
if for a, J, c, a\ b\ c' in (1) be substituted c, J, - e, a,/, a.
It was remarked that in the expression for the sum of the
squares of the residuals all the coefficients were ^ or — ^^
thus giving in effect the equation (1) = (2). It is curious that
the imsymmetrical equations (4) should lead to the synmie-
trical system of squares (1).
* The complete solntioii is given in the Solutions oj the Senate-Eowe Problems
and Riders fw 1878, pp. 16&, 166.
/
/
/
( 145 )
I ON SOME SYMMETRICAL FORMS OF
DETERMINANTS.
By B. F. Seott, M.A.
(Contiiiued from p. 138).
§ 13. Evaluation of the determinant
JL/ ^~ C» CV| C« Cm C» Cm • • • I I
Om Cm €Lm Cm Cm Cm • • •
Cm Om Cm Qfm Cm Cm • • •
Cm Cm Om Cm Ctj Cm »» •
Cj Cj Cj b^ Cj (Xj .,.
Cm C) Cm Cm Om Cm • • •
where all the elements are c with the exception of those
in the two lines one on either side of the principal diagonaL
This determinant
jD= 1, 0, 0, 0, 0, ...
Xa Cm Cvm Cm Ca «.•
X I Om Cm Cvm Cm . • •
L m Cm Om Cm Urn • • •
i.a Cm Cm Om Ca • • •
Multiply the first column by c and subtract it firom each
of the others,
2) =
1, -c
1,
1, b-c
1,
1,
0, X,
y, 0, X
0, 0, y
^ Cm "~ Cm • • .
a — c
• Vl 9 * * *
J a— Cj mm.
b-^Cj 9 •..
J b — Cj ..,
0, ...
0, ...
Sum ...
0, ...
— c
0, 1, 1, 1, 1,...
1, 0, Xj 0, 0, ...
1, y, 0, Xj 0, ...
1, 0, y, 0, a;,...
where, for brevity, jB = a — c, y = b^c.
VOL. VIII.
146 MR. SCOTT, ON SOME STMMETRICATi
We have then to evaluate these two determinants.
If
D.
0, », 0,
y, 0, as, I ,
0, y, 0, X
0, 0, y,
since i),=— ay, i>j = 0,
A« = (- ^Ty D^, = 0.
In the evaluation of the second determinant we shaV also
find it convenient to discriminate between the two eases of
even or odd degree.
First let the degree be even and let
0, 1, 1,
1, h
1, 1, 1
1, 0, X,
0,0,
0, 0,
1, y, 0,
ar, 0,
0, 0,
h 0, y,
0, a;,
0, 0,
1, 0, 0,
y, 0,
a;, 0,
1, 0, 0,
0, y,
0, a!,
1, 0, 0,
0,0,
y, 0, X
1, 0, 0,
0, 0,
o,y,
Expand by Cauchy's theorem (Salmon, Higher Algebra j
Art. 36) with regard to the elements of the last row and
column,
-y
1,
1, 1,
1, 1, 1
0,
X, 0,
0, 0,
y.
0, a;,
0, 0,
0,
y, 0,
X, 0,
0,
0, y,
0, »,
0,
0,0,
y, 0, X
0, ar.
0, 0,
y, 0,
Xy 0,
0, y,
0, »,
0,0,
y, 0, X
0,0,
0, y,
0, 0,
0, 0, y
FORMS OP DETERMINANTS.
147
To evalualb the determinant multiplying x observe that
it 18 of order 2n - 2. As it stands there is only one element
in the second row, erase the row and column in which it
stands, in the resulting determinant there will be only one
element in the third row, erase the row and column in which
it stands. Proceed in this way and the determinant
= x
.«-i
1, 1, 1, ...
y, Xj U, ...
^> Vj ajj ...
Let
K-
^1 ^? ^} ^i •••
y, a?, 0, 0, ...
0, y, a?, 0, ...
0, 0, y, Xj ...
•l-t
therefore
«;
.«-i
or
• x + y
Thus the determinant multiplying x is
X
x-\-y
Similarly, that multiplying y is
y.-> /-'-(-»)
Hence D^ = -xyD^-{xy)*
n-1
X
y
K" - (-y)"-'} -;fr:. ly""' - (- a^n*
aj + y' ' "' ' x+y
An equation of differences which gives
^ ^ ^^ {x-\-yY
But
therefore D^^2^-^'-'^r^ = - f-^-^'.
[x-^yY [ x + y j
l2
2n
148
MB. SCOTT, ON SOMB STMMETBICAL
Next, for determinants of odd order we csp. establish in
the same way the equation
«*+» ^ »«-» aj + y x + y
Solving thb, it appears that
-«••-«• 2(-a^)"
■*^' . -»-+i 2 (n + 1) (- xy)'
x + y
But jD3 = a; + y, thus
n _ a:""' + jT ' - (2n 4 1) (a; + y) (- a^)'
-"" (a; + 2^)*
Then, returning to the original determinant, its value is
(i) if of order 2n,
{{a-c)(c-b)}'
{0-0)"*'+ (5-0)""- (2n-\-l){{a-c)(c-b)}'{a+b-2o)-] ^
-0
(a + i-2c)'
'
(ii) if of order 2n — 1,
^1 a + 6-2c r
§ 14. Evaluation of the determinant
a, i, &, &, ...
C, &• Ca Ca • • •
Ctm Cum C» Co, • • •
jD =
of order 3n. The elements in the diagonal consist of a, 5, c
repeated n times in this order, and the rest of the row is filled
up with the next letter in cyclical order,
i) =
^> I7 1? 1> 1> •••
0, a, &, i, &, ...
v/, C, &, C, C, • • .
0, a, a, c, a, ...
— J, a — J,
- c.
— o,
, t-c,
» •'
, ...
) •'
J c "-aj ,.,
\
^
FOPMS OF DETERMINANTS.
149
by subtracting the first row multiplied by &, c, or a from
the others.
Thus, by Cauchy's theorem,
i> = [a-i,J-c.c-ar|l+nf r + T"^ + -^ R •
§ 15. In like manner
0, 1, 1, 1, ...
1, a, i, 5, ...
X| Ca 0» C, ...
X , CL» Cvrn Cm • • 9
= n[a--b,b'-^.(>-aY^^{a^+V'hc^—hc—€a'-ab).
The determinant is of order 3n + l, and, with the excep-
tion of the border, its formation is the same as of that in § 14.
The determinants of the present and preceding paragraphs
can plainly be generalized.
§ 16. In like manner
0, 1, 1, 1, ... 1
I J ^7 ^SJ ^8? •••
1, ttj, Xj ^3, ...
h ^1) S? ajj ...
when w = (a? - a^) [x — a^,..[x — aj
0, 1, 1, 1, ...
X , C»^ , £{/, X» ...
1| as, a,j a;, ...
1, Xf Xf a,, ...
du
(- ir —
^) ^f> ^8> •••
= tt- 2a,-
§17. This artifice of adding on a border evaluates all
determinants when the elements in a row or column are
all the same with the exception of those in the diagonal.
>->
150
MR. SCOTT, ON FOBlfS OF DBTB&MIKANTS.
Thus
«-
«l' J (aj -«,)', «J
, ...
y •••
a.
a„
, (a? — ag) , . . .
= x"-^w|
a;+S
a;
- 2a, j '
0,
• • • <
• • • •
1
• • • •
• • • •
• • • •
1
J
• ••
1
)
• ••
1,
(X
— a
.r,
<
>
<
J
• • •
1,
• • •
• • • 1
> • • • •
• • • •
[x
• • • <
-a
1 * • • •
.)',
<
• ••••• 4
• • •
• • •
• • •
= — X
.••-1
du
dx
where i^ = (oj — 2a,) [x ~ 2aj) . . (a; - 2a J.
For example, by the latter theorem,
0, 1 , 1,1
= (a
1, (y + »r, ^ 1 «'
1, a;* , (as+z)*, «'
1, a;' , y , (a! + 2)'
If s = a!+y+ « + M,
0, 1 , 1,1,1
1, {8-x)\ y" , «' , ««
1, a^ , (s-y)", «« , m'
1, a* , y , («-«)', «•
1, a;» , y , «" , (s-
«)'
= (a; + y + «)*(a?* + /-l-«'
— 2a?y — 2yz — 22fa?).
+2a5y«+2a;«w+2^j2;w+2a?yj3?
§ 18. If the element at the Intersection of the if^ row and
j^ column be a^tj while the diagonal element is as before.
«A >
= aj"-^z.ja; + S-^l
( X- 2a, j
a.
a.
a.
r
«H (^-^l^) «l«2 J ^l«8 > —
«1«2 ) (^-"^2)*J ^2^8 7 •••
«8^8 J (^-«8)*) •••
a,
29
a
87
«1«8 >
= - a;"-'w2
a,'
a? — 2a,- '
( 151 )
SOLUTION OF A MECHANICAL PEOBLEM
By Professor A, Q. Qreenhill, M.A,
The following rider, due to Mr. Hopklnson, was set in the
Mathematical Tripos of 1879, Thursday morning, January 2,
Number viii.
" A smooth wire is bent into the form of a circle radius a,
and rotates with uniform angular velocity g) about a vertical
axis through the centre which makes an angle a with the
plane of the circle. If a smooth bead slide on the wire,
show that the equation of motion of the bead along the
wire is
Q~Q S S S
^3 = ao)' cos"a cos - sin — a cosa sin - ,
where s is measured from the lowest point. Hence find the
position of equilibrium of the bead, and the time of a small
oscillation about that position.^'
To Integrate this equation, we must first multiply both
ds
sides by -j ; and then integrating with respect to tj
Cut
J [ -^ j = C— ^aW cosV coe^^—\-ffa cosa cos- ,
l-r I = w* - aV cos'^a (cos ^ — ) .
\dtj \ a a© cosa/
Consequently in the position of stable equilibrium of
the bead, cos- = ' — ^ (=cos7, suppose) and u is the
' a aar cosa ^ " ^"^ '
velocity of the bead relatively to the wire in this position
of equilibrium.
Therefore (-r) =w* — aWcosVfcos — cos 7] .
I. If u< a(o cos a (1 — COS7),
the bead will oscillate about the position of stable equilibrium
on one side of the lowest point of the wire.
or
»-•
152 PROF. GREENHILLj SOLUTION OF
IT. If aa> cosa (1 — cos 7) < u <aa> cos a (1 4 cos 7),
the bead will oscillate through the lowest point of the wire,
a position of unstable equilibrium, and beyond the two
positions of stable equilibrium on each side of the lowest point.
IIJ. If u>aaicosa(l + cos7),
the bead will perform complete revolutions of the wire.
To integrate the equation of motion again, put
tan^- = «, or 8 = 2a tsxi^z ;
. ^ ds 2a dz
therefore Tt'=lT?dt'
4a" (-^j = w* (1 + zy - aV cos'a (1 - COS7 - 1 + C0S7 «')'
= {m — aa> cosa. 1 — COS7 4 «' (w + am cosa. 1 4 COS7)},
{u 4 ao) cosa. 1 — C0S7 4 «* (t* — ato cosa.l 4 COS7)},
and therefore z is an elliptic function of t
L Let u<aft)cosa(l — C0S7),
and let s^j s^ be the extreme values of s^ then
„ «j _' u-\-a(o cos a (1 — cos 7)
^ a "" a© cos a (1 4 cos 7) — tt '
,- 5, am cos a (1 — cos 7) — tt
tan "5 "" = /^ r •
^ a u 4 aa> cos a (1 4 cos 7) '
or cos -* = cos 7 —
or
a acD cos a '
cos -s =5 cos 7H :
a ao cos a '
8 8
and, therefore, cos -^ + cos -^ = 2 cos 7.
' ' a a '
A MECHANICAL PROBLEM. 153
Then
= t?®' cos'a (l + cos ^A (l + cos j) (tan'i ^* - «') (;5'- tan'i ^A ,
or (^y = a)»cos^acos'i^cos'i Vtan^J J-^'V^'^ ;
and therefore
tani - = « = tani- SinKys,.
when ;fc'« = l-A« = i,
tan^i-*
and -^ =5 ft) cos a sin i -* cos i -■ .
The time of oscillation is then 221
When the oscillation is small we most put s^^a^^ and then
i'=lj A=0, Jr=i7r, and
• 2r=
or !r=
0) cos a sin ^7 cos ^7 '
ft) cos a sm 7
11. If ao) cosa(l — C0S7) <M<aft) cosa(l +COS7),
denoting by s^ the extreme value of «, we have
4a« ^^y + {aV cos-a (1 + cos 7)" - u^] (tan* J i^ - i5») (;?« + c»),
where, as before,
8 U
COS - = COS 7 - T ,
a aft) cos a
J , _ w — aft) cos a (1 — cos 7)
"" w + ao cosa(l + cos7)'
154 PEOF. GSEENHILL, M£CHAMICAL PROBLEM.
and therefore
1 + c" u
1 =0087 +
1 - c cuo COS a
Therefore, as before.
, «' cos' a cos' ^ -i
(1) r^^His'-^)(»-.A
and therefore
where
tan* - = « = tan i -* en iT-Ty,,
A:'* 1-A» e
W
tan'i^^
or A:*= — ^
tan'i-^+c"'
^, tan'i-^ + c*
and 7r^ = «' cos' a , a •
and the time of oscillation is 27, the bead now oscillating
through the lowest point of the wire.
III. If u>afi> cosaCl + co87), the bead will perform
complete revolutions, and
4a« (~y= {w'-a'o)' cos'a(l +CO87)'} (^» + J«)(«»+ c*),
, ,« u + afi>cosa(l — CO87)
when o = 77— r ,
u — a<o COS a (I + cos 7) '
, , t« — aft)COsa(I — C0S7) - ^
and c= 77- ^, as before.
w + ao)COsa(l + cos7) ^
i' + l w
6—1 acocosa'
1 + c' u
^ = 0087 +
1 — c aoicosoc
i
MB. SCOTT, ON A THEOREM OP PROF. CATLET'S. 155
V(o^ cos'a
»«'«'»" (ij-(T^W^C-^'")('*-^
and therefore
where
and
s t
tan^ - = « = Jtn jST-^ ,
K^ FcW cos'a
T* (i-6')(i-c»)'
and the time of a complete revolution is 2T.
NOTE ON A THEOREM OF PROF. CATLET'S.
By JB. F. Scott, M,A.
At pa^e 164, vol. v. of the Messenger^ Prof. Cayley gave
the followmg : " Theorem in Trigonometry.
If ^ + 5+(7+i^+G^ + ir»0,
then sin-J+T sinS+TF sinC-f -F, cosF, sinF « 0."
sin^ + O sin^H- O sinCn- (?, cos (?, sin Q
sin^ + H sin5+ ^ sin (7+ -ff, cos-fl", siniT
In seeking to establish this theorem the following method
occurred to me, which as it is applicable to all similar de-
terminants, is not without interest.
If we write a = e*^, &c., the first row of the determinant Is
uc
Hence we can take the factor r^^ outside the deter
2 i
Writing only the first row, the others being got by
changes, the determinant is now
7)_i (a'/''-i)(^r-i)(cy'-i) /'-n /'-i+.g+(7v
2' abcf ' f ^ f
aWcT - (o'J" + bV + cV)/*
^abcfgm
r
r
i
156
MR. SCOTT, NOTE ON A
Now add the second column to the last and divide by 2, then
subtract the last column from the second, thus
+ (a' + 5' + c')/«-l, /»,/*!.
Now multiply the last column by a*i* + iV + cV and add it
to the first, then the second column by d^-\'W-\-€? and
subtract it from the first, thus
= 2-wyA' {«wyA'-i} G^'-A») [v-f] (/'-^)
1 9»_A» h'-f f-q' ( ^., I \
= sin((7-J7) sin(J7-J?^ sin(-P- (?) sin(^+J9+C+jP+(?+^.
This method will be found to lead with great facility to the
evaluation of all determinants whose elements are sines and
cosmes.
The following examples may be given :
The value of the determinant
4aM>
\dt,
when
and
1
1
cos -4, cos -B, cos C7, cos D
sin -4, sin -B, sin (7, sin D
sin 3 J, sin 3^, sin 3 (7, sin32>
n sini [A^B] {cos(;S'+u4) +cos(fif+ B)
+ cos(/S+ 0) + cos(fl^+Z>)},
^A + B+ C+D, and in the product of the sines of
flferences we adhere to the order of the alphabet,
subtract any letter from one which follows it.
THEOBEM OF PBOF. CATLBT'S.
157
./•
x-:
^r
Similarly
sin u4, sin J5, sin 0, sin D
cos -4, cos -B, cos C, cos J9
sin2u4, sin 25, sin 2 (7, sin2i>
cos2u4, cos25, cos2C, cos22>
= 2*n sini (^-5) {cosi (-4 + J9-C- J9)
+ cosi(^ + (7-J9-J9) + cosi(^ + i>-5-(7)}.
We have also the general theorem for 2n + 1 angles
11 1
sin dj, sina,, ...sinag^^j
cos a
11
sin 2a
cos 2a
iJ
i)
sin na
cos na
1)
I)
= 2'^'n sini (a,-a4)(t>i),
and the following curious theorem :
The determinant
jD = cos a , cos (a+ J) , cos (a+2 J) ... cos {a+(ti— 1) h]
cos(a + J) ,cos(a+2i), cos(a + wi)
co8{a+(w-l)i}, co8{a+(2w-l)J}
vanishes for all values of n greater than 2. When n =s 2 its
value is - sin^i. The method applies equally to hyperbolic
and to some of the gudermannian functions, e,g.
cosh. A^ cosh J9, cosh G
sinh A^ sinh B^ sinh G
cosh 3^, cosh3J9, cosh 3 (7
= 4 sinh (5-0) sinh [G-A) sinh [A--B) cosh (^ + 5+0).
( 158 )
ON A CLASS OF DETERMINANTS, WITH A
NOTE ON PARTITIONS.
By J. W. i. GUtMer.
§ 25. The following remarks may be regarded as a con-
tinuation of two previous papers, " Expressions for Laplace's
coefficients, Bernoullian and Eulerian numbers, &c. as deter-
minants" (vol. vc. pp. 49-63), which contains §§ 1—13, and
"On a class of determinants" (vol. Vil. pp. 160-165) which
contains §§ 14—24.
§ 26. It may be observed that the determinant
%-Kj «2) «i) 1> —
«4""*4> «8) ^2J «i» —
which 18 specially considered ill the second paper, may be
written in a slightly different form ; for
«4 - *4> «8J «a? ^iJ •••
[n rows) =
K «2? «1» 1 )
K Sj «2J «1J
(n+1 rows)
(!)•
This is at once seen to be true by subtracting the second
column from the first in the second determinant.
We can prove directly thatj if
then
1 + ajO; H- a^aj* 4- agO;' + &c. 1^2 a ^ j
(w + 1 rows),
P.-
K ^21 ^M
K «8J %i ^
MB. GLAISHERy ON A CLASS OF D£TEBMINAKT& 159
for this value of P^ follows immediately from the system of
equations
- P, + «, = *„
-P3 + a,P,-a^, + a3 = &3,
&c. = &ci.
obtained by equating the coeflScients of the powers of x.
§27. Theorem. If
log (1 -f a^x + a,aj" + a^x' + &c.) = P^x - ^P^x* + ^P^x"" - &c.,
then
^ij 1 ) • ? • ) •••
2a^j a^, 1 , . , ...
Hj «8? ^1, 1 ) —
(n rows).
iVoof. By diflFerentiation
a^ 4 2g^ag -f 3a3a;' + &c. _ p _ p p
l + a,ajH-a,aj* + a,aj'*-|-&c.'" ' ^«^ + -^«"
whence, by § 26,
— &c.,
a.
1 ,1,
2a
a
- , a^, 1 , ...
3^
a.
, a^, a^j ...
(n rows) J
and, therefore P> as above.
§28. Theorem. If
exp (a^a; + a^aj* + a^a:' + &c.) x= 1 + PjX + P^aj' + P3aj' + &c.,
then
P=l
a,, 1, . , . , ...
3a., 2a., a., — 3, ...
"8}
"ti
"ll
4a^, 3a3, 20^, a^, ...
(n rows).
160 MR. aiiAISHEB, ON A GLASS OF DET£BMINANTS|
Broof. Bj differentiation and diyision
^ ^ 9 P, + 2P,x + SP.aj' + &c.
whence, equating coefficients.
therefore
a^
= P
19
2a, + a^F^
3a,+ 2a,P,+ a^F^
4a^ + 3a,P, + %aj?^ + a^F^
&c.
= 2P
s)
= 3P
89
= &C.,
= — a
19
aP- 2P
= — 2a
'89
2a,P,+ aii'.-SP,
3a,P, + 2a,P, + a,P,
= -3a
'89
4P,
&c.
= — 4a^
= &c.,
giving the above value of P^.
If the left-hand side be exp (a^ + a^ + a^ + &c.) instead
of exp(a,aj + aga3' + &c.), it is only necessary to multiply
throughout by expa^.
As an example, let a^o; + ajOj' + agOj' + &c. = e*, then, by
the theorem,
exp6* = l + Pjaj+P,»' + P3aj' + &c.,
where
P =
e
1 , —1, . , . , ...
n\
1 , 1,-2, . , ...
^11-3
2 j 9 ^ 9 -"^ 9 O, ...
3!' 21' ' ' "*
(n rows).
agreeing with the result in § 12.
\
MB. GLAISHEB, ON A CLASS OF DETERMINANTS* 161
§ 29. It Fesnlts from the theorems in the last two sections
that if
(n rows),
2a^j a„ 1, ., ...
H) «8> «8> «1> —
then e?^ = — ,
* n!
25.,
— 1, • , • , ...
^1) "2, . , ...
2^8, *!, - 3, ...
3fta> 2J„ ft,,...
5ind vice versd.
For example,
if
= a„
(n rows),
ft, = i(2«8-«i')7
and the theorem asserts that
«8 = i
Of-
-1
2a,-a,« , a, ,-2
V - Sa^ajj -f 3a„ 2aj — a,*, a^
Adding the second column, multiplied by a,, to the first
column, the right-hand side is at once seen to be equal to a,,.
§ 30. In § 27 let a^ = n, and the theorem shows that
log (1 + a: + 20:* + Sx^ + &c.) = P,x - ^P^x"" + JP,aj' - &c.,
where
P =
f 1' 1
2 , 1) 1> •) •••
V I i5« J« 1. •• a
4. o. ^, 1,«*.
(n rows).
Now log (1 + aj + 2x* + 3a)' + &c.) = log ^1 + T-^-p^
, 1-aj + aj* I
1+aj^
(l-x)« ""^^(l-a;)(l-a;'')
-log(l -a?) -log (1 -a:*) 4 log (1 + aj')
= a; + iaj' + &c. + aj* + ix* + &c. + aj'-^a;' + &c.,
VOL. VIII. M
162 MS. GLAISHEB, OV A CLASS OF DETEBlOHAKTSb
\i
and the coefficient of - a;*^ 0, 1, 3, 4, 3 or 1, according is
X E 0, 1, 2, 3, 4 or 5 (mod 6).
It therefore follows that the determinant
1*, 1, ., ., ... I (jirows)
" I 1| 1> •! •••
^ I 2| h h •••
4 . V. iB| X* ...
= 0| 1| * 3, 4, — 3 or 1| according as
X =0, 1, 2y 3, 4 or 5 (mod 6).
§ 31. This can be proved directlj as follows. Sabtracting
the second colonm from the first, the determinant becomes
.| 1| •! .|... (m rows)
'i ^1 1| •> •••
^1 ^1 1| 1| •••
Xv| V* jSm X| ...
^1 1| •! •> •••
7 11
■l.«jl if. J. Xl ...
^X| O. ^1 X. ...
(n — 1 rows),
Sobtract the second column, multiplied hj 3, from the first
colunni and the determinant
•I ^1 •! •> •••
^1 ^1 M •> •••
^1 ^7 h 1> •••
x^i 0| ^, 1| •••
*l ^1 •! •! •••
7 11
12, 2, 1, 1, ...
X«/« O. ^1 X, •••
(n — 1 rows)
(n — 2 rows)
MR GLAISHEB, UN A CLASS OP DETERMINANTS. 163
Subtract the second column multiplied by 4, and this
^j ^1 •) •) •••
4, 1, 1, ., ...
'^1 2j h h •••
12, 3, 2, 1,
• . •
(w - 3 rows),
and so on.
The changes In the first column are exhibited below
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121
1, 1, 2,
3,
4, 5, 6, 7, 8,
9,
10,
> 11,
12,...
0, 3, 7,
3,3,
13,
6,
21, 31, 43, 57, 73,
9, 12, 15, 18, 21,
91,
24,
111
27,
, 133,
) 30,
157, ...
t5o, •••
0,4,
4,
7,
4,
12, 19, 28, 39, 52,
8, 12, 16, 20, 24,
67,
28,
84,
32,
1 103,
f 36j
124, ...
40, ...
0.
3,
3,
4, 7, 12, 19, 28,
3, 6, 9, 12, 15,
39,
18,
52
21,
, 67,
> 24,
84, ...
27,...
0,
1, 1, 3, 7,13,
1, 1, 2, 3, 4,
21,
5,
3L
6,
, 43,
7,
57,...
o, .••
0, 0, 1,
25
144, 169,..
36, 49
)
., ., ., 4, a, 16,
Thus, the leading elements of the determinants are 1, 3, 4, 3,
1, 0, and then the determinant becomes the same as the
original determinant but reduced by six rows. This Is easily
seen to be tine generally, for from rt^ we subtract successively
w-1, 3n~6, 4n-12, 3n-12, w-5, that Is, 12n-36; so
that after the subtractions n^ becomes (w - 6)'*, and at the same
time the determinant Is reduced by six rows.
§ 32. Similarly, since
log (1 + 2aj + Sx"" + 4aj' + &c.) = - 2 log (1 - a?)
= 2 (ir + ia;" 4 Jaj' + &c).,
we have
(n rows),
ij Ij M •) '••
3, 2, 1, ., ...
O, U, Jly X, •*•
10, 4, 3, 2, ...
(the first column consisting of the trlanp^ular numbers) =—1 or
1 according as n is even or uneven. This can also be proved
at once by the method employed In the last section.
m2
164 MR. GLAISHER, OX A CLASS OF DETERMINANTS.
As another example, we see from the equation
that
^ T V ui -r»*c T wcv/i
'^'"^(l-x)"
1 9» 1*
(n rows)
ia*0 • ia* 1> '•••••
3.4*, 3», 2«, 1", ...
4.5', 4", 3', 2*, ...
*
Bs — 2 or 4 according as n is even or uneven.
Writing the first column of this determinant in the form
2'- 2', 3' — 3", &c., it is evident by transforming the deter-
minant as in § 26, that, if n > I,
{n rows)
* J ^ 1 • 5 • J •••
9* 9* 1*
A<a *•) *■ % •••••
q8 nil 98 1 8
U, U, iU, X, •••
4', 4^ 3*, 2*, ...
= — 4 or 2 according as n is even or uneven. This result also
follows at once from the latter part of § 26 ; for
H-2'a; + 3V + 4V4&c. _ 144a; + a?'
l+2''a; + 3V + 4V-f&c. - 1-aj'*
= 1 + 4a; + 2aj* + 4a?' + 2aj* + &c.
§§33—38. Partitions.
§ 33. The total number of partitions of any number ;?,
order being attended to, is 2**"* ; and if the partitions contain-
ing an uneven number of parts be treatea as negative the
number of partitions is zero. By the words "order being
attended to " it is meant that, for example, 3+4 and 4 4 3, or
1414 5, 14541 and 5 4 1 4 1 are to be regarded as
difierent partitions, so that if a partition of n which contains
r parts has a parts each equal to a, b parts each equal to ^,
c parts each equal to 7, &c., then this partition is to be counted
7* I
times, and therefore
a! bl c! ...
2
r!
a! bl c!...
^-y
r\
a! bl c!...
= 0,
the S's referring to all tlie partitions of n.
MR. QLAISHER, ON A CLASS OF DETERMINANTS. 165
Take, for example, the partitions of 5 ; these are
5, 4 + 1, 3 + 2, 3+1 + 1, 2 + 2+1,
2+1 + 1 + 1, 1 + 1 + 1 + 1 + 1,
and therefore
.-(.>+«o.(l|4;)-i|+l|=<,,
These theoremS| which result from the equations
1 1-x
l-(a? + a'* + aj" + &c.) " l-2a;*
l + (aj + a?* + a?' + &c.) ^
are well-known (see Sylvester, British Association Beportj
Edinburgh, 1871, Transactions of the Sections^ p. 24).
§34. In the same manner from the equations
we see that
r a\ b\ cl... n ' ^ r a\ b\ c!... n*
For example, if n = 5 these ^ve
l-H2lH-2l) + j(||4|)-i|] + l5-;=i..
If we omit the partition, which consists entirely of I's, the
second equation, in the case of n even, becomes
166 MB. OLAISHER, ON A CLASS OF DETERMINANTS.
§35. From the equation in § 30, VIZ.
log (l+a;+2a;'+ 3a;'-|-&c.) =- log (l-x) - log (l-aj')+log (1+a?'),
it follows that if p denotes the product of the parts, supposed
r in number, of a partition of n, then, if the order be
attended to,
^(-r?-o,' ' i,?»J
n' n' n' n n
(iO,
according as n =0, 1, 2, 3, 4, or 5 (mod 6).
If we consider only the different partitions (without regard
to the order of the parts in each) then in (i.) we are to replace
For example, if n = 5 the formula giveef
6-i(2!4 + 2l6) + i(||3 + |-!4)-ii;2 + iJJ = i,
5-10 + 7-2 + i = f
VIZ.
§36. By §27,
log (1 + sc + 2 V + 3V + &c.) = P,aj - iP,a!» + JP,®'
— &c.,
where
^n-
1
2
m+i ^
, . , ...
1 1
,-^,1,1,...
4"^', 3"*, 2"*, 1%...
(n rows).
and therefore this determinant
For example, put n = 4; since the partitions of 4 are
4, 3 + 1, 2 H- 2, 2 + 1 + 1, 1 + 1 + 1 + 1, this formula gives
= 4 {- 4~ + i (2.3"* + 4"*) - J.3.2*'* + i-H,
m+i
1 , i , . , .
-4 ,1,1,.
3»+i^ 2% 1% 1
J *W+1 fM Atfl ^ til
which is readily verified.
MR. aLAISHER, ON A CLASS OP DETERMINANTS. 167
§ 37. The formula iq the last section is still true if we
substitute a^, a j a, ... for 1"', 2, 3"*... and a„ 2a^^ 3a ... for
jm+i^ 2*^', 3*^^.. and if for fy a product formed from
1", 2*", 3"* ... we substitute tt, a similar product formed from
a^j a^j A, •••} VIZ.
.iSdj^ fltjj 1 ^ • ...
4a^, Og, dj, a^ ...
ir
(n rows)
=(-r„2(-rior=(-r„s(-r?^j^,r.
according as the order of the parts in the partition is or is not
attended to.
Also we have
^«> ^j) *■ J • J •••
«8) ^«? ^1) 1 ) —
^4) ^«> «ii «i? •••
(n rows)
rl
= H-2HV or (-rsH^^j^y^^
according as the order of the parts in the partitions is or is not
attended to.
§ 38. The formul« in §§ 33—35 would all be useful in
verifying that in writing down all the partitions of a number,
none had been omitted, but those in §§ 36, 37 would be of no
practical value for this purpose on account of the labour of
calculating the determinants. It is scarcely necessary to
remark that in general the determinant expressions considered
in this paper do not afford a good method of obtaining the
numerical values of the coefficient which they represent.
Erratum.— In § 24 (vol. vii, t). 165, line 3) for « log (!+«)= log x+CiX + Ac.'*
read " log log(l + a?) = logo; + Cix + <fec."
( 168 )
TBANSACTIONS OF SOCIETIES.
Londtm MathematietU SoeUCy,
Thursday, December 12th| 1878, 0. W. Herrifield, Esq., F.E.S., President, in
the Chair. Prof. W. Stanley Jeyons, F.B.S.J was elected a Member. The fol-
lowing communications were made : " On a Einematic Paradox (the Botameter),"
Mr. H. Perigal, F.BJL.S. ; '* On Forms of Numbers determined by Continuel
Fractions,'' Mr. S. Boberts, F.B.S. : *' On a Graphic Construction of the Powers
of a Linear Substitution/' Ihinoe Camille de Polignac.
Thursday, January 9th, 1879, C. W. Merriiield, Esq., F.BJ3^ President, in the
Chair. Dr. J. Hopkinson, FJt.S., was admitted into the Sodet^. The following
communications were made to the Society : *' On a Theorem in Elliptic Fimc-
tions," Prof. Cayley, F.B.S. (Mr. J. W. L. Glaisher and Prof. Smith spoke on the
subject of this pa^p^) ; " On a New Modular Eijuation," Prof. H. J. S. Smith,
F.B.S. : " On Coefocients of Induction and Capacityof Two Electrified Spheres,*'
Prof. Greenhill: "On Certain l^^tems of Partial Differential Equations of the
$1zst Order with siBTeral Dependent Variables,'' Prof. H. W. Lloyd Tanner.
B. TuoKSBy M JL., Eon. Sec.
ON THE STRESSES CAUSED IN AN ELASTIC
SOLID BY INEQUALITIES OF TEMPERATURE *
By J*. Sopkimon, F.JR.S,
Vabious phenomena due to the stresses caused hy in-*
equalities of temperature will occur to everyone. Glass
vessels crack when they are suddenly and unequally heated,
or when in manufacture they have been allowed to cool so as
to be in a state of stress when cold. Optical glass is doubly
refracting when badly annealed or when different parts of the
mass are at different temperatures. Iron castings which have
been withdrawn from the mould whilst stUl very hot, or of
^hich the form is such that some parts cool more rapidly than
pthers, are liable to breal^ without the application of bjxj
considerable external stress. The ordinary theory of elastic
3olids may easily be applied to some such cases.
Let uvw h^ the di3placeraents of any point {xyz) of a body
density p, parallel to the coordinate axes. Let if^y N^^ ^,
J*p T^^ 2\ be the elements of stress; i.e. Nfl is the tensLon
across an elementary area a resolyed ps^rallel to a;, the element
a being perpendicular to x ; 2\/8 is the shearing force across
an element /8 resolved parallel to si^ ^ being perpendicular to
V ; T^ is then also the shear parallel to y across an element
perpendicular to z.
* ^ Report of the British Associatien ioi 1872, p. 5^.
MR. HOPKINSON, ON THE STEESSES, &C. 169
If pXj p F, pZ be the external forces at [xyz)
dx dy d^ '^
§-f^f^'^=M «•
ax ay az
These are strictly accurate. Of an inferior order of
accuracy are the equations expressing the stresses in terms of
the strains of an isotropic solia
du
^.-«'«/'S
dv dw\ "**' ^ "
-, fdv dw\
where 5s»!j-+^+-T-=athe dilatation at the pomt. These
dx dy dz '^
equations are inaccurate^ inasmuch as they are inapplicable if
the strains be not very small, and as even then in all solids
which have been examined the stresses depend not only on the
then existing strains but in some degree on the strains which
the body has suffered in all preceding time (see Boltzmann,
Ahad. der Wissenaoh, zu Wien^ 1874; Kohlrausch, Fogg.
Annalerij 1876 ; Thomson, Proceedings cf Royal Society^ 1865 ;
some experiments of my own, Proceedings of Royal Society^
1879 ; Viscosity in Maxwell's Heat).
Assuming equations (2) we observe that as these and also
(1) are linear, we may superpose the effects of separate causes
of stress in a solid when they act simultaneously.
Equations (2) are intended to apply only to cases in which
when the stresses vanish the strains vanish, and in which the
strains result from stress only and not from inequalities of
temperature. The first limitation is easily removed by the
principle of superposition. We must determine separately the
stresses when no external forces are applied, and then the
stresses due to the external forces on the assumption that the
solid is unstrained when free and finally add the results. For
example, if we are considering a gun or press cylinder, we
know that internal pressure will produce the greatest tension
in the inner shells, and we can hence at once infer that If the
gun or cylinder be so made that normally the inner shells are
170 MR. HOPKINSON, ON THE STRESSES CAUSED IN AN
in compression the outer in tension it will be stronger to
resist internal pressure.
To ascertain the effect of unequal heating, assume that \fi
are independent of the temperature, an assumption of the
same order of accuracy as assuming in the theory of conduction
of heat that the conductivity is a constant independent of the
temperature. Let k be the coefficient of linear expansion,
r the temperature at any point in excess of a standard
temperature. If there be no stresses,
du __ dv dto
dx dy
&=*^5
therefore
^ (dw dv\ o
if there were stresses, but r were zero,
superposing effects we have
.N; = X5 + 2/*^-(3\ + 2/*)a:t I
fn (dw dv\
(3),
^dy dz,
Substitute in the equations of equilibrium
, X du a dx •»
dx
,. . da ,
dz
dx
7^ + pI^=0 ) (4),
dr „
=
where
7 = (3\ + 2fi) K.
If there be equilibrium of temperature v V = 0, and the
effect of unequal heating is exactly the same as that of an
ELASTIC SOLID BY INEQUALITIES OF TEMPERATURE. 171
external^ force potential ^ ; in this case we have
the equations
2/1 cP0
(5),
.S-«2
vVw=o,
still true and under the same conditions.
Examine the case when there are no bodily forces a-nd
when everything is symmetrical about a centre. The displace-
ment at any point is radial, call it U^ and the principal stresses
are radial and tangential, call them R and T.
The equation of equilibrium is
df/E)
dr
-2rr=0
(6),
and the stresses are expressed by
r= \e + 2/A - - 7T
r
g^l djr'U)
r* dr
p),
substituting
(X + 2/i)
^^+2i^«2-i= —
dr^ r dr r*J dr
(8);
therefore
r'U=
\ + 2/A
/rVc?r + ar° + 5
(9),
where a and h are constants to be determined by a knowledge
of -B or 17 for two specified values of r. This equation is of
course true whether there be equilibrium of temperature
or not.
The interior and exterior surfaces of a homogeneous spherical
shell are maintained at different temperatures to find the
resulting stresses.
Let r^, r^ be the internal and external radii, ^j, t^ the
172 MB. HOPKINSON, ON THE STRESSES CAUSED IN AN
internal and external temperatnresi then if r be the tempera-
ture at radios r,
r
where
and
(10).
Substitate in equation (9) and then in (7)
5=-
2fi
W^f'^/r^^'''^^^''^''-'-^
...(11),
write JS in the form
5 = ^ + ^ + g;
where
for we shall not require to find U] we find
whence
; (12)
i( will have a maximum or minimum value when
3r,V,
and its value then is
r; + r,r, + r^
»}
this is positive if ^^ > t^^ as we may see at once from physical
reasons.
p*
ELASTIC SOLID BY INEQUALITIES OF TEHFERATUBE. 173
if t^ > {,, this decreases as r increases, when r s r„ its value
becomes
ff [ - 2 {r, + r.) r, + (r," + r,r, + r.') 4 Q
The case when the thickness is small is interesting. Let
r^rssr^ + x^ then the maximum tension is
(X
neglecting the term - in comparison with unity, we see that
of two vessels the thicker is not sensihly more liable to break
than the thinner, a result at first sight contradictory to
experience. The explanation is that the greater liability of
thick vessels to break is due to the fact that, allowing heat to
Eass through but slowly, a greater difference of temperature
etween the two surfaces really exists.
Let t\^ t\ be the actual surface temperatures, we may
assume that, if t^ and t^ be the temperatures of the surrounding
media, the heat passing the two surfaces per unit of area wiU
befi-(^.-gandS,(<\-0.
Hence t\-t\^ ^A^h"') ,
using this in the equation last obtained we have a result quite
in accord with experience.
Ketuming now to equations (7) and (9), suppose the sphere
to be solid and to be heated in any manner symmetrical about
the centre. The constant h must vanish, and
= -4/i- +3(X + 2/i)a
* Thiis result was set bj me in the recent Mathematical Tripos Examination
(Friday afternoon, January 17, 1879, Question ix).
174 MR. THOMSON, VORTEX MOTION
now the mean temperature within the radius r is
irr T
therefore, since the pressure is zero at the surface of the sphere,
^= Q/\ 9 \ • \S^^^^ temperature of whole sphere— mean
temperature of sphere of radius r}...(14),
^=j-' ^ (").
— . {mean temperature of whole sphere — f t + J mean
3 (\ + 2/a)
temperature within the sphere of radius r)...(15),
Other problems of the same character as the preceding will
suggest themselves, for example that of a cylinder heated
symmetrically around an axis, but as no present use could be
made of the results I do not discuss them.
VORTEX MOTION IN A VISCOUS
INCOMPRESSIBLE FLUID.
By J. J, Thomson, Trinity College, Cambridge.
Let Uj v, w be the velocities of a particle of the fluid
along the axes of aj, y, z respectively ; V the potential of
the applied forces ; p the pressure of the fluid at any point.
Then if the usual law of fluid friction be assumed, viz.
that the tangential forces called into play by the viscosity
are proportional to the relative velocities of two neighbouring
elements of the fluid, the equations of motion are
(du ^ du ^ du ^ du] dV dp « \^ , x
'dv , dv ^ dv ^ dv) dV dp 2 ^ ,^.
(dw . dw dw dw) dV dp ., ^ r ^
IN A VISCOUS INCOMPfiESSIBLE FLUID. 175
where fi is the coefficient of viscosity, and p the density of
the fluid at any point.
Differentiating (1) with respect to y, (2) with respect to a?,
and subtracting, we get, if
^_dw dv
^~^" (fe »
du dw
y_^dv ^ du
dx dy ^
^jL^yjL^y^t^^ ^4. ^4. ^4. ^? _ MV* c>
dt fix dy dx dy dx dy dz p '
but since the fluid is incompressible
and since
du dv dio __ ^
dx dy dz" ^
dt rf?f d^ d^ i)?
5^ + ^^ + ^rfy-^^T.==:D^>
where jy has its usual meaning, the equation reduces to
similarly
D^ g dw dw ^dw ^fi ,
Dfi ^dv dv ydv ^fi J,
Di'^~€bo^'^d^''^d^ "p^'^^
D^ ^du du y,du __/M ,g
Di'^^d^'^'^Ty^^li "p^f'
these are the equations of vortex motion in a viscous fluid
and but for the term on the right-hand side are exactly the
same as for a perfect fluid.
For small motions these equations reduce to
5? = p^^'*"-
176 MB. THOMSONi YOBTEX MOTION
These equations are the same as those which occur in
conduction of heat. Hence we see by analogy that if we
have a quantity of viscous fluid acted on by conservative
forces, and if at any time the motion be irrotational then
any vortex motion uiat may be in the fluid at any future
time must have come from the boundary of the fluid.
We also see that if vortex motion be set up in any part
of a viscous fluid the motion throughout the fluid inmiediately
becomes rotationaL
If tiie medium in which the vortex motion takes place,
which, according to Sir W. Thomson's theory, constitutes
matter, were viscous the qualities of bodies would not be
permanent, the viscosi^ would tend to make all bodies
possess the same properties.
Motion in two dimensions.
Let us consider the case in which every particle on the
fluid moves in a plane parallel to the plane of a;y, and let us
suppose that initially there was an indefinitely long straight
vortex filament whose axis coincided with the axis of z^
in this case
f=0, f/ = 0, 1(7 = 0,
and the equation t
Df ^ dw dw ^dw „«,
becomes ^ 57 ~ M'V%
since each particle of fluid describes a circle round the axis
of 2; in a plane parallel to that of xy and since the motion
is symmetrical round the axis of z
Bt'dt*
where r is the distance of the particle from the axis of ;;.
The equation of vortex motion thus becomes
d^ ft /<?»? . 1 rf(r
p \dr* r dr) '
dt p
IN A VISCOUS INCOMPRESSIBLE FLUID. 177
of which the solation is
r=«;
2irkt
(Fourier, Theorte de la chaleur^ Chap, ix.) where p^ = strength
of the original Yortex filament, and exp u denotes e* ; also k
is written in place of — .
To find the velocitjr at any point.
Let 8 be the velocity at a point whose polar coordinates
are r, d, since the point describes a circle of radius r,
tt = — «sin0, v = 8cos0.
dx dy
dv ^ dv sind du . a ^^ ^^^^
^ds s
dr r
1 d . .
., . d.. ^^P [" 1m) ^
therefore -y {sr) = r ^ ^ L ;
dr^ ^ 2irkt ^°'
therefore sr = - cj, —-———+ (;,
when ^ = 0^
IT
«f — >
Trr
therefore «= A |i_exp(- ;^)| .
We see that except when < = the velocity is zero at the
axis of «, a result which forms a remarkable contrast to the
case of a perfect fluid when the velocity is infinite at the axis
of z. The discontinuity which exists is a consequence of
our initial circumstances being discontinuous, which obliged
us to take a discontinuous function as the solution of the
equations of motion.
VOL. viil. N
178 MB. THOMSON, YOBTEX MOTION
To find the rate of diminution of the kinetic energy.
The rate of dimination of the kinetic energy
where s is the velocity,
if neither fM nor ^= ; hence, we see that so long as /li is not
the rate of diminution of the kinetic energy does not depend
on fjk.
The vortex motion at a point distant r from the axis
of 2? is greatest, when
dt '
i.e. when
27rkt ^* 2irke
tie. when < =
4Jc
i.e. the time which must elapse before the vortex motion
at a point attains its maximum varies as the square of the
distance of the point from the axis.
Suppose we have initially a single circular vortex in an
infinite mass of fluid; to find the distribution of vortex
motion, the motion being so small that squares of the
velocity may be neglected.
Take the axis of the filament as the axis of z^ let r be
the distance of a point from the axis of Zy 6 the an^le
measured in a plane perpendicular to the axis of z^ which
this distance makes with a fixed line.
u,
Let, as before, - = A:.
i
IN A VISCOUS INCOMPRESSIBLE FLUID. 179
The equations of motion are
f=*v'f (1),
5 = *V'1J (2),
Since the motion is symmetrical round the air of z^ we
may put
f=s — «sin^, fj^scosOj
where s is independent of 0.
Making these substitutions and transforming from a;, y^ z
to r, 6^ Zj equations 1 and 2 reduce to
we may put
§-'g-^ <»).
^, d*» 1 <& /A 1\ ' ,,.
*^*° ^» "^ ; Jr + * U ~ i^j °"^ ^ ^'
where A is anj constant, since the solution of (3) with the con-
ditions that, when ^ = 0, s = except when z = is
^^p (" ht}
^jT^^-g^ (function of r),
and since the solution of 4 is
« = e~**
» = e7, f r A /t] (function of z and t\
where J^ is Bessel's fimction of the first order, we see
summation being taken for all positive values of hy since
f*
•'0
N2
180 MB. THOMSON, YOBTEX MOTION
and since for large valaes o£B
we see I e/j (ra) t^, (r6) rcfr = ;
hence if we mnltlpty each nde of equation (5) by
and integrate from to oo we get if a be the radios of the
original Yortez filament
[ "'■ (' ^/l) * '
c = s^ddrj where s^ is the value of s for the initial vortex ; let
7 = P. Now
/V- (W) ,*. «■ (^)*+ fi- (l - pL) j;. (JM).
if JS be very great, =— v , since then
'^. (^) = >v/(^) '
when B is very great the difference of successive roots of
cTj (i2) = 0, =7r, hence ^'=-n; hence, changing from summa-
tion to integration
.s = c«
Now I exp (- A;P«) '^^ = i jc: ;
1.
IN A VISCOUS INCOMPRESSIBLE FLUID. 181
therefore
let i>=
exp I — -y- )
Since
•'i IrV^J -- 2:2^ V^ - 2S "*" 2X4:6 "*"••• 2-.nl(n + l)I ■*■" V '
we have
^arD( {f-^a^D 1 / / a*. . 2aV\
16" t 2:4 "*'2* V2l3!■*■2l3I■*■2!Y!>/-^/"'•'
■*■ ^'^" PU!(w+1)1 "^ (7i-l)!wl2! "^ (n-l)l(7i-2)!2!3! +-J^} »
hence
_ ^^P V 4^J or n _ f^4a' 2
*""^ V(27rAi) 32 t(Ae)'' 2.4 (A^"
1 f r" g* aV2 ^ 2.3 )
■*"2* V2r3T"*"2l3!'^2!2!; (A:e)* "*■"•; •
We see that there is no vortex motion at the axis of 0,
and that the distribution of vortex motion is similar in all
planes perpendicular to this axis.
Since k and t only enter through the product Tct^ if equal
vortex rings were initially in two fluids A and jB, then the
distribution of vortex motion in the fluid A at the time t
would be the same as the distribution in the fluid B at the
time TT ^9 where ha and hb are the values of k for the fluids A
and B respectively.
( 182 )
NOTES ON DETEBMINANTS.
By JR. F. 8eoU, M.A.
§1. Let
^« =
0, 1, 1, 1, 1
(n rows).
ly C, a, Oy
1, l^ C| a,
I, 0, h^ c, a
1, 0, 0, 6, c
WherOi with the exception of the border^ the elements in the
principal diagonal are equal to c, in the Imes on either side of
it a and h^ the rest are zero.
By expanding with respect to the elements of the last row
and column we may establish the following equation of
differences for D.
ii9
0^-00^,^-01)0^^
n-\
'Stab «i*^-t;
where u and v are the roots of the equation
«* - c« + aJ = 0.
Solving this we obtain
- _ u*-ft?* nc (u* + O
•"■ (a + ft + c)' (a + J + c) (w-t?)*
2a Jn %r^ + v'
.«-i
a + J+c (w — t?y
(-«)'+(-&)'
(a+6 + c)*
§ 2. In establishing ihe eauation of differences for D^ it
was necessary to evaluate the detenninant
a =
1, 1, 1, 1, 1
c, a, 0, 0,
h^ c, a, 0,
0, J, c, a,
0, 0, J, c, a
(w rows),
V
MB. SCOTT, KUTES ON DETERMINANTS.
183
and it is readily proved that
« =
a
(-IP J («•-»") + «"*'-»'
>*N-1
* a + b-^c u — v
§ 3. It is known that
a + &+ c
=JP«-i?*(«lJ«8)---0»
where ^ (o^i a,, ...aj denotes as asual the product of
(«a "•«»)} •••(«« "Oj
and^^i is the sum of the products n — t at a time without
repetition of the quantities ttj, a^, ...a^.
The simplest way to prove this seems to be to equate the
coefficients of a?' in the identity
a^ , a, , ...ttj, 1
O OS •••«») ^
I do not think it has hitherto been noticed that this
theorem may be extended. If in the array
we erase the r columns which contain the w"*, v% t(>*', ...
powers of the letters, and denote the determinant of the
remaining columns by Z)^ , . ... we can shew that
n
•»,»,»,
?*(«„«,,... aj,
by forming an identity as above and equating coefficients.
We might also evaluate the determinant D^^ „ ... by the
method given by Cayley, Salmon, Higher AlgeSra^ p. 290.
But the above method has the advantage in this instance of
giving a determinant on the right of order r instead of w, and
in applications r Is generally less than n.
184
MR. S(X)TT| NOTES ON DETERMINANTS.
§4. By substitutmg the exponential valaes for the sines
and cosines, we can show by aid of the first theorem in § 3
that for 2n angles
sin aj...sin a^ f =2
cos aj...cos a^
sin 2aj...sin 2a^
cos2aj...cos2a^
sin na^...sinna,
cos na^. ..cos Tia,
,«»-S"+l
n sin J (^i — ^i) ^i
{i> k).
Where iS=2cosi K + «8+-«-H»i.""^»+i'""-"^8j
is formed by dividing the 2n angles ipto two sets of n in all
possible ways, and taking the cosine of half the difference of
(2n) !
the sums of these two sets. Thus i9 is a sum of i , ,
terms.
§5. By aid of the theorems in §3 and the theorem for
expanding the determinant formed by compounding two
rectangular arrays, we can shew that :
(a-a'r, (a-/3'r,...(a-\'r
(/g-aT, (/3-/8'r,...(/3-\'r
(X-a'P, (\-/8'p,...(\-X'P
Where here and in what follows a, /3, ...\; a', ff, ...X' are
any two sets of n quantities, and (7, is the product of all the
coefficients in the expansion of (1 + xf.
(a-a')",...(a-V)*
t\n
a
= ^?*(a,/3,...\)r»(a',y3',...V)/-
1
(X-o')",...(X-X')
Where
' a — <x', fit — yo , ...a — X
/ =
A» "^ fit • A "- fS ^ •••A#"~ A#
is a function formed like a determinant, only all the si^s are
+ instead of ±. (I take this notation from a " Smith's rrize "
paper, Jan. 30, 1873, set by Prof. Cayley).
MR. SCOTT, NOTES ON DETERMINANTS.
185
If u^{x-'OLy){x-py)...{x"\y\
V = (a; - ay) {x - /3'y). . .{x - X'y),
Using. the notation of invariants, Salmon, Higher Algebra^
§ 151. In like manner we obtain the covariant determinant :
(a -a')", ...(a -V/, (a -a;)*
(X - aT, ...(X - X')*, (X - xj
(x— ol) J ,, •(x — X ) ,
(a -a')"*S ...(a - XT', (a - a;)
=(-l)"C7,?l(«,A-^)
.«+l
= (- 1)
«+l ^n+l
C.
(\ - a'rS . . .(X - \'P, (\ - «)"*'
(a: -a'rS ...(«->'')"*',
Where
ot — ct, ct— /8,...a — Xjflt — a;
xi:Ka')/3', ...X')«t;J.
/=
X — Ct, X — p», •• .X ~" X , X — a?
a? ~~ OL J X "~ p J •• tX ~" A
§6. By slightly varying the form of the arrays we
compound, we get other theorems of the same kind, e.g.
(a-aT, .-.(a-X')*, 1
(X-a')*, ...(X-X7, 1
1 ... 1
(a-a'r\...(a-X7«, 1
?i(a',/8',...X').
G
(X-aT**, ...(X-X')"*\ 1
1 ... 1
Where
(X — (x,...(X""X,l
x?i(a',/8',...X')/'.
2" =
A"-OC, •••A""A<, 1
IX — CX, ...X —
1 ... 1
= (12)"'*^. — .
186
MB. SCOTT, NOTES ON DETEBMINANTS.
By aUowing a, ^, 7, ...X; a', ffy ...X' to coincide, we get
invanant and covariant determinants for a single quantic.
§7. If
where
Then
7=^ iff.
2D^
Where 2)= | a,-^ | is the discriminant of u and 8 is the snm
of the n determinants obtained by substituting for each
column of i) in succession the corresponding column of the
discriminant of t; ; k is supposed to be incapable of becoming
negative.
This is readily proved by transforming u and v by a
simultaneous linear transformation! each to the sum of n
squares.
By substituting pa^ for the coefficients of Uj and then
differentiating with respect to p^ we get other integrals.
§ 8* If there are two sets of n quantities a, b^ ...Z; a, ffj .,,\
and if
Then
and
cfj„ •••»!•
1 ... 1
= 0if« = n(r-l) + 3,
= 0if« = n(r-.l) + 2.
These two theorems include most of the determinant
relations in geometry. For example, if we have two groups
of points, fl?, y, z^ ^, 17, (T, and if a^ denote the ratio of the
sauare of the line joining the r^ point of the first, and the
«^ point of the second group, to the square of the parallel
semi-diameter of the ellipsoid
aj* y* ^* — 1
a (T
Since
a o c
H
PEOP- SYLVESTER, NOTE ON CONTINUANTS.
187
We see that
^ii» •••^is
^Sl> •••^66
= 0,
a relation between the lines joining two. groups of six points
in space.
it is also easy to shew that
^It) •••^16
^51) •••^86
= 0,
if the five points of one group lie on an ellipsoid
{x-pf ly-qf [o-rr _.
o" + h* ^ c- ~ '
nmilar and similarly situated to the given one.
If the ellipsoid become a sphere we can replace Or. by the
squares of the lines joining the points.
It may be remarked that such space relations connected
with an ellipsoid are not really more general than those
connected with a sphere ; for they are what the relations in
an ordinary space become when it is altered by a homogeneous
pure strain, such that the sphere
aj'+y* + «' = -B'
becomes the ellipsoid
^« + J2 + ^ - 1-
NOTE ON CONTINUANTS.
By Professor /. /. Sylvester,
To find the number of terms in the cumulant or
continuant (^i) 0,9***0) "^^ ^^7 pi^oceed as follows:
l^ There is the term a^, a^, a,, ...a^.
2^ The number of terms of the first order of degradation,
i^e. obtained by leaving out any pair of consecutive elementS|
isn-l,sayw„,j.
3*". The number of terms of the second order of degradation
obtained by leaving out any two pairs of such, i.e. by leaving
188 PROP. STLVBSTEB, NOTE ON CONTINUANTS.
oat the first and second and some other pair of those that
follow the second, the second and third and a pair of those
that follow the third, the third and fourth and a pair of those
that follow the fourth and so on, is
and, consequently,
(n- 3) +• (n- 4) + (n- 5) +...
(n-2)(n-3) _
„ — 2 > ®*y» **»»«•
4*. The number of the third order of degradation is in
like manner
_ (n-4)(n-5) (n-5)(n>6)
t.6. j-^ + j-g +...
^ (n-3)(n-4)(n-5)
1.2.3 *
and so in general
_ (n-r)(n-r-l)...(n-2r-l)
~ 1.2...r
Hence, the total number is
,4.r« ,x , (n-2)(n-3) (n- 3) (n--4)(n^5)
i + («-i) + 172 + 17273 +••••
Fer^/fco^ibn. In general
(;8in5 + cos5)"- (i sin0- cos5)*
=2 cosg{(tsingp+(n-2)(isingp+ ^'^"'^|f^"^^ . (tsin^P+...},
for we know that
cos 5 {(sin^p - (n - 2) (sin ^)-»+ (^-^)(^-^) (sin^p-fic.}
Bcosnd, or sinn^, according as n is odd or even.
Hence, putting
tsin^ + cos^ = ^+i\/5|
so that isinO — cos tf = J - ^ V^,
2fsind = l,
I
PROP. SYLVESTER, NOTE ON CONTINUANTS. 189
and cosd^^/5^
l + l^ i;+ 12 ^ 1.2.3 ^'•'
But because
if tt^ is the number of terms in (a^, a^, •••«J)
with the initial conditions
^0=1) ^1 = 1-
Solving this difference-equation, we shall obtain
«.=^5{(4+iV5r-(i-iV5n,
agreeing with the preceding result.
Corollary 1. The value of the continuant of the n^ order
Im7« wCa •••wC). is
which admits also of the clumsy representation
l{^ + i V(aJ* + 4)}*^^ -{ix'i V(aj' + ^)r'] - V(aj' + 4).
Corollary 2. The value of the pro-continuant of the n^
J /« /I « /I « ^. . sin(w+l)^
order (2 cos&, 2 cosff, ...2 cosa), is \ ^ ^ .
^ ' ' '^ sma
By the pro-continuant is to be understood what a con-
tinuant becomes, when in its representative xleterminant, the
oblique line of negative units are all changed into positive
units so that the matrix has two precisely similar bands of
units one above and one below the diagonal line and in
opposition with it.
Corollary 3. The integral of the partial-difference equation
limited by the conditions
_ Uix'-y)
^^ ^'*^''n{x-2y)ny'
1
I
( 190 )
EQUATION OF THE WAVE-SUEFACE IN
ELLIPTIC COORDINATES.
By Professor Cayley.
The equation of the wave-sorface
oc* 5y* C2*
when transformed to coordinateB p, j, r, sach that
a:« v^ ^ ,
=0,
a? . y . ^ .
CB« y« «»
—a+r "b+r -c+r
(that is to the elliptic coordinates belonging to the quadric
a;* «/* z*
surface f- -^ H =1), assumes the form
— a —6 — c ''
(j + r — a — J — c) (r+j? — a — J — c)(jp + 2 — «- ^i""^)=0,
(Senate House Problem, January 14, 1879).
In fact Pj 2} ^ are the roots of the equation
a? f z^
+ -71-7 = ^^7
we have therefore
(w — ^) (w — j) (t« — r) = (t« — a) (tt - J) ( w — c)
— a^(w — 5) (tt — c)--y*(ti — c) (m — a) — »*(u — a) (m — J);
whence, writing for shortness
A=^a + b-\-c , P=j? + j + r,
C=ahc , R=jpqr^
we have
aj*+ ^4- 0' = P--4,
5ca;'*+ cay'+ abz^ = B- (7,
PROP. CATLETf, EQUATION OP THE WAVE-SURPACE. 191
and thence also
a(J + c)aj'+J(c + a)y« + c(a + J)«* = 5(P--4)-(5-(7),
The equation of the wave-surface is
(Ac — {a (i + c) aj* + J (c + a) y + c (a + h) »*}
+ («'+y' + «')(«a^ + Jy* + c»")=0,
and by the formula just obtained this is
that is ^»-2^»P+^(P»+ (2)-(P(2-ii) = 0,
thatis M-(2 + r)}{4-(r+j?)}{u4-(p+j) = 0,
or, substituting for A its value a + & + c, and reversing the
sign of each factor, we have the formula in question.
It is easy to see that, taking a, i, c to be each positive,
(a>&>c), and assuming also p>q>T^ we obtain the dif-
ferent real points of space by giving to these coordinates
respectively the different real values from oo to a, a to 5, and
J to c respectively. Hence
greatest, least value, is
2 + r, a + 5, a-\-c^
J> + J, 00 , a + J,
so that r +jp, 2? + J, may be either of them=a+J + c, but
q + r cannot be = of + 5 + c, that isy + r = a + J + c does not
belong to any real point on the wave surface. We can only
have r + jp and 2? 4- ? each=a+6 + c, if2? = a + c, j = r = o,
and these values belong as is easily shown to the nodes on the
wave-surface; hence, the equations r-\-p = a-^h-\-c and
p^q=za + b + c being satisfied simultaneously only at the
nodes of the surface, must belong to the two sheets respec-
tively. And it can be shown that p + r=a + b + c belongs
to the external sheet, and j» + j = a + 6 + c belongs to the in-
ternal sheet. In fact, for the point (0, 0, V«), which is on the
external sheet, we have ^ = a + c, g = a, r = 6, and therefore
P + r^a + b + c: for the point (0, 0, V^), which is on the
mtemal sheet, either
(^= J + c, 5 = 0, r = &) or (jp = a, j = J + c, r = c),
according asi + c>aor6 + c><z, but in each case
p + q==a + b + c.
( 192 )
TRANSACTIONS OF SOCIETIES.
London Mathematical Society,
Thursday, February 13th, 1879, C. W. Merrifield, Baq^ F.R.S., President^ in
the Chair. Sir. J. Cockle, F.B.S., was admitted into the Society. Mr. B.
Hargreaves, Fellow of St. Johns College, Cambridge, and Prof. W. B. Story,
Leipzig, of the Johns Hopkins University, Baltimore, were proposed for election.
Dr. Hirst, F.E.S., communicated a paper by M. Halphen "On the number of
conies which satisfy five independent conditions". Sir J. Cockle gave a con-
struction for making Magic Squares; Prof. Ca^ley pointed out tluit the con-
struction had been given in Leyboum's Mathematical Repository. Messrs. Harley,
Henrici, Boberts, Hart, and other gentlemen took part in a ddscussion upon the
subject, and gave different constructions. Prof. Henrici spoke upon Frames.
Prof. H. J. S. Smith, F.R.S., read two papers, "On a Modular Equation" and "On
the formula for four Abelian functions answering to the formula for the four theta
functions." Mr. J. J. Walker, MA., communicated a "Quaternion proof of
Minding's theorem."
ThuiSday, March 18th, 1879, C. W. Merrifield, Esq., F.R.S., President^' in the
Chair. Mr. J. D. H. Dickson, M.A., was admitted into the Society. Mr. B.
Hargreaves, M.A., and Prof. W. B. Story, Ph.D., were elected Members.
Hr. Donald McAUster, B.A., Fellow of St. John's College, Camlnidge, was
proposed for election.
Prof. Cayley, F.R.S., spoke brieflj^, but in high praise^ of the late Prof.
Clifford's work as a mathematician, instancing more particularly the papers
" On the Canonical Form and Dissection of a Riemann's Surface," " On Mr. Spot-
tiswoode's Contact Problems," and " The Classification of Loci." The Chairman,
the Rev. A. Freeman, M.A., and Dr. Hirst, F.R.S., added a few remarks on the
loss the Society and the mathematical world generally had sustained, and
expressed the hope that steps would be taken to secure the publication, if
desirable, of any mathematical papers Prof. Clifford might have left.
Dr. Hirst made a statement respecting the "De Morgan Memorial" Medal to
be presented to the Society (to be award^ by the Society in such maimer as shall
hereafter be determined). It appeared that the Bust and the Die for the Medal
hid been executed by Mr. Woolner, and that after all claims had been made a
small sum would be still required to complete the amount requisite for the Medal
Fund. A subscription Ust was commenced at the meeting, and Mr. Tucker was
einpowered to receive the subscriptions of such members as should be desirous
of contributing to the special fund. Copies of the Medal were exhibited at the
meeting.
The following communications were made : " On Differential Equations, Total
and Partial ; and on a new soluble class of the first, and an exceptional case of
tiie second," Sir J. Cockle, F.R.S. ; " Discussion of two double series arising from
the number of terms in determinants of certain forms," Mr. J. D. H. Dickson, M.A. ;
" Two Geometrical Notes relating to Surfaces of the Second Order," Prof. H. J. S.
Smith, F.R.S.
R. TUOKBR, M.A., Mon, Sec.
THE END OF VOL. VIII.
PRINTED BT W. METCALFE AND SON, TRINITT STREET, CAMBRIDGE.
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