UC-NRLF
$B ma 3bE
GIFT OF
George Davidson
1325-1911
A'^
/Tt^-^/^/'A
MATHEMATICAL TRACTS.
No. I.
DETERMINANTS,
EELATING TO*^ THE
MODERN HIGHER MATHEMATICS:
TRACT No. 1.
DETERMINANTS.
BT
i
w
Rev. W. J. yWRJGHT, A.M.,
MEMBER OF THE LONDON MATHEMATICAL SOCIETY.
" That vast theory, transcendental in point of diflaculty, elementary in regard to its
being the basis of researches in the higher arithmetic, and in analytical geometry."
— (M. Heemite, quoted by Prof. Stlvestee in Phil. Mag. 1852.)
LONDON :
C. F. HODGSON & SOIST, GOUGH SQUARE,
FLEET STREET.
1875.
My acknowledgments are due to R. Tucker, Esq., M.A., Honorary-
Secret ary of the London Mathematical Society, for valuable assistance
rende red in passing these sheets through the press. — W. J. W.
^Jj-^i^^-^eC&^trx^ /ji-^SEc
CONTENTS.
CHAPTER I.
Definitions
Formation of Determinants
Minors
Fourth Order
Circle through Three Points ...
Multiplication of Determinants
CHAPTER n.
Minors as Differential Coefficients ...
Skew Symmetrical
Theorems
Orthogonal Substitutions
Laplace's Equation in ^
Determinants from Roots of Equations
Pairs of Imaginary Eoots
Theorem of Malmsten
Simultaneous Differential Equations
CHAPTER HL
Functional Determinants
Multiple Integral
The Jacobian
The Hessian
CHAPTER IV.
Study 1st — Discriminant in Investigating Loci
Study 2nd — Foci of Involution ...
Page
9
10
17
19
21
25
28
31
33
36
41
44
47
48
52
55
57
60
62
65
70
rwi.^d2d21
INTRODUCTION TO THE SERIES.
DuEiNQ some intervals of foreign travel, and consequent
interruption of formal ministerial labor, I resolved to
begin tlie preparation of a Series of Elementary Tracts
upon the following subjects in the Modern Higher Mathe-
matics ; viz —
Trilinear Coordinates.
Invariants.
Theory of Surfaces.
Elliptic Integrals.
Quaternions.
Upon further reflection, I have concluded to introduce
the Series by a treatise upon Determinants, brief and very
elementary, but sufficiently inclusive and rigorous to sup-
port and explain the references to this theory which are
involved in the ordinary exposition of the first three sub-
jects of the proposed list.
In undertaking this labor, I hope to turn the attention
of the students of my country, especially those who are
desirous of becoming Mathematicians, to these studies,
which at present lie considerably beyond the usual " Sci-
entific Course,^' even in our best colleges, but which the
demands of Physics and higher Engineering must soon
11 INTRODUCTION.
bring within it. I purpose, therefore, to give a strictly
elementary view of the principal developments of the Pure
Mathematics since the year 1841. I mark this year, not
only because it is the proper initial point at which to
begin the proposed survey, but because the year itself
was remarkably rich in mathematical productions.
Jacobi, in that year, exhibited the versatility of his
genius, whose power twelve years before had been proved
by his " Nova Fundamenta" in giving to the world his
celebrated Memoirs " De forrnatione et proprietatibus De-
terminantium'^ and De Deterniinanfibus Functionalibas/'
which have been the bases of all subsequent labors in the
Theory of Determinants.
In the same year, Dr. Geo. Boole laid down the prin-
ciples out of which has grown the Modern Higher Algebra.
In the year 1841, also, was published, in the seventh
volume of '^Memoires des Savans etrmigers^^' the full text of
the general theory of the Abelian Functions, although what
was known as Abel's Theorem had appeared much earlier.
Before the close of that year, last and perhaps least,
but confessedly of immense influence on the British Uni-
versities, was published Gregory^s "Processes and Ex-
amples of the Differential and Integral Calculus."
At this period Modern G eometry was unknown. Indeed,
till the appearance of Townsend's volumes, in 1863, it is
believed that the only work in the English language on
this subject was that of Dr. Mulcahy, and in any language
that of Chasles, " T/aite de Geometrie Superieure/^ which
had then been published but little more than a decade.
Mathematicians have not only introduced a new lan-
guage, which, taken in connexion with the new processes.
INTRODUCTION. Ill
makes modern mathematics absolutely unintelligible to
one who has for a few years laid aside such studies, but
also new functions, whose theory is regarded as a high
subject of research. It would be simple pedantry to
attempt an illustration of this in terms of the language
itself; but we may select a function which is well known,
and, ascending briefly the steps by which it has reached
its present development, observe something of the spirit of
modern mathematical analysis.
Take the theory of Elliptic Functions. Before the
middle of the last century, mathematicians began to
investigate the solutions of problems depending on the
rectification of elliptic arcs.
Undoubtedly the first definite progress in the right
direction was the discovery of Euler, which is recorded
in sec 7. oi Novi Comm, Petrup. for 1758-59, and which
gives the integral of the differential equation
mdx ndtf
{a-^hx-i-cx'^ + dx^ + ex^y (a-\-by-\-ci/-j-dy^-{-ey^y
The next step was taken by Lagrange, who published,
in the fourth volume (p. 98) of Melanges de PhilosopJiie et
de Mathematique de Turin/' sua. d priori solution of the same
general equation which Euler had solved tentatively for
special cases.
In 1775, John Landen published in the ^^Philosophical
Transactions^' his theorem, showing that any arc of an
hyperbola is equal to the difference of two elliptic arcs.
The extension of this theorem relating to the general
theory of transformation is still the subject of research
among mathematicians, among whom especially may bo
mentioned Richelot (see '^Die Landensche Transformation/'
b2
IV INTRODUCTION.
Konigsburg, 1868, also in several volumes of Grelle's
Journal.)
In 1786, Legendre's first paper upon Elliptic Integrals
was presented to tlie French Academy ; and from that time
onward, for a space of nearly fifty years, till his death, this
subject chiefly engaged his attention; and when, in 1825,
he presented to the Academic des Sciences the first
volume of his " Traite des Fonctions ElUptiques/' it was
supposed that the resources of the Integral Calculus in
this direction were exhausted.
About this time, however, the young Norwegian Abel
appeared upon the field ; and, by bringing into his analysis
the general Theory of Equations, was enabled to show that
what had been done was but a small part of what might
be expected ; and immediately extended the boundaries
of knowledge by proving his theorem for the com-
parison of all Transcendental Functions whatever, whose
differentials are irrational from involving the second root
of a rational function of the variable x. This is not the
place to describe Abel's theorem ; but the great research
bestowed by modern mathematicians upon the Abelian
Functions serves to show the spirit and line of a particular
analysis, and the interest which attaches to a subject,
which, under continual expansion for more than a century
by minds of the highest mathematical power, still suggests
for itself a much greater amplitude.
In the complete works of Abel, by Holmboe, we see the
ease and power of that remarkable genius, for whom the
principal mathematicians of his age, Poisson, Cauchy, and
Legendre, foresaw the wreath of an enduring fame. 0^
the labors of Jacobi in this direction, whose work.
INTEODUCTION. V
*' Wova Fundamenta,'^ appeared in the same year of Abel's
death, 1829, it is not my intention to speak. Had Abel
reached the patriarchal age of Legendre, he would still be
living to write theorems ior future generations. Abel died
before he had completed his twenty-seventh year.
In caMer 23 of the " Journal de VEcole Poly technique/'
and in the 9th volume of Liouville, and in the 18th and
19th of Gomptes Bendus, we find Abel's work proved and
elucidated by Hermite and Liouville. In these journals,
and in Gomptes Bendus since 1843, the contributions of
MM. Serret and Chasles would need especial study. So
also *^' Theorie der AhelscJien Functionen/' by Olebsch and
Gordon, Professors in the University of Giessen (1866), and
" Theorie des Fondions douhlement periodiques et des Fonc-
tions elliptiques/' by Briot and Bouquet (1859).
It is not necessary to mention the greater number of
distinguished Continental writers upon Abelian functions.
Neumann of Halle, and Eiemann of Tiibingen (1863-4),
Ivory, Bronwin, and Cayley, of Cambridge, are some of
the well known writers upon these functions.
The student, however, should not fail to study the papers
of Konigsberger (Grelle, Vol. 64) and Weirstrass on the
solution of HyperelHptic Functions {CrelUj Yol. 47) ; nor
should a paper by Rosenhain, in " Memoires de I'Institut
par divers Savans/' be omitted, as also a report by Russell
on Elliptic and HyperelHptic Integrals before the British
Association, from 1870 and now in progress.
But what is the use of such studies ? If the array of
illustrious names herein given do not suflSciently guaran-
tee their importance, let me say that it is by such abstract
and difficult labors men become mathematicians. What
VI INTRODUCTION.
then? Well, suppose that it is shown that the secular
inequalities resulting from the action of one planet on an-
other are the same as if the mass of the disturbing planet
were diffused along its orbit in the form of an elliptic ring
of variable but indefinitely small thickness, and that it is
inquired, what is the attraction exerted by such a ring
upon an external point ? The problem involves eventually
two elliptic integrals, as Gauss shows, of the first and
second kinds.
The final application, then, of the higher analysis must be
the sufficient answer to all cui bono inquirers. Take, for
instance, the original BesseFs functions in L^), YJ^), and
«=^I
Jn (z) = -- I cos (z sin ia — n(u) du),
hitherto mostly in the hands of German mathematicians,
and successfully applied to the solution of physical pro-
blems in heat, electricity, and the investigation of aerial
vibrations in cylindrical spaces. A good example and
illustration of this function may be seen in " 8tudien ilher
BesselVschen FunMlonen/' by Dr. Eugen Lommel, a paper
in Crelle, Yol. 56, and one of high value by Strutt of
Cambridge.
If the utility, then, of advanced modern mathematical
study is not to be doubted, what provision can be made
for its wider diffusion ?
Now, the work of reducing the higher mathematics to
the comprehension of ordinary readers, while confessedly
a difficult and generally a thankless undertaking, has in
some cases been attended with unlooked-for success.
Bowditch's notes upon " Mecanique Celeste/' side by side
with his translation ; Mrs. Somerville's paraphrase of the
INTRODUCTION. Vll
same original work, and the excessive elementary labors of
the Jesuit Fathers upon the Principia, were and are rewarded
with the strongest expressions of appreciation. And there
can be no doubt that similar labors will, in some circles,
always be regarded with favor. Students must early know
the goal, else their ambition may come too late. The equip-
ment of a mathematican is now a very different thing from
what it was thirty, or even ten, years ago. There should
be some way by which, in very early years, the broad field
of modern mathematics could be entered. Determinants
should be taught constantly with common Algebra ; Qua-
ternions with Geometry ; Trilinear Coordinates with the
Cartesian; and Invariants, Co- variants, and Contravariants
with the general Theory of Equations.
One grand principle should never be forgotten : the
educational value of a subject is greatly modified by the
the hands which administer it.
This is conspicuously true in mathematical teaching,
whether by books or lectures. Let this be suggested.
Every high subject has its easy elementary side, and there
it may be pierced. The works of Cremona, Helmholtz,
Tait, Sylvester, Clifford, and Cayley, may, in some of their
elementary forms, be commingled with ordinary mathe-
matical studies j and thus the ancient tasks of the student
will be expanded and enlivened by fresh contributions
from the great teachers of the world. Inspiration is
needed for study, and study must deepen the inspira-
tion.
The fundamental equations of Quaternions in i, j, Jc
are easily exhibited to a class in Geometry in such manner
as to become a source of real pleasure to them ; and thus
Vlll INTRODUCTION.
they may be incited to learn the power of an instrument
which bids fair to stand unrivalled in the field of mathe-
matical physics.
The rich stores of research and discovery entombed in the
volumes of the learned societies of Europe, and in the ma-
thematical journals, are something enormous; and my
object is to bring, in a more elementary form, some of
the more important subjects into a wider notice.
In regard to this tract on Determinants, it is very ele-
mentary, and intended to be more suggestive than ex-
haustive.
The works consulted in its preparation embrace the
entire literature of the subject ; viz. — The theory and
practiceof Determinants, by Baltzer,Brioschi,Spofctiswoode,
Salmon, Trudi, Dodgson (the two latter hardly worth con-
sulting) j the numberless papers in Crelle and Liouville,
and in the Proceedings of the Eoyal Societies, from 1841 ;
also a short account of Functional Determinants in the
Analytical Mechanics of Prof. Peirce, of Harvard; and the
chapter devoted to the subject by Todhunter, in his
Theory of Equations, and those of Boole, Ferrers, and
Whitworth.
W. J. w.
15, Eegent Sqtjahe,
London, W.C. ; 1875.
ELEMENTARY DETERMINANTS.
CHAPTER I.
PRINCIPLES.
1. Definitions. — The common and general expression for a
determinant of the nth order consists'of the arrangement of n^
quantities in n rows and n columns, as follows : —
ail
^2 2
Cli n
^n n
or, more briefly, 2 (± an «2 2 «^n«)>
which is to be understood as expressing the sum of the
1.2.3 ... n products obtained by fully permuting the n
suffixes, so that each product shall include all the suffixes, and
the several products differ from each other by at least one
variation of these suffixes.
Every variation in the suffixes introduces a change of sign.
The letters of the expression
S(± an 022 a«n),
taken from the diagonal of the square, are called the leading
letters ; and these, together with all the others of the deter-
minant, are called the constituents.
The products themselves, when formed, are called the ele-
ments*
* Called by Laplace resultants {Hist, de VAcad. 1772).
10 ELEMENTARY DETERMINANTS.
Coiisfcifcuents are called conjugate to each other, when, con-
sidered in reference to their respective rows and columns,
they hold the same positions.
Reserving other definitions till we have made some develop-
ment of the subject, let us seek some simple illustrations in
the formation of determinants.
2. Let us assume a determinant of two places, or the second
order, as
^2 62
Writing together the leading letters ai 62, we have the first
element, or product, and permuting the suffixes we obtain the
second 0-3 bi, since, by definition, a variation of the suffixes
gives a change of sign.* These two products taken together
form the eccj)ansio7i of the determinant. Hence we write
3. Connecting now these constituents with variables, let us
find the conditions of co-existence of the homogeneous equa-
tions of the first degree
aiX-\-hiy =^ a2X-\-h>yj
and suppose c to be their common value, then
a.2£» + &2 2/ = ^•
Eliminating y and x, we have
(ai 52— ^2^1) ^ = ^2^—^10 I
and {aih2—a2hi)y = aiC—a2C)
Observe (1) that the coefficient di^a— «^2&i is common to both
* Laplace has not only stated the rule for the change of signs hy
disarrangement, which he refers to M. Cramer, hut proves the more simple
rule of Bezout by permutation of the suffixes. (Mist, de I' Acad. 1-772,
p. 295.)
ELEMENTATiY DETERMINANTS.
11
variables ; (2) that this coefficient is identical with the value of
the determinant
as given above ; it may therefore be written in that form. The
same remark will evidently apply to the second members
h^c—hiG and a^c — ttgC, and we may write (1) as
«! 5i
X ==
c hi
^2 &2
G hz
»! 6i
y =
G a^
^2 ^2
G di
1
(2).
If now we regard c = 0, the second members of (1) and (2)
vanish, and we have simply
^2 h
= 0,
which must be interpreted as the condition of the co-existence
of the two equations, when their second members vanish.
4. It will be sufficiently evident, from what has preceded,
how a determinant of the second order, as last written, is to be
expanded ; viz., by multiplying together the letters of each
diagonal, beginning with the upper left-hand corner, and
connecting the products with the negative sign. Observing
this rule for forming the products, it plainly can make no
difference with the result, if we write the rows as columns ; as,
tti 0-2
h, h.
ax hi
^2 ^2
5. It is also evident that the sign of the determinant will
change when the rows, or columns, are interchanged j as,
ai hi = —
hi a^
= —
0^2 &2
^2 &2
&2 «2
ai hi
* Laplace " Sur le calcul integral.
12
ELEMENTARY DETERMINANTS.
6. Let US now examine a determinant of the third order.
Cii hi Ci
= ai
h C2
+ &,
c, aa
+ Cl
^2 &2
dz h C2
h Cs
C3 %
% &3
«3 h C3
Each of these three determinants is formed by omitting in
succession the row and column which contain ai, hi, Cj ; i.e., by
writing in the above order the remainders of the second and
third columns, the third and first, and first and second. The
further expansion may be written out by the rule already
given for the determinant of the second degree, that is, by
diagonal multiplication. Otherwise, we may write down the
leading letters ai, Z>2, Cg, and fully permute the suffixes. The
sum of all the products thus obtained, with their appropriate
signs, will express the true expansion. The rule for the signs
being, as has been stated, that every variation of the suffixes
yields a change of sign, or that an even number of permutations
gives Si plus, and an odd number, a minus sign. The per-
mutation of the suffixes of the diagonal letters (% &2 ^3) gives
six products, which result corresponds with products obtained
from reducing each of the three equivalent determinants as
written above ; as,
aih2C3—aih^C2-i-hiC2a3—hiCsa2-\-Ciazhs — Ciash2.
7. It is easily shown that, if two rows, or two columns,
become identical, the determinant vanishes ; as,
= 0,
tti ai Cl
=
ai hi Cl
a^ ^2 C2
(h h c^
^3 % Cg
a^ &3 C3
which is perhaps sufficiently obvious without multiplying out
at length.
8. We shall now proceed to show how this determinant
arises. Having shown how a determinant of the third order
may be reduced, the determinant itself being given, let us
ELEMENTARY DETERMINANTS. 13
seek, inversely, to construct a function, or its equivalent
functions, by an actual process of elimination, such that its
several products shall be identical with those of the given
determinant.*
Let us seek, for example, the condition of the co- existence
of the 'equations
aiX + hiy + CiZ = a2^ + ^22/ + ^^2 2! = CLs^ + ^sy + c^z.
If the common value be zero, then
aiX-\-hiy-\-CiZ = 0 "\
a2X + h2y-\-C2Z = 0 > (3)«
azX + h3y + CiZ = 0)
As to the manner of solving these equations so as to exhibit
the required condition, two methods, at least, are open to us.f
(a.) We may multiply the second of the equations by I, the
third by m, and add ; then, whatever the value of the variables,
I and m may be so taken as to cause two of their coefficients
with which they are multiplied to disappear — -that is, two of the
coefficients of the second and third equations ; and, since the
equations are simultaneous, that of the third must vanish
also. The equations will now contain only two unknowns, I
and w, whence these may be determined from the second
and third, and their values substituted will give the desired
condition. So far as this relates to elimination, it is similar to
the method employed by Laplace, referred to in a note below.
{b.) Otherwise, by eliminating alternately y and z from (3),
(a2h3 — ash2)x + (c2hs — h2Cs)z — 0,
{a2Ci— a^G^x-\- (1)20^—1 zC'dy = 0,
* This was also exhibited by Laplace, and its application to the resolu-
tion of linear equations. It might be of interest to compare the method
of Lagrange, in his Memoir on the " Movement of the nodes and inclination
of the orbits of planets," with the theorem of Malmsten for finding particular
integrals by determinants.
t fcJee Ferrers, Salmon, and Tait, on Determinants.
14
ELEMENTARY DETERMINANTS.
which, remembering to change signs in transposing, may
evidently be written
y ^
Z>2C3— C2&3 a^C2 — a^Cs a2&3 — ^362
Dividing now the terms of the first of (3) respectively by these
equals, we have
^1(^2^3 — C2&3)+^l('^3C2 — fl^2C3)+Ci(tl2&3 — %W =0 (4).
Had we divided in the same manner the terms of the
second and third equations of (3), we should have found
identical relations ; and, since the equations are simultaneous,
we have therefore found the required condition. If now we
perform in (4) the multiplications indicated, we shall have six
products identical with those obtained above.
We see also that (4) may be written (Art. 5)
0,
ay
&2 C2
+ h
1
C2 0^2
+ Ci
^2 62
hs C3
C3 ag
ag &3
and hence also
a, h, Ci
^2 1)2 C2
= 0.
0
h h Cz
1
9. If the products are written out, by permuting the suffixes,
it is only necessary to observe the cyclic order ; thus, if we
have a-i (^2 C3) the next function of the order or line a must
be a^ih^Ci), and the third a^ (&1C2).
Also that, while {a^h^c^^ is of course identical with itself, it
indicates the determinant equally in each of two positions ; i.e.,
«! hi Ci
=
«1 ^2 ^3
a^ Z>2 ^2
h 62 h
% ^3 C3
Ci C2 C3
The one = (^i&aO = ci\(J)2C^-\- 'bi{c2a^-\-Ci{a2l^,
the other = (^162^3) = ci\(b2C^ + diQ^z^^i) -\- a^^hiC-^-
Taken together the products are equal, but the corresponding
ELEMENTARY DETERMINANTS.
15
terms, after the first, are dissimilar. The reason of this remark
will be obvious, when it is considered that the products may
be derived otherwise than by permuting the suffixes.
10. We are in a position now to illustrate one or two im-
portant uses of determinants as a system of notation. The
equation, for instance, of the straight line passing through two
given points, may be written as a determinant
111 =0.
y 2/1 2/2
In this form it is easily remembered ; while, for practical use,
greater brevity and clearness will be ensured. Suppose one
of the points, as (x2y2), is changed to the origin; then, since its
coordinates must vanish, the determinant becomes
0.
1
1
1
y
2/1
0
X
Xi
0
Now, we know the equation of a line passing through the
origin and a given point to be ^ = ^ aj, which must be the
Xi
value of the determinant if our notation holds true. Hence
we write
111
=
2/ 2/1
2/ 2/1 0
X Xi
aj a?! 0
= 0;
and we may thus here state what will be found true generally,
that if all the constituents but one of a column or row become
zero, both the column and row which contain that constituent
may be erased from the determinant.
A second illustration may be found in the expression for
the area of a triangle in terms of the coordinates of its vertices,
the axes being rectangular.* This may be written as the
Salmon's Conies, p. 30.
16
ELEMENTARY DETERMINANTS.
foregoing, with an additional suffix, as,
1
1
1
2/1
2/2
2/3
»i
x^
X
= 0.
If, now, two of the points, as (xi ?/i), (x2 1/2), be connected with
the origin, the coordinates evidently of the other vertex become
zero, and we have, therefore, simply
yi y2 = 0.
Xi X2
Other illustrations will occur to the reader.
11. If any row or column he multiplied hy any quantity, the
determinant is multiplied hy that quantity.
a h c
=
ai hi Ci
% ^2 C2
xa h c
050.1 hi Ci
xa-i 62 ^2
ax hx ex
«! hi Ci
a^ &2 ^2
(!)•
A negative sign placed before the determinant is equivalent
to interchanging one row or column with another parallel to it.
(2).
a h
=
«i hi
=
h a
ai hi
a h
hi ai
It follows, from (1), that
and, from (1) and (2), that
x^ xy xz
=zx'
1 y z
X yi Zi
1 2/1 ^x
x 2/2 ^2
1 ^2 2^2
a h c
=zah
1 c 1
a h c\
1 Ci 1
a h C2
1 C2 1
All these results are so nearly self-evident, or so easily verified,
that it is only necessary to write them down.
ELEMENTARY DETERMINANTS.
17
12. Minors. The determinant
a he
= a
&1 Ci
+ &
ci ai
+ c
til &i
«] &i Ci
Z>2 c,\
Ca ^2
^2 h
«2 ^2 C2
may be written more briefly as
..(1)
•where A represents the primitive determinant, and A, B, and G
the several minors formed, as is evident, by omitting in turn
the column and row which contain a, 5, c. The determinant
may also be written A = aA + a^Bi + 0^2 ^i? where A, i?i, and Gi
represent the minors when the rows of the determinant are
written as columns —
a tti «2
= a
hh.
+ a.
62 &
+ ^2
h hi
h hi &2
Ci C2
C2 c
C Ci
C Ci C2
..(2).
In comparing (1) and (2), it will be seen that the first minor
in each of the two sets of minors is the same, but the others
are unlike. It is important to observe this difference. The
practical use will be seen in the solution of the following
equations :
ax -\-hy -k-cz = e,
aiX + hiy + CiZ = ei,
a^x + hiy + CiZ = e^.
Multiply the first by A, the second by J5i, and the third by Ci,
and add, and we have Ax = Ae + Biey + Gie2, since y and z
vanish. In like manner the values of y and z may be found.
It might be necessary to some readers to see the entire
process written out ; thus,
aAx -hhAy -{-cAz = e A,
ayBiX + hiBiy + CiBiZ = ej^i,
a2GiX-\-h.j,Gii/ + C2G1Z = e^Gi ;
18
ELEMENTARY DETERMINANTS.
or
a
l,h.
a+h
\h.
y + c
hh
z= e
hK:
Cl C2
Ci C2
Ci C2
Ci C2
«!
h^l
x + h
hh
y + <^i
h^l z = ei
h i
C'i c
C2 c
C2 c
C2 c
^2
h h.
x-i- h
bbl
y+c2
&&1
Z = 62
5 &1
C Ci
C Ci
C Ci
C Ci
Adding and combining, we have
a di a2
x +
h hi &2
y +
C Ci C2
z =
e ei e.
h 61 h
h h, &2
C Ci C2
h hi 62
C Ci C2
C Ci C2
6 &! &2
C Ci C2
The coefficients of 1/ and z having two parallel lines identical
vanish, and we have, changing the rows to columns for the
final expression,
a h c
X =
e h c
«! hiCi
ei hi ci
cbi h^ C2
62 &2 C2
13. If the constituents of any determinant he resolvahle into the
sum of n other constituents, the determinant is resolvahle into the
sum of n other determinants.
Let A = a^ + &-B + cO, where A, B, 0 have the meaning of
Art 12. Increase a, h, c by x, y, z, respectively,*
Ai = (a+x)A-{-(h + y)B+(c + z) G
= (a-\-oe)
Hence
Ai =
hi ci
+ (}+y)
Ci ai
+ (c + z)
ai hi
h C2
C2 ^2
«2 h
a-\-x cfci «2
=
a ai a>
+
X Qi a.>
h-Yy hi h.,
b hi h2
y h h
c +
Z Ci C2
c c
1 C2
Z Ci C2
If we had used the sign of multiplication, or division,
* Salmon's Deter, p. 10.
ELEMENTARY DETERMINANTS.
19
between the quantities a and x, b and y, &c., we should have
reached results already pointed out in Art. 11.
14. Since identical parallel lines in a determinant cause it
to vanish, we might infer the same result if two given lines
differ only by a constant factor ; as,
= 0.
ace a
ax a
a a
a a
So also, having in mind the proof of the last Art., we might
show that, when the sum of several lines differs from the given
lines only by constant factors, the same result will follow ; as,
la ■\- max a ai
lb-]- mil b bi
Ic + WlCi c Ci
In the same manner, if to any line we add the sum of the
other lines separately, or increased by constant factors, the
determinant will vanish ;* thus,
= 1
a a ai
+ m
(Xi a ai
b b h
&i b b.
G C Ci
Ci G Ci
lab + na -\-mh a b
1 tti bi nai+mbi ai bi
1 ^2 ^2 na^ + mbi a^ ^2
And, as the last determinant vanishes, the remaining one is evi-
dently the original. Hence we may easily verify the following :
\ + na -\-mb a b
l + ^o-i-j-m&i «! bi
l + woa + mSa Ci2 &2
a b G
ai bi Ci
a^ O2 Cq,
&c.
« — (& + c) be
«i— (&1 + C1) bi Ci
^-Z— (^2 + ^2) ^2^2
&c.
15. Determinants of the fourth order.
di a^ % 0^4
&i ^2 &3 'bi
C\ C2 C3 C4
di ^2 ^3 C?4
a—a2 b — 62 C—C2
% — ^2 &1 — &2 Ci — c^
^2 ^2 ^2
&C.
= 0
ttl
bi
Ci di
^2
h
Ca <^2
ag
h
C3 ^3
a^
h
C4 d^
expresses the condition of the co-existence of four homogeneous
* Spottiswoode's Deter, p. 18.
c 2
20
ELEMENTARY DETERMINANTS.
equations of tlie first degree when the second members vanish
(Art. 8). We may regard it as the sum of four determinants
of the third order, each of which gives three other partial
determinants, and each of these in turn gives two products.
The whole number of products of a determinant of this order
will therefore be
1.2.3.4,
a result identical with the number obtained by permuting the
suffixes, as a-Ji^cS^. This, as a determinant, may be expressed
as four partial determinants,
ai (&2 C3 d^, ^2 (^3 C4 (?i), «3 (&4 Ci £^2), (^i (^1 C2 d^.
This result may be obtained by the actual solution of four
equations with four variables, or the law of formation as seen
in the case of three unknowns would enable us to write
ai hi Ci di
(1% O2 C2 0^2
ttg &3 C3 d^
a^ &4 C4 d^
ai
(h
a.
a.
h
h
h
h
«i
«2
C3
e,
d,
d.
d.
d.
= «!
'2 '
■ C3
. d.
. d.
+ ^3
h, .
h, .
. da
with this exception, we could not tell what signs to write
before them.
These must be determined by considering the number of
permutations which arise between (Xi (62 03(^4) and ai^hsC^di).
First, considering ai (hiC^d^), which we assume tobepZ^^s, we
exchange suffixes with a and h, which gives the required suffix
for a. Then, let h and d exchange, which gives di, finally b and
c, when the required element is reached in all three permuta-
tions ; and therefore the sign must be negative, by definition,
since the number is odd. The third element is obtained from
di (^2 C3 f^4) by permuting the suffixes of a and c, and h and d,
an even number ; and therefore the sign to be prefixed is plus.
ELEMENTARY DETERMINANTS.
21
The fourth element with three permutations is found, in the
same manner, to be negative ; and thus the whole number of
products of this, and any other determinant, of any order,
assuming the law of formation to be general, may be written
out at once.
16. Before proceeding further, it may be interesting to work
one or two examples for the sake of illustrating the reduction
of determinants of the third order as exhibited under Art. 14,
and show how the same principles may be applied to those of
four places.
Ex. 1. — Let it be required to find the equation of a circle
through three points, say (2, 3), (4, 5), (6, 1).
We shall evidently obtain three equations by substituting
successively these coordinates of the three points in the
general equation x'^-^y^-{-2ax-\-2hy + c = 0, viz.,
4a-f 6Z; + c =-13,
8a-f-]06 + c=-41,
12a+ 2&H-c=-37.
To obtain a, h, c we have
4 6 1
a =
8 10 1
12 2 1
-2 2 0
a=2
-14 0
3 11
8
-2 2
-1 4
a = 8
-13 6 1
-41 10 1
-37 2 1
-13 3 1
-28 2 0
-24 -^2 0
-7 2
-6 -2
13
3'
Explanation. — The A, co-
efficient of a, has the two
factors 2 and 4. The bottom
row subtracted from each of
the other rows gives zero for
a constituent in two places,
which, by Art. 8, causes A to
reduce to the 2nd or lowest
order.
The absolute term of the
equation is first factored, and
then the upper row is taken
from each of the others, when
it, like the other, reduces to a A
of 2nd degree.
For the other values of the unknowns, we write for the
determinant, which has been found to be —48, successively,
22
ELEMENTARY DETERMINANTS.
A&
1 =
-13 1
4
-41 1
8
-37 1
12
8
3'
and Ac =
-13 4 6
-41 8 10
-37 12 2
61
From these values the required equation can be formed. The
reductions of the right members of these equations are
effected in the same manner as for the value of a, almost by
simple inspection.
Ex. 2. — Take the quadric (given by Dr. Salmon, p. 168 of
his Solid Geometry')
7ajH62/H52^— 4?/;3— 4a;?/ + 10aj-f 4?/ + 62 + 4 = 0,
differentiate it with respect to its variables, and we shall have
7a;-2v/ + 5 = 0,
— 2aj + 67/-2;2 + 2=:0,
-2?/ + 5;3-L3=0,
5.r + 27/-|-3;? + 4 = 0.
The determinant will then be, when written at length,
7 -
_2
0
-2
6 -
-2
0 -
_2
5
5
2
3
2 -
-2
0
-4
6 -
-2
-3 -
-2
5
1
2
3
0
-6
-6
-3
0
14
10
18
0
4
14
15
1
2
3
4
Explanation. — Obtained by taking
the last column from the first ; next,
twice the bottom row from the first,
and adding- four and three times the
same to tho second and third rows ;
then thrice the last column from the
first and second.
It is to be especially observed that
the sign changes in the determinant
when the factor 12 appears. The
reason is obvious, since the whole
determinant is
«i {h c 3^/4) - a^ (^3 ^4 <^i) + % {h^ Cj d^
and Oi = Oc, = a^ — 0,
while ^4 = 1, and therefore in this
case the determinant reduces to
ELEMENTARY DETERMINANTS.
23
= 12
= 12
3 3 1
7 5 6
2 7 6
= 12
0 0 1
-11 -13 6
-13 -8 5
-11 -13
-13 -8
= -972.
Ex. 3. — To find an expression for three points in in-
volution.
Substitute, in the determinant
1 1 1 =0,
Xi t^2 '^3
2/1 2/2 Vs
Xi = ai-\-a2, X2 = hi + h2, X3 = Ci + C2
IJl = ^1^2, 2/2 = &1&2, 2/3 = C1C2.
and we have
111
(i\ + a2 hx + h^ C1 + C2
cii^a &1&2 C1C2
= (ci — a^ (hi - C2) (tti — J2)
+ (C2— ai)(&2— Ci)(a2-&i).
K^
Ex. 4. — The following solution is given of the determinant
proposed by Dr. Salmon (p. 12, Deter.)
25
-15
23
-5
0
0 -
0
-5
= +6
-15
-10
19
5
30
-20
14 -
5
23
19
15
9
9
-5
04
-50
1
15
24
14
9
-5
Explanation. — The sum of 2nd and 4th
columns is taken from the Ist, 9 times last
row is added to the first, once and twice
the last are taken from 2nd and 3rd, the
sum of 2nd and 3rd columns is taken
from the first, the last row is added to
the 1st and 2nd, and finally twice the 2nd
row is added to the last. Sign changes
twice — 1st, when —^^.{biC^d^ alone
remains of the determinant ; and, 2nd,
when hy a still further reduction the
determinant becomes —a2{a^h-^.
0 80 -36
-12 -23 29
0 -70 72
36 X -600
= -194400.
8-1
-7 2
24
ELEMENTARY DETERMINANTS.
Ex. 5. — Notation.
I
m
n
h
m,
ni
k
m^
^2
0.
Tliis determinant expresses what will be at once recognised
as expressing the elimination of x, y, z from
Ix -\-my +nz = 0,
liX-\-miy + niZ = 0,
Z2 a? + ma y + 7^2 2J = 0,
or the condition that three straight lines may be parallel to
one plane.
Ex. 6. — Let Si, 82, 83 be three circles of the form
8,= (x~aiy+(y-iO'-ol = 0;
find the circle orthotomic to these three.
Let 8 be the required circle ; and, since 8 and ^1 are to be
orthotomic, we must have
^a,-ay + (hi-hy = cl+c'',
and, by eliminating a, h, and {a^ + P — c^) from the four resulting
equations, we have the determinant
0.
^+/
X
y 1
<+K-<
ai
Z>i 1
< + !>l-cl
a^
62 1
< + i>l-i
^3
63 1
17. Multiplication of Determinants. — We have now to
determine the product of one determinant by another. This
we may accomplish by the method of transformation.*
* Salmon, Spottiswoode, and Tait.
ELEMENTARY DETERMINANTS. 25
Let US take two systems of linear equations,
ax +hj -{-cz = Vf
ayx + hiy-hciz = v^,
aa a; + 62 2/ + ^2 ^ = '^2?
and dv + evi -{-fv^ = 0,
(ZiV + eiVi+/iV2= 0,
d^v + e^ Vi +/2 V2 = 0.
Substitute the values of v^ v^ , &c, in the second, and, collecting
terms, we shall have
{ad + ea^ -j-fa2 ) x-\- &c. = 0,
{adi + ei tti +/i a^) x + &c. = 0,
(ac?2 + 62 ci\ 4-/2 ^2) ^ + &c. = 0.
The condition of coexistence of these equations (Art. 8) will
be the determinant
ad -{-aie ^a^f hd +&ie +^2/ cd -\-Cie ■\-C2f
ad^-\-aiei-\-a2fi bdi + hie^ + h^fi cdi + CiGi + c^fi
ad2 + aie2 + a2f2 hd2 + hie2+h2f2 6(^2 + 0162 + ^2/2
= 0...(1).
But it is evident that these two systems of equations may
be treated as one, since the variables they contain are common
to both ; and we may inquire the condition of coexistence of
these six equations
ax -{-by +CZ —v = 0,
aiX + hiy-\-CiZ — Vi ^0,
ttg a? + Z?2 2/ + C2 2! — -yg = 0,
dv +evi +fv2 =0,
diV + eiVi+fiV2 = 0,
c?2 1; + 62 ^1 +/2 ^?2 = ^•
Here we may say, as before, the condition of co-existence of
SSK- '
26 ELEMENTARY DETERMINANTS.
these equations is expressed by the determinant
a h c -1 0 0 [ = 0.
aj hi Ci
(Xi2 O2 C2
0 0 0
0 0 0
0 0 0
This determinant may evidently be written
A B =AxD,
0 D
1
0
0
0
-1
0
0
0
-1
d
e
/
d.
61
/l
d2
62
/2
or
a h c \ X
d e f
ai hi ci
d\ Oi /i
a^ hi C2
^2 62 /a
(2);
but this is no other than the condition expressed by (1) ; and
therefore we say that (1) and (2) must be equal.
That is, the product of one determinant by another is a
determinant whose constituents consist of the sums of the products
ohtained hy multiplying each column of the one determinant hy
the rows of the other.
Ex. 1.
cos a cos h cos c
cos «! cos hi cos Ci
cos ^2 cos hi cos C2
2 _
10 0
0 10
0 0 1
= 1,
Ex. 2. — The equations of four planes intersecting in a point
are
Ix -j-my -\-nz +d =0,
liX + miy-\-7iiZ-{-di = 0,
Z2 a; -f wi2 ^ + ^2 2! + (?2 = 0,
kx + m3y-^niZ-^d2= 0,
ELEMENTARY DETERMINANTS.
27
and the determinant formed is evident ; but if two of the planes
pass through the axis of z, we shall have
= 0,
I
m
n
d
h
m^
th
d
h
W2
0
0
h
ms
0
0
which is simply the product of the two determinants
= 0,
k ^2
n d
111 di
which may be multiplied by the rule.
Suppose, however, we wished to interpret the latter equation
geometrically, in which case we see that either
= 0.
k ^2
= 0,
or
n d
k W3
01, d^
The first supposition marks the coincidence of the third and
fourth planes ; the second that the four planes intersect some-
where in the axis of z.
28
CHAPTER 11.
FORMS OF INVERSE AND SKEW DETERMINANTS.
18. Minors as constituents and as differential coefficients.
We have already seen that a determinant may be written
briefly by the aid of its minors as
A = aA-\-hB-\-cC.
But since in any determinant we can interchange parallel lines
and obtain the same result with a change of sign, when the
number of such interchanges is odd, we can write a deter-
minant of the third order, as above,
A = aiAi + hiBi + CiCi,
as the result of interchanging the first and second rows, and
A = a^ A2 + h2 B2 + C2 G2
for the like process between the first and third, or evidently,
in general,
A = rti,^i, + 0^2cAc Ci„cA^c (1),
where c is 1 n.
If now we write
ABC
A, B, Gi
A2 B2 G2
we have what is called the inverse or reciprocal of a deter-
minant of three places, that is, a determinant consisting of the
minors corresponding to the constitue^its of the given determinant.
19. If, now, we differentiate (I), of the last Art., in respect to
to c^ic, we must have
dA J, dA AS
= ^ic, -;i = ^c, &C.
daic da%c
ELEMENTARY DETERMINANTS. 29
That is, if we differentiate a determinant in respect to anj
constituent, the corresponding minor will be the differential
coefficient.*
Hence, for a determinant of the nth. order, we may write
d^ , dA dA ...
While this is a more cumbrous notation than that which it
replaces, it has its advantages, which will become more ap-
parent ; for example, it enables us to distinguish, at once,
between those determinants which do and do not, identically,
vanish.
Since a determinant is the same in the sum of the products
(Art. 14), whether we expand in the order of the rows or
columns, we may write
^A . dA dA
^ = ^kl T^-+^ 2 1 Cikn 1 •
ddki dak2 da^n
It is equally evident, from what has preceded, that
dA dA dA _ ^
^Ik -j rC('2k-^ (^nk-j • — U,
daic da^c da,,c
since in these products we have in fact introduced into the
given determinant a line parallel and identical with some other
line, and therefore the determinant in such form vanishes
identically.
This may be explained briefly thus : from what has pre-
ceded, it is manifest that, when we take the sum of the pro-
ducts of any line — that is, the sum of •the products of all
the constituents of that line by their corresponding minors —
the determinant subsists ; but if the minors do not correspond
with their constituents, the determinant vanishes identically ;
hence, in general,
dA ^ dA dA f.
«is ^— + a2s 3 — CLns — = ^•
dau da^c cine
* The notation followed here is the same as that of Jacobi, Baltzer,
Spottiswoode, and Brioschi.
30
ELEMENTARY DETERMINANTS.
20. We shall fail, perhaps, of our object unless we descend
to special cases.
Let US take a determinant of four places
ail •
^2 1 a 12
an
(X24
The first minor is obtained by erasing one row and one column .
the second minor by erasing two rows and two columns.
Let Ai 1 = the first minor, and ^22 the second;
then
but
dai:
= Ai,
d'A
dan
da-ii da^;
0.
=A
22 )
So also, when we take the second differential in respect to
either the first row or column, the result must be the same,
since A^ 1 does not contain any one of these constituents.
Hence, in general, we may write
d'^A r. d'A
daii da^i
= 0 or
dax 1 dai „
= 0.
21. Since an interchange of two lines efiects a change of
sign, we must indicate a corresponding change in the ensuing
differential coefficient.
Thus, while
d^ A
At 2)
dai 1 da2 2
an exchange of a-^ 3 with a^ o, ov a^i gives
d^A d'A d'A
= A,
dai 1 <^^i 2 ^^1 2 ^(^1 1 da^ 1 da^
since in either exchange the second minor is not affected, or in
general
(V-A
rPA
da^c dakk
da,k duk,
ELEMENTARY DETERMINANTS.
31
Evidently no a priori proof is needed here ; a simple induction,
as above, is sufficient : or, in other words, the theorem demands
only a clear statement, when its truth is at once obvious.
22. In the case of a symmetrical determinant (Art. 1, def.),
when a-i i = cti 2> we shall find, on difi'erentiating the determi-
nant in reference to any conjugate constituent, that the difier-
ential coefficient will be doubled, since the constituent function
is supposed to enter twice j as, if cti 2 = ^2 1 and cii 3 = cig 1,
«! 1 (Xi 2 Cf>ii
0/2 \ ^22 ^2 3
^3 1 ^3 2 <^3 3
^11 Cti2 (l\:
C(j\2 ^2 2 ^2 1
^13 ^2 3 ^^3 :
and we have
d^
= 2 J-i 2, and, in general,
^^ = 1 and
daj,c
dA
= 2^,.
23. In the case, then, of a skew,* as
the following, when the terms of the
leading diagonal are zero, and the
conjugates are of opposite signs, as
(Zj 2 = — ^2 1 and ain = — CLm;
in which case
consequently
dA
dai^
0 a^
^2 1 0
da2 1
— -4i„ — An\ — 0,
«i«
when the determinant is of the third and every odd order.
When the determinant is skew and of an even order, we
shall have
^12 = — -^2 1 and - — = 2Aci,.\
da^
* Salmon, p. 30 ; CreUe, Vol. 51, p. 264.
t Baltzer, p. 13.
32 ELEMENTARY DETERMINANTS.
That is, wlien the skew symmetric determinant is of an odd
degree^ it vanishes ; hut if of an even degree, its differential
coefficient in respect to any constituent function is equal to twice
its corresponding minor.
24. Referring again to equation (1), Art. 19, we see that,
since -= — = -4ic is a determinant of the n—1 order, it may,
da^,
as such, have an expansion similar to that equation. If, for
example, the original determinant were of the fourth order,
- — would express a determinant whose outer row and
daic
column had been erased, in other words, a determinant of the
third order.
Let us take, then, - — to represent generally a determinant
daj,i
of the n—\ order, and suppose
. dA , dA dA
A = aci 1- ac 2 -^ — a,n - — ,
da^ 1 da-c 2 da^ „
to represent a determinant of the n"" order ; then, if we differ-
entiate this equation in respect to % 1, the left member will be
identical with the proposed expression for the determinant of
the n — 1 order ; that is,
dA d^A , d'^A d^A
da^ 1 " dac 1 da^ 1 " da^ 2 <^% 1 dag „ da^ 1
The same equation, differentiated with respect to a,, 2 ^^^
aj, n , will yield similar expressions for determinants of the
n — 1 order,
d'^A
dA _ ^ d'^A ^ d^A
dak2~ ' ^ dac 1 da^^ ' '"^ ^ da^ 2 dak 2
.. a<
dA d'A
'. — = a,i ' — +
• a.
' dac n daj, 2
d^A
dac u dau «'
ELEMENTARY DETERMINANTS.
33
25. Remembering that the determinant subsists when the
constituent function, and the function of the differential
coefficient, as factors, are identical (Art. 18), we may write
but
^ = CL\c-^ ho^2c -. —
dai c aa2 c
^ dA , dA
dai c da^ ^
dA ^
dObn c '
dA
^j» 1 ~^ »
when c and 1 are different.
We shall continue to use the differential notation, and apply
it to the minors of the reciprocal of a determinant, as
dA dA dA
dai 1 dai % dai n
dA dA dA
aa<i I aci<i 2 aai ^
dA dA dA
dan 1 dan 2 dan «
We might use a different notation, as
^11 -o-i 2 -^1 n
An 1 A,
but we prefer to familiarize the reader with the one we have
adopted.
26. We now propose the following theorem : — Any deter-
minant other than skew, multiplied iy its second differential
coefficient, is equal to the difference of the products of the dfferential
coefficients an, ai2, ct.21? ^22? taJcen as conjugates.
Confining, for the present, the demonstration to a particular
case, let us write
dA
^ dA . dA
0= ^11- — -\'Cl2\:^ —
ctoi2 da-ii
an
da^ 2*
34
ELEMENTARY DETERMINANTS.
aai 2 wa2 2
0 = ai4-- l-<X2 4^ —
Multiply these equations by
d^^ d'A
^4 2
dA
da^2
dA
da^.
d^A
dai 1 da2 1 dui i da^ 2 ' dai 1 da^ ^
respectively ; and, adding the results,
/ ^^ d'A ^ ^^^ d'A ^^^ d'A \ dA
\ daiidttii daiida2 2 danda^J da^
1^^ dA ^^ d'A ^ d^A \
\ doiiduzi dan dci^i dai^daij
d'A
dai 1 da^
dA
da^
(a ^'^ \a ^'^ a ^'A \ dA
^ \ dai 1 <^^2 1 dai 1 da^ 2 dai 1 ^^2 4 ' da^ j
The right mem|per may be reduced as follows : — The first
parenthesis becomes, by making one, interchange of suffixes,
(a ^'^ +a ^'^ a -Jlj^—]
\ daiida^i da^dai^^ dai^daiJ'
T> i. A dA . dA
But A = aii- — + ^123 —
aai 1 aai 2
dA
dai^
, dA d^A , d'A
and —- — = an l-ai2i -. —
aa2 1 clai i da^ 1 dai 2 da^ 1
therefore — - — is the value of the parenthesis.
da^i
«14
d'^A
dai 4 da2 1
dA
In the same manner, the second is found equal to — — ; and
daii
so also the third and fourth, without change vf sign. That is,
the values of the third and fourth parentheses appear to have
ELEMENTARY DETERMINANTS.
35
the same sign. The essential sign must be determined from
the rule of signs.
In this case we remember that
The parentheses after the second therefore destroy each otl^er.
The multipliers used interpose to change this order in the
first and second, and hence we write as the result
d'^A
dai I da^ .
dA dA___dA_ dA
daii da^i da^i da^
This proof might, it is evident, have been made general. It
is now, however, in a form to be readily verified.
27. Theorem second. — A determinant formed from the first
differential coefficients of the given determinant may he expressed
in terms of the given determinant^ and is equal to that determinant
involved to a degree one less than its numher of places.
I
Let
an «! 2
^21 ^22
^2,
^»1 ^n2 ^nn
be the given determinant.
Its first differential coefficients, taken in order and arranged
in square form, will then be
(iA
dan
dA
dai2
dA
da^n
d^
da^i
d^
da2 2
dA
da^n
dA
da,, I
^A
dan -2,
d2
dA
' da^n
36 ELEMENTARY DETERMINANTS.
Let now A^ be multiplied by A, and we shall have
AAi =
d^ , d^ c?A , c?A
^11^^ \--"0.in^ — • a2i- h... a2»i —
daii ct<^in dan da^
^ cZA , (?A ^A . dA
%i3 r... ttin^ — »2ii r... ftin^ —
ao-ai aa2n aa2i o^ti^2»
c?A ^ „
a„i- — +... &c.
dan
&c. &c.
cZA , c?A
o^ii:} f-... ami — ■
•&c.
&c.
d^ ,
da^i
Observing these products, it will be seen that all except
those of the leading diagonal vanish identically ; and hence
we have
AAj
A 0 ... 0 = A'
0 A ... 0
0 0 ... A
or . Ai = A"-^
which was to be proved.
28. We shall now begin to introduce, as we proceed with
the general theory, some of the geometrical uses of determi-
nants.
Mr. Spottiswoode, in Yol. 51, p. 262, and Prof. Cayley, in
32nd Vol. of Grelle, have discussed the subject of orthogonal
substitutions in connection with skew determinants.*
We have already given a definition of a skew determinant;
we will now show how to effect an orthogonal transformation
of the third order, and express the values of the nine direc-
tion-cosines in terms of three independent variables, or in
general how to connect n"^ quantities by ^n (n + 1) relations,
* On the number of linear substitutions, see Journal de VEcole Folytech-
nique, Tom. 22, 38 cahier.
ELEMENTARY DETERMINANTS. 37
iw (n— 1) of them only being independent. Let ns, for ex-
ample, write the following linear equations :
x = aiiU + ai2'v-{-ai2Wy
y = a2iU + a22V + a23W,
z = a3iU + a32V-{-a3sf^f
and a derived system
X — aiiU + a2iV + asiWf
Y = tti 2 w + ^2 2 v + ag 2 w;,
Z = aiaW + ^asV + aggW,
where we will suppose aik=: — Uki and «,»=!;
therefore, by addition, we have at once
x + X = 2u, y + Y=2v, z-hZ=z2w.
If, in the first system, we find the values of u, i', ty, which
we do by multiplying the equations respectively by
dA dA dA
dtti I dOi I da^ I
and adding, when
. dA , dA ,dA
Aw = - — aj + -- — y + ~ — z,
dan da2\ dcizx
and, by a similar process, we obtain
. dA ^ dA ^ dA
Av = - — x-\- - — 2/ + -— 0,
dai 2 da2 % da^ 2
Aw = - — Xf &c. ;
dai2
whence, by substituting for the values of u, VjWj u = — — , &c.,
we obtain
AX= (2 4^- a]^ + 2 P-y + 2
\ dan I "<^'^2i
aai2 ^ CLCL22 I CLCL32
A^ = 2-— a; + 2-— 2/+ 2- A)z.
dais "«23 ^ "^3 3 /
Treating the second system in the same manner, we find
Au = -= — X + &c. ,
dan
da2i
Aw = - — Z + &c. ;
da^i
and also, by substitntion, taking value of x and instead of X,
we find
\ ' doii I dai2 dcfiz
A2/ = 2-^X + &o.,
da2i
Az = 24^X + &c.,
dag I
or, more symmetrically,
a; = C11X+C12 Y+Ci3-Z^'
y = CiiX+CiiY+CisZ
Z = C3]X-HC32 Y+C33Z .
(1).
and
where
X = Ciia5 + C2iy + C3i2!
Y= Ci2X + C22y + C3
Z = Ci3X+C2zy+c,
'83«-)
2-^- A —
cZdii " da22
Cii,
-A
dA
(2),
C22, anjd — — -^ = Cia, &c.
ELEMENTARY DETEKMINANTS.
39
Now, if (1) and (2) are connected by an orthogonal substi-
tution, we must have, by Solid Geom.,
0, &c. &c.
That is, the suras of the squares of the direction-cosines = 1,
and the sums of their products taken two and two = 0, when
the axes are rectangular. But these results immediately
follow, if we substitute (2) in (1).
Proceeding now to give to c values corresponding to any
given case, we see that the determinant must be analogous to
the following *
A =
1 n — m
-n 1 I
m --1 1
= l + ZHmH<
and, forming the minors,
Im—n
lm-\-n nl—m
l + m^ mn -|- 1
mn — l l-\-n^
and
dA .
o A
^A
l^l^ + rn' + n'
2(hn+n)
— C22 — ^'ssj
dai2 __ ^ ._
A ''' l + l'-^m' + n''
&c. &c.*
^ where Cj 1, Ci 2, &c. represent the values of the nine direction-
cosines in the given transformation.
29. Ex. 1. We may find an illustration of what has gone
before in the following well-known geometrical relations.
* The values of I, m, n are a tan ^Q, h tan ^0, c tan \e, where the system is
revolved through an an^le 0, the direction-cosines of the old axes being
a, b, c. {Crellc, vol. 51, ^. 263.)
40
ELEMENTARY DETERMINANTS.
Suppose Zmw, Zimi^i, Za^Tig^a the din^ction-coeines of three
right lines in reference to their three rectangular axes ; ai, ag, a^
the angles included between them :
P + m^ + ri^ = 1, III +mmi -j-nni = cos ai,
Z^ 4- mj + Wj =1, ZZg -\-m1n2 +nn2 = cos ctj,
Z^ + m^ + ^2 = 1, Z,Z2-|-'W?i?% + ^^i^2 = <^os as-
Now we are enabled to write
I 771 n
II Wi TZi
2 ^2 ^^2
1 cos ag COS ^2
COS «3 1 COS til
COS a.z COS «! 1
Z m n
2
10 0
h '^h ^1
0 1 U
Z2 mg tia
0 0 1
For the above equations are true for every value of cii, a.^,
and therefore true when ai &c. = 0, as
= 1,
which conforms to the condition, and is true when the lines
are at right angles to each other, giving a determinant which
has already been noticed (Art. 16).
Ex. 2. Another illustration is afforded by a determinant
which is related to equations of a higher character than we
had purposed to introduce at this stage of our progress, but
we will just notice it.
Suppose a function of Z is expressed in the following deter-
minant,
a— I d e
d b-l f (1),
e f c-l
and suppose this function be multiplied by a function of ~Z;
we may then write as the result
/(- 0-/(0 =
A-l'' B
D B-P
JE
F
(^).
F G-r I
ELEMENTARY DETERMINANTS. 41
Determinant (1) expresses an equation of frequent occnrrence
in mathematical physics, as an instance of which the reader
may examine Laplace's equation in g on the secular inequali-
ties'of the planets (Mecanique Celeste, Bk. II. sec. 56.)
Are the roots of such an equation real ? Special cases had, of
course, been resolved by the older mathematicians, as Cauchy
and others ; but the method by Sylvester (PMlosojoMcal Mag.
1852), depending upon the rule for the multiplication of deter-
minants, is more simple and elegant. The method is shown
above in (2), when/(Q •/( — 0 is given, in which we find by
expansion
A = a''-{-d' + e\ D = ef+d (a + h),
B = ¥+f+d\ E=fd + e(a-^c),
C = c'+f +e\ F=ed-\-f (b + c).
With these values (2) becomes
I'-Ll'-^MP-N (3),
where, if L, M, and N are essentially positive, then, according
to Des Cartes' rule of signs, we must have an equation for P,
and therefore for f(l), whose roots cannot be of the form
of (l—Jpy = — 2/^ ^^d therefore negative, but must be essen-
tially real. The only question to be considered is, what is
the essential sign of Jv, M, and JV? In the expansion of (2),
we shall find that the L of (3) is equal to
a'' + h^ + c''-\-2f + 2e^ + 2d\
M= (ah-dy + (ac-eyi-(hc-fy
+ 2 (af-^edy + 2 (he-fy+(cd-fy,
and N= a d e
d b f
6 f C
where L, M, N are, it is evident, essentially positive.
Ex. 3. It might be well to mention one peculiar case in the
multiplication of determinants, as exhibiting or suggesting an
easy treatment of a large number of theorems. It may be
42
ELEMENTARY DETERMINANTS.
found in Grelle, Vols. 39 and 51 ; it is also given by Salmon
and Brioschi.
Suppose
i + t~i=o
the equation to a conic, a, h the semi-axes.
If, now, we take any three points on the curve and form a
triangle, its area could be expressed at once by the determi-
nant given in Art. 8, in terms of the co-ordinates of its vertices ;
and similarly, in this case, the determinant
1
1 1
y
h
h h
X
Xy £a
a
a a
immediately suggests itself as expressing twice the area of the
a
given triangle, = db 2 — , iS being = to the area of the triangle
whose points are given (xy)^ (a^i^/i), fe^/a)- If now we square
this determinant, or multiply it by
= =f2
8_
2aV
^ y. -I
a b
^ Hi -I
a b
^ Hi —I
a b
we shall obtain a symmetrical determinant, as
O'l ^ g _ _ 48^
h b,f ~ a'})''
9 f ci
where S = \ab (- aib^Ci + a^f^ + big^-\- c^h^ - 2fyhy ;
and since, if the points are on the curve, we have
2 «,2 ^2 2 ^2 -.2
5+1-1 = 0' ^'+^-1 = 0, and ^^ + || -1=0,
ELEMENTARY DETERMINANTS.
43
aj = &i = Ci= 0, and likewise 8 = \ab (^2/^^)*,
wMch is the value of the determinant
0 h
9
h 0
f
9 f
0
where^=^+2^-l, , = ^» + &' -l,/=^+M-^-l,
a" b" a-
which can be reduced as follows :
Let c, d, e represent the sides of the triangle, and 0, D, E the
parallel semi-diameters respectively.
Then, from the nature of the ellipse, we have
>y
G^ a
D^"" «^ "^ Z>^ '
2 — ^2 "r
but feL^V^^l=^'
= 2(1-?-^)'
E
6'-
and corresponding values for — and -^i> which differ from the
other values of h, g, and/ by only the factor 2 and the negative
sign.
Therefore, by substituting, we have
therefore
45P
d?
2D^
2E'
d^ e^
W 2E^
0
Si = i.ah
cde
4iO^D''E^^
ODE
30. It must be borne in mind that the examples here given
are simply for illustration, and to satisfy the reader that the
u
ELEMENTARY DETERMINANTS.
principles employed are capable of wide application in all
Co-ordinate Geometry.
Two theorems will now be added, whicli tbe reader will be
able to prove in a manner more or less general.
1. The square of a determinant of an even order can he ex-
pressed hy a shew symmetric of an even order.
2. While a symmetric shew of an even order does 7iot vanish,
its inverse is a symmetric shew determinant.*
31. Let us now consider briefly determinants arising from the
roots of equations.
It is well known that, by Sturm's theorem, we find the number
and places of real roots — that the imaginary roots enter by pairs,
and are equal in number to the variations of signs of the leading
powers of x in all the functions.
Let
1
C2
-1 ^n-1
^2
be the determinant formed from the roots of the equation
Substituting c for x, we wiite
c^+c^n-icr'+&c.
&c. &c.
2C-^..Oo=0,
= 0,
'■+ ...=0.
* These theorems have, in fact, already been exhibited, but their appli-
cations to linear equations generally will be seen in Crelle, Vols. 51 and 52,
and, for earlier investigations of the theory of Substitutions, see Euler, Vols.
15 and 20 of Novi Commentarii Acad. Fetrop. Compare also the formulas
given by Rodigues in Liouville, tom. 5, with those of Euler here cited in
N. G. A. P. under Be motu corporum rigidoruni.
I
ELEMENTARY DETERMINANTS. 45
Let these equations be multiplied by any indeterminants, as
fCi, K'a ... <c„ , and assume
Kicl + K^cl + K^cl + &c. =v (1),
also '^1 + ^2-1- 'fn = 0,
K1C1+K2C2+ JCnCn = 0,
'^iC^^ + 'c.c-H K,,Cl'' = 0',
whence, by a short algebraic process, we shall find
0,= -l(^,c»;+fC2C^ K^cl) (2).
By differentiating the given determinant and employing
the value of v, we have, from the determinant,
dA . , dA , , dA s .
s;f' + 5^-''^+ d7<^^'
and, from (1),
_ dA V _ dA v^ dA V
"^"■^'A' "'"dd'A' "''"d^'A!
which values, substituted in (2), give
0.
which evidently represents the sums of the combinations of
the roots taken n—s and n — s.
Let us now seek for A in terms of the involved roots by
their differences.
Let s = n—1,
and 0 (x) = (x—Ci) (aj — Cg) ... (ic — c„) ;
and since /Cj, x-g, &c. are any values
V . V V
*■! — — 7 — \J '^2 — 7 — \i ...... '^w — T~7 \»
^l(Ci) ^i(C2) (pi{Cn)
1 1 dA 1 1 dA
therefore
^i(cO A dc^^-'' 9i(cn) A (ZC^'
46
but
ELEMENTARY DETERMINANTS.
dc'l-'
,«-. ,n-.
= Ai.
Let (J)' (x) designate
therefore
(aj— C2) (x—cs) (aj-c„),
1 _ 1 dA,
(pi (ca) Ai del
Similarly, if we put
d^i _ A dA2 _ .
n-2
dA
dc„_i
n-2
A„.i = 1,
and 02(«) = («— C3) ... (a; — O, ^aW = («— ^4) ... («— c„)j
there will result
1 1 ^A, 1 1
^^(Ca) Aa dcj ^' 0n-2(c„.i) A„_i'
whence, by multiplication, member by member,
A = 0'(<^i) ^1(^2) 02(^3) ... «?>n-2(c„-l)
= (Ci — C2)(Ci-C3)...(Ci — c„)...(c2-c„)...(c,»^i -O (3),
which is the product of the differences of n roots expressed as
a determinant.'
All this is easily generalized as follows : —
If, in
A =
C2
cl o\
^1 ^2
w,
we consider that this determinant would vanish if Ci = C2, and
that therefore Ci — Co, must be a factor, and what is true of these
ELEMENTAET DETERMINANTS.
47
two roots is true of all the others considered two and two ;
hence we are enabled to write (3) at once.
Or we might prove trae generally the method which is here
exhibited as a special case,
= (Cl-C3)(C2-C3)
Ex. : Prove that
1
1 1
=
0
Cj 02 Cg
< < ^l
C1-C3
-Ca)
1 1
C1 + C3 C2 + C3
=
0
^2"" C3
C3
1
1
1
1
Cl
Ca
Ca
C4
^;
<
^I
^'5
r-t
<
^'t
r,J
i
2
a
4
= (c,
(Ci-C3)(C2-C3)(c3 — Ci).
C4) (Ca - C4) (Ca - C4) (C2 - C3)
(Ci — C2) (Ci + C2 + C3 + C4) .
32. If now we proceed to form the square of (4) of the last
Art., we may write the result
80 Si 8,,.
81 82 Sn
8n-\ 8n
8„
where 80, 82, 8n, &c. express the sumof the first, second,
and nth powers of Cj, Ca, &c.
Thus, for example.
1 1
80 8i
81 82
= (ci-c2y.
In the same manner we shall find
^0 ^1
8,
^1 82
8,
S, 8,
8,
= 2(Cl-C2)'(c,-C3)'(C3-Cl)^
&c. &c.
where 2 = the sum of the products.
These determinants are of great practical value in the theory
of equations, inasmuch as, with their aid, as with the functions
of Sturm, we determine the number of variations of signs, and,
48 ELEMENTARY DETERMINANTS.
as stated at the beginning of the preceding Art., this determines
the number of pairs of imaginary roots.
But if these determinants are all positive, there will be no
variations, and consequently all the roots of the equation will
be real.
To those acquainted with the general theory of equations
these hints will be sufficient to show the bearing of* determi-
nants upon this subject ; the real object in this and the pre-
ceding Art. being to prepare the way for the solution of linear
differential equations by the use of the determinant notation.
33. When n~l particular integrals are given, to find the n*^. *
Let us take the general linear differential equation, coefficients
being constant
0+^£^^+ 4^^^=« -(i)-
If we separate the signs of operation from those of quantity,
the part involving only signs of operation and constants may
be considered as an operation performed on ?/, as
/(£)»=»•
From which we get ?/ at once explicitly, if we are able to
perform the inverse operation
This we cannot easily do in its general form, but we can con-
ceive the operation f li-) to be made up of certain binomial
operations, and then perform the inverse operation for each of
these. We will, however, in this case proceed in a different
manner.
Let us first assume the n particular integrals, that is, values
* See Malmsten, in Crelle, vol. 39.
ELEMENTARY DETERMINANTS.
49
that will satisfy (1), as yi, 2/2 ...^/n; coefficients now being
variable.
Proceeding as in Art 3] , and placing
'^•12/J +'^22/2 ICnVl =^
f^i^i +'^22/2 K^yl =v
we obtain
•^i2/r'+'^22/r' '^nv:
0
Ar=-^(K,y^^ + K,y;+...K,rJ
C^h
Solving, as before, for the values of tbe indeterminates
»:i, K2 ... K-,,, and substituting in (2), we find, since
A = ■
y\ y\
vT" yV
y\
yV
that
In differentiating this determinant, we get A', or
r/A
^A
A' n ^i^ I n "^
dy:-'''
(4).
therefore
Resuming now equation (1) :
Let us suppose the n—1 particular integrals 2/1, 2/2 > ••• Vn-i
are known, f
Let y = yiKi +2/2*^2 + 2/n-l'^n-l
0 = 2/1 '-''l + 2/2 4 + Vn-l f^'n-l
0 = y\K[ +2/2 '^2 +•>... 2/n-i<-l
o = 2/r.; + 2/2"'.;+ 2/::?Cj
(5).
* r, n, « — 1 do not, of course, indicate powers,
t Crelle, vol. 39, p. 94.
50 ELEMENTARY DETERMINANTS.
Solving, we find
2/2 2/3
yl y\
y. 2/
2/n-l
2/n-l
ylil
: d=
2/3 2/4
2/3 2/i
2/3 VT
2/1 2/2
2/; 2/^
yV yl'
yn.2
2/L2
i/n-2
:: AiiA^:
A..1.
If now we differentiate successively equation (5), remem-
bering the assumed relations between the n—1 functions, we
shall have
dx
2/1 '^'1+2/2 '^'2+ y'n-\*^n-\:
2/'l''^'l+2/2'^'2 +
yn-\f^n-\
d''-'y
dx''-'
dry
dx"*
2/r
''^•i+2/r'^2+-
yl
+.
/rs + 2/rs
+
...
•^ ?» - 1 n - 1
y>
',+^^2+
yl-
i^
-1
+2(»/r'< + 2/r'< + CK.J
which, substituted in (1), give
+[^(;/rn+2;/»-']s + [^(2/;::D+22/;::!]<-i = o-
Let «■', = PA,, tj = UA-i K„= UA,, ;
ELEME NOTARY DETERMINANTS. 51
whence
k'; = u'a,+ua[, k:=:U'a,+ ua:, i^:=u'a,+ua',.
But yr\ + yr\ + CiVi
famishes the determinant
and
dA
dx
Vi 2/2
ur yV
2/1 y-i
y\ y\
j\ y%
yn-i
yl-i
yT-\
yn-i
y\-i
•'n-l
= ^,
therefore ^/P' K + yV K + yl:^ K^, = 0,
and 2/r-^ A^ - 2/r'^ + Cl A^_^ = A';
' therefore JJ'A + UAA + 2UA' = 0,
By integrating this equation, we have
u =
A'
or, substituting for JT the values of k, and A„ and integrating
again, ic, = L^^ e-/^''^ dx.
If we write
we have
A = -±- =S±
A
yn-2
yi-yi'
., = (_!)-! [dA_^,-fA..^^^
J dy'l~-
e2
52
ELEMENTAEY DETERMINANTS.
Returning now to equation (4), we see that it can be written
^' where A = Ce-f""^'.
Ar =
Ce-f^
A single instance is thus given in full, that the reader may-
judge for himself of the practical benefits of the determinant
notation in conducting intricate analytical operations.
34. In the solution of simultaneous difierential equations— r-
that is, a system of equations with but one dependent variable,
in which some form of this variable, as a function of the in-
dependent variables, must be found to satisfy all the equa-
tions— there is no reason why determinants may not be em-
ployed to effect the elimination (if this method be preferred to
those of D'Alembert or Lagrange) as in the case of ordinary
linear equations.
If, for example, we have three simultaneous differential
equations of the form,
d
Ex. 1
dt
X + hy + cz = 0,
ax + — y-i-cz
dt
0,
ax-\-Vy-\--~ 0 = 0.
ctt
The condition of co-existence is the determinant
L* \df I \ dtl
d
b
c
It
d
1
a
dt
c
y
d
a
dt
I. e.
-—. — (ah 4- ac + h'c) — - + ah'c -\- a be a; = 0,
Ldr dt J
and we can proceed to integrate at once this equation, givii
rise to only three arbitrary constants.
ELEMENTARY DETERMINANTS.
53
Ex. 2. — Let us take four simultaneous equations.
The equations of Airy, for determining the secular variations
of the eccentricity and longitude of the perihelion, will serve
as an illustration :
— u + aiV—a^v = 0,
»!«— — v — a^u = 0,
(Ml
dt
u -\-hiv' —h^v = 0,
hiu — — v'—h^u = 0.
dt
The determinant for eliminating the variables w, v, u\ v\ is
therefore
d
dt
a,
-aj
0
dl
d
dt
0
—a
0
-h
\
d
dt
-h.
0
d
dt
h
d"
d'
^^4 + (^1 + ^1 + 2a2&2) ^2 + (aA-a2hy = 0 ;
which can readily be integrated, and is, of course, symme-
trical for either of the variables. This may be regarded as
the equation in u.
35. One other example upon this point, and then we shall
proceed to another subject.
Suppose we have a pair of linear partial differential equa-
tions, as
-J- dx + —- dy = 0,
ax dv
dU ^^ dU
dx dy
= 0,
54
ELEMENTARY DETERMINANTS.
where Z7 and V are functions of x and y ; then the condition of
the dependence of these functions is expressed by the deter-
minant dV dV ^
dV
dx
dV
dy
dU
dx
dU
dy
This leads us to the consideration of what are caWed functional
determinants ; and the general proposition is that, when afunc-
tional detenninant of a system of functions vanishes, it expresses
the condition of dependence of the functions ; that is, we may test
the dependence of functions in a manner analogous to that
which we have employed to determine the co -existence of
linear equations.
55
CHAPTER III.
FUNCTIONAL DETERMINANTS.
36. As this subject is supposed to present some difficulties,
and is of the highest interest in connection with geometrical
researches, we shall seek in the first place to exhibit some
of its principles in a very elementary form, and then proceed
to show the field of application.
Suppose we have a series of functions v^, V2 ... v,, of as many
variables aji, x^ ... ot^^ and by virtue of the relationship of these
functions we are enabled to find
f(viV.i ... Vn) = 0,
in other words, that they are connected by an equation which
vanishes identically. This relationship is expressible as
a determinant* (Art. 35)
dvi dvi dvi _ r.
dxi
dx2
dx,
dv2
dxi
dv-j,
dx2
dv.
••■ dx.
dvn
dv,.
dv,
dxi
dx-i
dx,
Then we say -^i, v^ ... v^ are
To fix our thoughts by an illustration, suppose
Vi = x + 2ij-{-z,
V2 = x—2y + Sz,
V3 =z 2x1) —xz + 4iyz — 2z^f
* On this subject, see Jacobi, Crelle, Vol. 22.
Vol. 51. Pierce's Analytical Mechanics.
Spottiswoode, Crelle^
66
then
ELEMENTARY DETERMINANTS.
0
dvi
dx
dvi
dy
dvi
dz
dv2
dx
dv2
dy
dv2
dz
dv^
.dx
dvs
dy
dvs
dz
becomes xy
2
-2
2y—z 2x + 4iz ~-x+4iy — 4iZ
reducing, we find
^4,(^^x + 4^y-4^z) + 8(2y-z)-2(2x+4z) = 0,
wMcb, vanisHng identically, sliows the functions Vi, ^2, v-^ to be
dependent.
37. Let us suppose the connecting equation to be
If now we differentiate this equation in respect to any one of
the functions Vi, v^ ..- v,, under consideration, regarded as
functions of the variables ajj, X2 . . aj„, we must have, in general,*
dF_dFdxidFdx2 ^ ^n.
dvr dxi dvr dx2 dVr ' ' ' dxn dv^ '
when F = Vr the left member of this equation = 1. And, in
general, if we replace F by v„ we shall have, when s = r,
1 =
dv.
dxj . dVg dx^
dxi dv^ dx2 dvg
dv,
dx,,
If, however, s is not equal to r, we must have
0
_ dvj^ dxi dVg
dxi dVr dx2
dx2
dvy
dvs
dx„
dx_„
dv.
dx^
dVr
(!)•
(2).
* Jacobi in Crelle, Vol. 22.
ELEMENTARY DETERMINANTS.
In the same manner,
dXr dVi , dXr dVn _ ^
dVi ' dXr dVn dXr
dxr dvi , dXr dVn Q
dvi dxg dVn dxg
57
(3),
(4).
By means of (1), (2), (3), (4), we are enabled to solve a
system of equations analogous to the following : —
dv
dv
Vdx'^'^'dx,
dvi , dvi
dx dxi
dVn I dVn
dx ^ dxi
y>
dv
dXn
dvi
dXn
dx^
(5).
If we multiply these equations by
dx da; dx o n
, , (KC. &c.,
dv dvi dVn
and add, we shall have, by virtue of (3) and (4),
dx
dx , dx
dv dVi
dxi .
dxi
dvi
CiX„ , UXn
dv dvi
dv^
dxi
dv^
, dx^
dVn
(6).
It is evident that, if the given functions v ... v^ are in-
dependent, and y = .yi = y,, = 0, then t^ = % = it„ = 0; or, in
other words, if the given functions are independent, then (5)
and (6) reduce in turn to 0.
38. The question now arises, how shall we express the
solution of such systems as the above in the ordinary language
58
ELEMENTARY DETERMINANTS.
of determinants P If we examine the solution of system (5)
of the preceding Art., we shall see that what might be
denominated the modulus of transformation is the determinant
A =
dv dv
dx ' dx^
dv
dx,.
dvi dvi
dx dx-i
dvi
dx,,
dVn dv,,
dx dxi
dVn
dXn
or
^dx ^ dx^
dv dv
A dx„
^Tv
= A»,
dvi dvi
A^*
dvi
••• •• •
. dx . dxi
dv^ dv,,
^dx„
dVn
since, manifestly, writing
dx dx
dv ' dvi
dx
dVn
= A',
dxn dx^
dv dvi
dx,
dVn
we must have
AXA' =
1 ...
(!)•
Hence the notation to be adopted, which is all that is required,
is sufficiently evident. If now we differentiate A' in respect
to any one of its constituents, as --^, we shall have — 7—=^ ;
dvj. , dXi
dvj,
but, in consequence of (1), we are enabled to write A = A'-—-*
7 dxi dxi
d—-
ELEMENTAKY -DETERMINANTS.
where ^4= the corresponding minor, and therefore
A'
dvk
In the same manner, in general,
dA . dxv ,1 o dA! dA dv^ dxj
— — = A —\ therefore —-- . -— - = — ~* . -,^
ndVi dVi jdXi jdVi dXi avi
d-—^ * d ^ d—-
dxj, dVk dxk
The same course of reasoning may be applied to the con-
necting equations. That is, if
/. = o /,.= o
connect the variables Xi, x^... x,, with Vi, I'a . . . v,, ; then, inversely,
if we find from /^ = 0, &c., the values of Vi, % . . . v^^ and
substitute these in the same equations, these will vanish
identically; or, since we may eliminate n—\ of the variables
from these equations, each may be treated as the function of
a single variable and the given functions ; therefore
dfk dvi dfk dv2 dfk dv,^ df\
dvi ' dXk dV'i ' dXk dv^ ' dxj, dx^
If A; = 1, 2 ... w, we shall obtain n equations, from which,
eliminating the differentials, a linear partial differential
equation will arise, which shall be satisfied by the primitive
equation under consideration, as f^ = 0.
Proceeding in a manner similar to that for obtaining (1),
we write, finally,
\ dvi dv2 dVnl \ dxi dx^ dx,,/'
The general application of these principles to the trans-
60 ELEMENTARY DETERMINANTS.
formation of multiple integrals, as
I VdvidVi dv„,
where tlie functions Vi ... Vn are connected with the same num-
ber of other variables Xi ... x^ by equations similar to those
assumed above, will not be considered.
It may, however, be remarked that, in transforming from
one set of variables to another, the formula of transforma-
tion
dx^
Ydvi dvr,
dxi ' dx„ '
dxi
reduces at once to
•n
Ydv^..
•n
. . . . dvn = V.A dxy .
•
....<&„,
•n
and Vdxi..
•
•n
. . . . dxn = V. A' dvi .
•
dv^^.
On this subject, see Baltzer, p. 64.
39. The Jacohian. — The determinant already considered,
dU,
dxi
dU,
dXn
dUr,
dU^
dxi
'" dx.
which, after Jacobi, is called the Jacobian, and generally de-
noted by /, has received considerable attention in the theory
of elimination. The principal proposition is that, if a system
of homogeneous equations be satisfied by a set of values, these
values will satisfy both the Jacobian and its differential in
regard to all the variables.
Let us take a system of three equations
Ui = 0, 1*2 = 0, 1^3 = 0 ;
ELEMENTARY DETERMINANTS.
/ will then be written
/Ini. rill- rltt^
01
dui dui dui
dx dy dz
du^ du^ du^
dx dy dz
and let us assume, what is not difficult to prove, that
dui . dui , dui
ax dy dz
du2
dx
diu
+ 2/
du^
dy
, du2
dz
dx dy dz
Solving for x, we have. Art. 12,
(1) Aa; = Z7i awi + Z/a ai*2 + TJ^ au^,
where Z7i, ZJg, &c. = the minors. We see here that, if Wi, %> "^3
vanish, the determinant vanishes.
Differentiating (1) in respect to x and ?/,
(2)
., dA dUi , dU2 , dUs
ax dx dx dx
\dx ax dx J
(3)
dA dUi ^ dU2 1 dU^
dy dy dy dy
\dv dv dv I
dy
But the first parenthesis = A, Art. 12, and the second paren-
thesis = 0.
62
ELEMENTARY DETERMINANTS.
Again, introducing the supposition Ui = U2 = U3=z 0, we
dA dA
see that — and — must vanish, since (2) and (3) in this
case, in consequence of (1), reduce to 0.
The application of this principle is obvious ; for if we have
three equations homogeneous in the second degree, their /
will be of the third, and each of its differentials of the second,'^
and these three new equations will be satisfied by the values-
common to the given equations. We have then
wj = 0, — =0,
u.^ =. 0,
dA
A ^-^ A
z
to eliminate .t^, y"^, z^, xy, zy, xz ; and therefore the ellminant^
that is, the eliminating determinant, can be formed.
When the given equations are of the third degree homo-
geneous, / is of the sixth, its differentials of the fifth ; and by
using Sylvester's dialytic process, we can eliminate the twenty-
one quantities of an equation of the fifth degree.
40. The Hessian. — We will now show how to form this im-^
portant determinant. Let V be any homogeneous function of
71 variables, analogous to (a, Z/, c, d'^x, yY, and, taking its
second differential coefficients in respect to each of the variables^
we write, for the special case.
H
ax + hy hx + cy
hx + cy cx-\' dy
This is called the Hessian, after the late Dr. Otto Hesse, of
Munich.
The degree of the determinant will be w(jd — 2), where
2) = the degree of the function, and n the number of variables^
ELEMENTARY DETERMINANTS. G3
If we connect the variables x and y witli two others u and z
by the equations x = eu -{-fz,
y = e^u+fiz,
calling the transformed function F', and taking its second dif-
ferentials, and indicating the Hessian thus formed by H' we
may write H' = Hx A^,
where A = e f
^1 /i
In orthogonal substitutions, A^ = I, and 11= H\
Hesse has shown, in the use of this theorem, that if V= 0
be an equation to a plane curve of the nth. order, the vanish-
ing of the Hessian indicates the condition by which the curve
reduces to a pencil of n right lines ; and in like manner, if
V= 0 be an equation to a surface, this surface reduces to a
cone when H vanishes.*
* Crelle, vol. 42, p. 123.
04
CHAPTER IV.
SOME APPLICATIONS.
41. In proceeding to the common applications of what has
been explained, it will be necessary to introduce some of the
terms of higher algebra ; thus,
(a, h, cjo;, yY
is called a binary quadratic, which, written fully, is simply
ax^ + 2hx7j + cy'^,
and since it is a homogeneous function it is called also a
quantic. If any quantic is to be considered apart from nu-
merical coefficients, it is written
(a, h, c'^x, yy.
Let us now take the first "expression, and linearly trans-
form it, substituting x = Ix -j-my,
y = l'x + m'y,
and we shall have Ax"^ + 2Bxy + Cy^
as the transformed function. If, now, we compare the Hessian
of the given and the transformed expression, we shall find the
relation given in the last Art. to be true, viz.,
or, in full, AC-B' = (ac-b') (Im- I'my.
Now H and H' are called respectively the discriminants of the
given and transformed quadratic.
ELEMENTARY DETERMINANTS.
65
42. When a qnantic has been transformed as above, any
function of its coefficients is called an invariant. Hence ac—b^
is also, by definition, an invariant ; and, in general, a quantic
of the quadratic class, irrespective of its variables, has no
other invariant than its own discriminant, and in such cases
the two terms indicate identical functions. Now, when we
take the Hessian of any quantic, or what is sometimes called
the second emanant, we obtain the covariant of the quantic,
that is, a function of the coefficients involving the variables of
the given quantic.
43. Study first. — Let us write the quadric surface
ax^ + hy^ + cz^ + 2exy + 2fxz -f 2hyz -\-2gx-{-2iy + 2hz-^d = 0 ;
the discriminant will then be
a e f g =0 (1),
e b h i
f h c h
g i k d
which may be formed in the manner already described, or we
may transform to any parallel a^es drawn through xy'z' by
writing x + x' for x, &c., and we shall find certain relations
connecting the coefficients a, b, &c. with a, b', &c. ; in other
words, that there are functions of the given coefficients equal
to the same functions of the transformed coefficients.
By taking the difierentials in respect to each of the variables,
we shall find the new coefficient of x to be
2(ax-{-ey-^fz+g),
and the condition that this general equation shall represent a
cone will be the determinant of the following equations.
ax + ey' -\- fz + g = 0
ex -\- by -\- hz -\- i =0
fx -{• hy -{- cz -\- h = 0
gx + iy -\-hz ■\-d=. 0 ^
(2).
65
ELEMENTARY DETERMINANTS.
The determinant of whicli is the same as (1) ; the coordinates
of the new vertex satisfying each of the above equations.
Forming now the first minors of this determinant, we have
h h i
— e
h i e
+ f
i e h
-9
e h h
h c k
0 k f
k f h
f h c
i k d
k d g
d g i
g i k
and the second minors
ah \ c k -{-ah
I k d
k h
d i
+ ai
h c
i k
&c.
Considering the first and second minors, we see that
I h i
h c k
i k d
Xh =
h i
i d
X
h h
h c
h h
i k
since
b h i
h c k
i k d
X h + hH''- hH' = (hc-li') {bd^i") - (hk-hiy ;
and we shall find, in general, that any first minor, multiplied
by a constituent, is expressible in terms of the second minors,
formed from this first minor.
Thus we shall find E ; i. e., second first minor, or
hie X c +fhk'^ —fMc^
c kf
k d g
Also A^ or
= ef
c k
—
h c
.
/"
h c
k d
i k
yk
h h i
. d =
c k
.
h I
—
h k 1
h c k
k d
i d
i d\
i k d
(3),
ELEMENTARY DETERMINANTS.
67
and
e
c kf
k d g
. d + * — * =
e g
.
c k
—
h k
. /^
I d
k d
i d
g d
If, now, A and ^ = 0, (4) becomes
h k
.
/^
=
e
1
,
c k
i d
g d
i d
k d
and (3) reduces to
c k
. b i
h k
1
k d
i
d
i d
. fk
=
e g
.
h k
g d
i d
i d
(4).
h i
fh -
e g
,
h k
i d
g d
i d
I d
And these, multiplied together, give
I b i
I i d
but minor F^ i. e.,
e b i .c24-* — * =
f hk
g i d
hence, on the supposition that J. = ^ = 0, we have F =0.
Continuing our analysis, we find that, when ^ = 0 = J^, or
^ = i^=0, wehave A = 0.*
In general, we write the following, analogous to (2) and (3),
dA d^^ d'^
da db da dd
/ d'A y
XdadiJ *
d^A d'A
d'A d'A
da
dA _
de dd de da db da di db dg
d_A ^ _^ ^ d'A_ _ d^ ^ d'A
df dd df da dc da dk dc dg
* For an extension of the geometrical applications, herein considered,
to tangential coordinates, and the determination of circular sections, see
Philosophical May., vol. xiv., 4th series, p. 393.
68
Assume
ELEMENTARY DETERMINANTS.
eE-fF-\-gG = 0
hE-cF + W = 0
iE -JcF+dG = 0
(5),
which gives the determinant for the elimination of G and F^
e -/ 9
h — c k
i —k d
F = F.E=0; i.e., F=0;
therefore F=0 and G = 0.
By equations similar to (5), as
aA-\-fF-gG = 0,
&c. &c.,
we may show that AB = 0, from which it follows that, if 5 = 0,
H=I=0', or, when A = 0, F = G = 0.
It can be shown that equations (5) are true when A = 0
and ^ = 0 in all cases.
Let us now, in view of these suppositions and results, con-
sider the nature of the surface given at the head of this Article.
Suppose
(fA
da dd
0,
dbdd
= 0, and
dcdd
= 0,
and consequently
d'A
dddh
0 ; and suppose also a to be nega-
tive ; then, multiplying the surface by a, subtract {ey -^fz + gY,
and finally let ax = ax-{- ey +fz + g, and we shall have
dc di db dk
z-\-
d'A
db dc
= 0,
an equation to a parabolic cylinder.
The three latter suppositions applied to the surface reduce
it to
dc dd
a^^- + -^4^^f + 2[
dddh dcdil dbdd
db dk db dc
ELEMENTARY DETERMINANTS. 69
(Pa
This equation, multiplied by , adding and subtracting
CiC CtCu
{^Jh^-^idS^ and finally making
d'A ,^ d'A ^ d?A ^ ^ d?A
dc dd do dd dd dh dc di*
we obtain
If, now, we multiply this by — , add and subtract * ( tt ) »
J , dA n dA , dA 1
and put -~ z tor -- 25 + — - , we nave
dd dd dh
2 d'^A dA ,2 , / d'^A y dA ,2 , fdA^ ,2 , ^'A . ^
dcdd dd
which is the equation to an ellipsoid when A is negative, and
dA d'A , ,, ...
—7, -, — r-, both positive.
da dc ad
When A is positive, and ^-, — — - either one or both negative,
^ dd dc dd
this equation represents a hyperholoid of one sheet. If A be
negative it represents a hyperholoid of tivo sheets^ if it vanishes
a cone. Also, if -7-, = 0, it is the equation to an ellijptic or
dd
d^A
hyperbolic paraboloid, according as ^^-^ is positive or negative.
In the same manner it represents an elliptic or hyperbolic cylinder
when 4^ = ^ = 0, and -f|. is positive or negative.
dd dk dc dd
To find the plane perpendicular to the chord to which it is
conjugate; i.e., the diametral plane.
70
ELEMENTARY DETERMINANTS.
Let Ij m, n be the direction cosines of the chord, the plane
in question will be, from equations (1),
l{ax-\-ey -\-fz-{-g) -\-m{ex-\-'by -\-hz-\-i) +n{fx-{-hij -)rcz + lc) = 0,
provided Z, m, n are proportional to the coefficients of thfe
variables x, y^ z ; and we shall have, in that case,
^ la + 7ne-[-nf =pl ^
• U-\-7nb-jrnh= 2'>in> (6);
Ic -h mh -\-nc = pn )
therefore, to find p, we have the determinant
a—p e f
e b—p h
c h c—p
The value of p being found and substituted in equations (6),
we shall obtain the values of I, m, »i, and thus be able to find
the three diametral planes of the surface.
We conclude this study with the simple remark that we are
not here concerned with teaching Modern Geometry, but with
an exercise for the practice of Determinants, and to indicate
their use in the investigation of loci.
44. The Jacobian, which has already been described, de-
serves, on account of its importance, a special consideration.
Study second. — (a) Let V and Vi be two functions, homo-
geneous in the second degree, / is then
= 0,
dV
dV
dx
dy
dV,
dV,
dx
dy
which, under the conditions mentioned, determines the foci of
involution of two pairs of points.
ELEMENTARY DETERMINANTS.
71
(5) Let 8i = 0, S'2 = 0, /S3 = 0 be three circles, and w = 0 the
equation to the circle orthotomic. The polar of any point on
u (^?/;3), with regard to each of the given circles, will pass
through a single point. Let zt^, U2, %, &c. represent the dif-
ferentials -7-, &c., then
da:
lui -\-mu2 +WW3 = 0,
Ivi -\-mV2, -\-nVi = 0,
Iwi + mwa + nw^ = 0.
The determinant of which is a Jacobian and = u, the equa-
tion of the circle orthotomic required.
(c) If we proceed to the conicoids, as F, Fi, F2, the equa-
tions of the three polars will be
v^x + v;'y + v;"z = 0;
therefore
= /,
dr dv" dv"
dx dy dz
dY[ dV^ dVT
dx dy dz
dv; dv; dv;'
dx f*'i dz
and is a curve of the third order, in other words, the locus of
a point whose polars, in regard to F, Fi, F2, meet in a point.
{db) It is easily shown that two conies always intersect in
four points.
Let Fand Ft intersect, and through these points draw F2.
Then the / of the three conies is the equation to the curve
which cuts F2 in six 'points.
(e) If we form the Hessian of lV-\-mVx-^nVi\ then, if we
72 ELEMENTARY DETERMINANTS.
examine the coefficients of I, m, n, we shall find them invariants
of F, Fi, F2, one of which vanishes whenever ZF+mFiH-wF2
represents two planes ; the other vanishes (as shown by Prof.
Cayley) when any two of the eight points of intersection co-
incide, and their / is a curve of the sixth order, when F, Fi, V2
represent quadrics, and this curve is the locus of a point whose
polar planes meet in a line.
London! Printed by C. F. Hodgson & Son, Gough tqiiare, Fleet Street, E.C.
i
^^4^/-^ ^^ct^ct:^^^
%:
MATHEMATICAL TRACTS.
isv.. n.
TRILINEAR COORDINATF^.
I
TEACTS
RELATING TO THE
MODERN HIGHER MATHEMATICS.
TRACT No. 2.
TRILINEAR COORDINATES.
BT
Eev. W. J. WEIGHT, Ph.D.,
MEMBER OF THK LONDON MATHEMATICAL SOCIETY.
'E06Aa> aoi flirelv Sxrirep ol yeufxerpai.'*
' Plato : Gorgias.
LONDON :
C. F. HODGSON & SON, GOUGH SQUARE,
FLEET STREET*
1877.
My acknowledgments are due to R. Tucker, Esq., M.A., Honorary
Secretary of the London Mathematical Society, for valuable assistance
rendered in passing these sheets through the press. — W. J. W.
CONTENTS.
CHAPTER I. Page
Condition of Concurrence 9
„ Parallelism 10
„ collinearity 14
„ Perpendicularity 16
Straight Line THROUGH A Given Point 20
Distance between Two Points ... 22
Perpendicular Distance of A Point FROM A Line 24
CHAPTER n.
Tangential Equation to Intersection of Two Right Lines ... 29
Tangential Equation to Point at Infinity 30
Triangular Coordinates ... ... ... ... ... ... 31
Excursus on Imaginaries 37
CHAPTER III.
Transformations of Coordinates ... ... 42
Concurrence of Straight Line and Conic 44
Excursus 45
Self-Conjugate Triangle 48
Directed Line upon the Curve ... ... ... ... ... 49
Inscribed Triangle 54
CHAPTER IV.
Inscribed Conic 59^
Brianchon's Theorem 61
Polar of a Point in respect to a Conic 64
Coordinates of Pole ... ... ... ... ... ... 65
Conic breaks up into Eight Lines 67
Equation to the Asymptotes 68
Nene-Point Circle 71
Polar Reciprocals 74
Reciprocal of a Conic 76
PREFACE TO TRACT NO. II.
Ministerial and other duties have prevented the earlier
appearance of this Tract. The delay has afforded an
opportunity to those persons who have become acquainted
with the proposed plan of this Series of expressing their
opinion upon the merits of such an undertaking.
A considerable number of Professors and Amateurs
have been pleased to signify their approval of this effort,
and to give me more than deserved commendations. I
have no object in referring to this, except so far as to
certify that the purpose in view is a good one, and that
the means adopted, while novel, are likely to prove in a
fair measure successful. I take this opportunity of again
urging upon those to whom these Tracts may come the
great importance of the study of the Modern Mathe-
matics, not only in their various subjects as educative
instruments, but also as the best media of investigation.
The extent and value of the new methods, together with
the duties of those capable of teaching them, are happily
expressed in a letter to me from M. Hermite, dated Paris,
October 28, 1876, who will probably pardon the liberty I
take with his communication, on the ground that the fol-
lowing extract is of public importance : —
IV PREFACE.
^^ Les vues exposees par vous, Monsieur, dans la pre-
face de cet ouvrage [Tract No. I.] sur les obligations
qu^imposent a I'enseignement les grands progres de la
science de notre temps, je les adopte pleinement, et,
autant qu'il m'a etc possible, j'ai essay e de m'y con-
former dans mon Cours d' Analyse de PEcole Poly tech-
nique. Une grande transformation s'est deja faite et
continue encore de se faire dans le domaine de F Analyse ;
des voies nouvelles plus fecondes et je crois aussi plus
faciles ont ete ouvertes, et c'est Toeuvre de ceux qui
veulent servir la science et leur pays de discern er ce
que les elements peuvent recevoir de Pimmense ela-
boration qui s'est accomplie depuis Gauss jusqu'a Kie-
mann.*'
I am also indebted to Prof. Benj. Peirce, of Harvard,
for a communication in reference to tlie form of Laplace's
equation for secular perturbations, referred to on p. 41 of
Tract No. I.
Without detracting from the value of the Ancient Geo-
metry, it is believed that a considerable portion might
be omitted, if necessary, to give place to the Modern,
and that our regular college curriculum would be greatly
enriched by such substitution.
In any event, I hold it to be the duty of every teacher
of Geometry, whether in the form of analysis or synthesis,
to incorporate in his instructions large masses of the New
Geometry, unless, indeed, there happens to be a Chair
devoted to this especial science.
In presenting Trilinear Coordinates, it is not proposed
to supersede the Cartesian, nor even to regard them as
inseparable from them ; but to show (as Dr. Salmon has
PREFACE. V
shown) the peculiar province and power of each. In this
Tract it has not been thought necessary to advance far
in this comparison. The student will quickly see where
he can most advantageously employ the one or the other,
'„or, leaving both^ press into his service the Triangular or
Tangential Coordinates,
All that could be attempted in a work of this size is
to give a syllabus of the more common equational forms,
and to exhibit, in as simple a manner as possible, their
genesis.
There are other systems of Coordinates which space did
not allow me to exhibit ; the quadrilinearj which involves
four straight lines as lines of reference, is one of some
importance.
Another form of Coordinates I will just mention, the
'description of which has been communicated to me by
Rev. Thos. Hill, D.D., LL.D., late President of Harvard.
These Coordinates consist in defining a curve by express-
ing the length of a perpendicular let fall from the origin
■upon a normal as the function of its direction. Thus, if
6 represent the angle contained by the perpendicular and
the axis of X, then p =f{d). These are known in this
/orm as Watson's Coordinates. Dr. Hill has modified
[this system, and succeeded in achieving some very in-
teresting results. (See Proceedings of the American
Association for the Advancement of Science, 1873 — 75.)
For my first interest in the subject of this Tract I am
indebted to a paper read before the Royal Society of
Edinburgh in 1865, and pubHshed in the Messenger of
Mathematics of the year following, the author of which.
Rev. Hugh Martin, D.D., has exhibited in that paper
VI PEEPACE.
mucli of the power and originality whicli characterise his
well-known treatise upon " The Atonement.^'
It may be said, however, that works upon Modern
Geometry do in general suggest the treatment of their
subjects by the method of Trilinear Coordinates. They
do, indeed, suggest far more than has been attempted
here. In the works of Mulcahy, Townsend, Salmon,
Ferrers, Whitworth, the recent volumes of Dr. Booth,
Carnot, Steiner, Serret, Eouche and Comberousse,
Bobillier, Cremona, Briot and Bouquet, Chasles,
together with the journals Annali di Matematica
jpura ed applicata (of which Cremona is co-editor),
Comptes Bendus des Seances, that of Crelle and Bor-
chardt, Nouvelles Annales de MatJiematiques, may be
found much that leads to, and much more that leads
beyond, that which now follows.
Books, at best, are but poor substitutes for the living
teacher. Under familiar, oral teaching the difficulties
which otherwise too frequently envelope the student
rapidly disappear. Hence I would again emphasize the
importance of admitting these subjects to our colleges as
parts of the regular course.
Since the publication of Tract No. I., the heads of two
of our leading Universities have made haste to inform me
that some parts of the Modern Mathematics I am endea-
vouring to enforce and popularise are taught in their col-
leges. I profoundly wish that these exceptions were made
the rule.
W. J. W.
Chambersbtjrg, Pa.; 1877.
TRILINEAR COORDINATES.
CHAPTER I.
FUNDAMENTAL EQUATIONS.
1. The fundamental equation of tlie straight line in Tri-
linear Coordinates is
la + mfj + ny = 0. ^
2. The apparatus for expressing this conception consists of
a triangle of reference, whose sides are called the three lines
of reference.
3. The angular points of this triangle are indicated by A at
the vertex, B at the left, and G at the right ; the lengths of
the sides opposite these angles by a, h, c ; and the perpen-
dicular distances of any point from BG, GA, AB by a, /3, y.
The distance a may be described as reckoned downward or
upward from the given point, /3 to the right, and y to the
left.
4. We may say, in general , that the position of a point in a
plane is known implicitly when its perpendicular distances
from any two sides of the proposed triangle are given. Its
perpendicular distance from the third side is then given by
these data, for manifestly
2A-(/36 + cy)_
u,
a
where A = area of given triangle.
8 TRILINEAR COORDINATES.
6. By attention to tlie figure, which scarcely need be drawn,
We are clearly presented with the equation
aa + 6/3-l-cy = 2A (1),
which is found by taking the sum of the areas of the three
triangles APG, APB, BPG, % % '^ respectively.
Z ^ ii
Observing that -— = r sin J., 77 = '* sin B, ^ = r sin (7,
z z z
(1) may be written
a sin J. + /3 sin 5 + y sin (7 = — = V,
r
where r = radius of the circumscribing circle. These equa-
tions hold, whether the point is situated below BG, within the
triangle, or above the vertex.
In the first case, by convention, aa is regarded as negative ;
in the second, each term as positive ; while in the last aa is
alone positive.
6. It will be observed also that the point is equally deter-
mined if the ratios of the three perpendiculars are given, for
we see at once that each ratio determines a locus, which is a
line drawn through the angle upon which the point is situated.
The point sought is at the intersection of these lines.
7. Before proceeding further, it may be well to exhibit in
full the process for deriving the equation of the straight line
(Art. 1).
Let Pi, P2 be the given points ; a^ljiyi^ ('■2P272 their coordi-
nates ; and Pi P2 the straight line whose equation is to be
determined. Take any point P on this line, and let its co-
ordinates be a, (^j y ; then, by similar triangles,
PPi : PP2 :: ai-a : a— 02 : ft— /3 : /3— ft : yi— y : y-yz;
or, taking the last two ratios, we are immediately presented
with the two determinants (D. 2; i,e., Tract No. I., Art. 2)
TRILINEAR COORDINATES.
Pi n
A 72
7 /3
71—72 A— A
7i «i
=
a y
1 72 "2
«i — «2 71 — 72 1
«2 A
=
A — /^2 «! — 02
likewise
If now we multiply these equations respectively by a, /5, y,
and add, we shall have at once
A 71
+ /3
7i «i
+ 7
"1 A
A 72
72 "2
"2 A
0
(1)-
Let now these determinants in this last result be repre-
sented by Z, m, n respectively, and we have
Za + m/3 + WY = 0 (2).
And gince Z, w, n represent constants, and since also a, /3, y
are the coordinates of any point of the line, this equation ex-
presses, as before stated in (Art. 1), the conception of the
general equation of the straight line in trilinear coordinates.
Cor. 1. — This is also plainly the equation of a straight line
through two given points.
Cor. 2. — The ratios represented by Z, m, n are manifestly
constant whatever the position of P on the locus, which in-
volves also the deduction that this locus must be a straight
line.
8. The condition of concurrence. — Let the straight lines be
Zia + Wi/34-%y = 0 (1),
l^a-^-m^P + n^y = 0 (2).
These equations, regarded as simultaneous, must have a, /3, y
as the coordinates of a common point. To obtain the ratios,
we are presented with the determinants (D., Arts. 10, 12)
1 1 1
=
Ml fli
+
% ^i
+
Zi mi
Ix mi ny
m2 n^
n.2 h
k 1^2
k niz n-i
b2
10
TRILINEAR COORDINATES.
otherwise
a
: /3 : y ::
^2 ^2
;
^1 ^1
^2 ^2
•
^2 ^2
Hence the trilinear ratios of the point of intersection are de-
termined.
Cor. — The general equation of a straight line passing
through their point of intersection may be written
Za + m/3 + wy = fc (Zia-f Wi/3 + %y) (3),
where h is any constant; for the locus of (3) must pass through
every point common to the loci of (1) and (2).
^ . 9. Three straight lines, as
Zia + mi/3 + %y = 0,
ZjO+ma/S + War = 0,
Zga + Wa/S + JZay = 0,
present the determinant (D. 6)
li nil Ui =0
Z2 TYli 712
as the condition that three straight lines shall have a point in
common.
10. The condition of parallelism. — Let the two straight lines be
la + mp-^ny = 0,
Zia + Wi/S + ^iy = 0.
Let us find the condition of parallelism.
Suppose a, /3, y, /, g, h the coordinates of two points in the
former ; ai, ft, yi, /i, ^1, hi the coordinates of any two points in
the latter.
If these lines are parallel, the geometry of the figure requires
a-/ : fi-g : y-h :: cii—fi : /3,-^i : yi-^.
Let us seek an expression for this in terms of the constants
of the given equations and the triangle of reference.
TRILINEAR COORDINATES.
11
Remembering (Art. 5) that
aa + 5/3 + cy = 2A,
and consequently af •\-l:)g + ch =■ 2A,
we have a{a—f)-\-h(^io—g) + c{y—K) = 0 ....
Also, since la-\-m(i-\-ny = 0,
and If ■]-mg + nh = 0,
we obtain I (a—f) +'m (P -g) +n (y—h)
Equations (1) and (2) give the eliminant (D. 39)
(1).
(2).
1
1 ]
u
=
0,
I m n
a h c
from which we derive the ratios
likewise
m n 1
h c
•
n I
c a
•
I m
a b
y
hence
Ml Hi
b c
•
rii li
c a
*
li mi
a b
>
m n
h c
'
n I
c a
•
a
m
b
'-
7}
b
C
•
f
c
^1
a,
-
a
mi
b
Multiplying each of these ratios by Zi, t^i, Wi, and remember-
ing how they were derived, we have, by restoring,
?i
These are (D. 6) the expanded determinants for
m n
+ Wi
n I
+ ni
I m
b c
c a
a b
"M
mi til
-\-mi
Ui li
+ %
1
b c
c a
li mi
a b
h
mi
Ui
::
li
mi
I
m
n
i>
mi
a
b
G
a
b
12 TRILINEAR COORDINATES.
By (D. 7) the riglit-liand determinant vanishes, and hence
Zl lUi Til
I m n
a h c
= 0*
is the condition of parallelism ; or, by reverting this determi-
nant, it can be written (D. 12)
aA + hB + cC = 0 (3),
and in this form is easily remembered.
11. Excursus on the straight line. — We have obtained (Art. 5)
the equation
a sin J. + /3 sin B-\-y sin 0 = — = a constant ;
r
and therefore we may write
la-{-m(j-\-ny-\-k (a sin J.+/3 sini^ + ysin C) = 0
as the parallel of the line
la -j- wz/3 -|- ny.
This follows from the analogy of the Cartesian coordinates,
where, it will be remembered, two lines differing by only a
constant are parallel. Also, if two equations are so connected
that their difference is ever a constant, their sum represents
their parallel and is situated half-way between them.
In the last Art., equation (3) is the result, in fact, of elimi-
nation between three equations, one of which is the impossible
equation aa-\-hj3 + cy = 0 ;
impossible at least in any finite conception, since we have
* That this determinant may rigorously be equated to 0 is evident from
ths consideration of the ratios, when it will be seen we have been, in fact,
concerned with only one equation.
TRILINEAU- COORDINATES.
13
proved it equal, in every position of the origin, to the area of
the triangle of reference. Here again, after the analogy of
the Cartesian, of which trilinear coordinates may be regarded
as a particular case,* we may interpret
aa + hP-\-cy = 0
as a line situated at an infinite distance from the origin, or we
may say that every straight line may be regarded as parallel
to the straight line at infinity.
Thus, analytically :
The ratios (Art. 8) a : j3 : y express the relations of the
coordinates of the point of intersection of two straight lines.
The actual values are evidently given by the three equations,
aa -\- bP + cy = 2A,
Zla + ^l/3 + »^ly = 0,
Zatt+WaiS + Way = 0,
where
2A
mi
ni
a b c
^2
n.
li mi ni
I2 m^ n2
Writing A for the minor in the one case, and Ai for the
determinant in the latter, we have
2AA
When a becomes infinite, Ai becomes zero. But this ex-
presses the condition of the straight line at infinity ; that is,
the point of intersection lies at an infinite distance.
But this is also the condition of parallelism of two straight
lines.
* Salmon's Conies, p. 64.
14
TRILINEAE COORDINATES.
The determinant, therefore, to represent parallel straight
lines, may be written
A =
= 0,
h c
h c
ma ^2
which identically vanishes, and where it will be seen the ratios
I : m : n are merged in, and have become identical with.
a
b : c.
12. The condition of coUinearity . — Let the three points
"lAyi) "2/32725 "aft 73 be determined in the same straight line.
We see it is only necessary to accent the a, /3, y of equation (1),
(Art. 7), change Qi to ag, ft to ft, &c., and we can write the
condition at once
Ol
^2 72
+ ft
72 "2
+
71
"2 ft
/33 73
73 "3
"3 ft
),
«! ft 71 =0,
«2 /^2 72
«3 ft
73
0.
By (D. 6)
which is the condition determining three points in a straight
line.
The following well-known theorem will illustrate this : —
Let P be a point within the triangle of reference. Through
this point let straight lines be drawn from A, B, G to meet the
opposite sides respectively in Ai, j^i, Gy ; these are the angular
points of a triangle whose sides, when produced, will meet the
corresponding sides of the first triangle in three points which
lie in a straight line.
Suppose /, g, h the coordinates of the point P ; a, /3, y those
of any point, as ^i. Then a = 0 ; and, by similar triangles,
g and li will be the ratios.
Ai will therefore be represented by 0, g, h.
For Pi, g of course is 0, / and h its ratios, since (o '. y '.'. g '. h.
Hence, in the same manner, B^ is represented by
/, 0, h.
TRILINEAR COORDINATES.
15
1.9 A
+ fi
h 0
+ 7
1 0 h
hf
[ence the line joining Ai, Bi is (Art. 7, Cor. 1)
0 3 I = 0.
/ol
Otherwise agh + phf—yfy = 0 (1),
which may be written, the line
ising the coefficients only to represent the line.
Recurring again to equation (1), (Art. 7), we see that, if
'ij 72 are each = 0, we must have, for (2) of the same Article,
Oa+O/3 + wy = 0 (2).
But this condition attaches to the line AB, which is there-
fore represented by 0, 0, 1.
The intersection of A^ 7?i and AB is therefore the concur-
rence of (1) and (2), which (Art. 8) is the point
fh, -gh, 0;
)r, by ratios, /, —^,0.
|- BG and Bi Ci will intersect, similarly, in
0, g, —7i;
-/, 0, h.
md AC, A^Ci in
[ence, since
f-9 0
-/ 0 h
0 9 -h
= 0,
bhe lines (AB, A,B,), (BC, B,G{), (AC, A^Ci) meet in points
'^hich are collinear.
13. Another illustration of the use of these coordinates is
Found in the proof that the straight line joining the middle
)oiuts of two sides of the triangle of reference is parallel to
ihe third side. If the points be taken on BC and AC, then
16 TRILINEAR COORDINATES.
equations (1) and (2) of fhe last Article will represent the lines
to be considered, remembering only to accent two of tte co-
ordinates, when (1) becomes
gK ¥h —fig)
and (2) 0 0 1.
Substituting these in the determinant of parallelism (Art. 10),
we find the required expression
am— hi = ahfi—hghif
a
h
c
I
m
n
0
0
1
by giving m and I their values ; and since, if the given lines
are parallel, h = hi, we may write
afi = hg,
which accords with the geometry of the figure.
14 The condition of perjoendicularity. — The more common
method of determining this condition is by establishing, in the
first place, the angular relation of a given straight line to two
of the sides of the triangle of reference. For this purpose the
internal bisector of one of the angles may be used as an axis.
A line drawn through the vertex A, for instance, may be re-
garded as known when its inclination to the bisector of this
angle is determined. The equation of such a line evidently is
concerned with but the two coordinates /3, y.
Two lines thus drawn may be represented by
t(^ = sy (1),
and /i/3 = Sjy (2),
and their angular relations to the internal bisector of the
angle Ahj 0 and di.
If now these lines be conceived as drawn parallel respec-
tively to the given lines,
la-\-m(D-\-ny = 0,
lia-\-mi(i-j-n^y = 0,
TRILINEAR COOEDINATES.
17
Those condition of perpendicularity is sought, we may write,
'regarding only tlie ratios of /3 and y,
(ma — lb) /3 + {na — Ic) y = 0,
(mia—lib) /3 -j- (n^a—\G) y = 0,
which are of the form of (1) and (2).
Equation (1) may be treated as follows : —
sin(y+0) : sin(l^-f^) \: t: s.
This, by composition, division, alternation, and reducing, be-
comes
tan Q : tan — :: t—s : ^ + .
Li
tan di : tan— :: t^—S]^ : ti-\-Si.
Similarly,
But the condition of perpendicularity in general is
tan 0 tan 01 + 1 = 0;
therefore, by reduction and supplying values, we get
mmia^-^-nnia^ + Ui^ (&^ + c^— 26c cos J.)
— (nli-\-nil) (ao—ah cos^) — (Imi-j-l^m) (ah — ac cos J^)
— (mni+min) {o? cos J.) = 0,
rhich, remembering that
Z>^-|-c^— 2&C cos J. = a^, c—h cos A = a cos B,
h—c cos A = a cos G,
)ecomes
III — (jnui + miTi) cos A + mwi — {nli -\- n^) cos B ♦
+ nni—(lmi-\-lim) cos 0 = 0,
Ihe condition necessary.
General Exercises.
1. To prove whether perpendiculars upon the opposite sides
leet.
We perceive that the perpendicular divides any angle of the
18 TRILINEAR COOEDTNATES.
triangle into parts wtich are the complements of the remain-
ing two angles.
Therefore the equation of AD is
cos B .p = cos C . y,
or, more fully,
/3 : y : : sin CAD : sin BAD :: cos (7 : cos B.
Similarly, cos J. . a := cos B . /3,
and cos 0 . y = cos A . a.
If we write the equations of these perpendiculars in order,
we see that a does not appear in the first or AD, 13 in the
second or BE, and y is wanting in the last or CF-, and re-
membering that these are the coefficients of a linear equation,
as, Oa + cos B .jj — cos (7 . y = 0,
&c. &c.,
and remembering also that, by Art. 8, the problem is simply
elimination between these three equations, the condition of
concurrence, as we have already seen, is presented by the
determinant
0.
0
cosi?
—COS G
COS J.
0
cos G
cos^
— cos jB
0
2. On the sides of the triangle of reference, as bases, are
constructed three triangles, similar and so placed that the
adjacent base angles are equal, and each base angle respec-
tively equal to the vertex most remote ; thus :
A,BG = AB^G = ABG,, B,GA = BC^A = BGA„
and G,AB = GA,B = GAB^ ;
then will AAi, BBi, GGi cointersect.
Since the point A^ falls without the triangle of reference,
but within the angle A, the ordinate a must be negative. The
same applies to /3 at the point Bi, &c.
We first seek the perpendiculars on a, &, c from Ai, which
are, in order,
S.sinG,, /S.sin(O+(70, S, . sin (B-\-BO,
TRILINEAR COORDINATES.
19
where
8
^^i^^i, and s,-^^^^
sm J.1
sin A
Dividing these by the first to obtain the ratios, we have for
the coordinates
of^i,
of A,
ofOi,
where / represents
— 1, h 9,
h -1, /,
sin {A + ^i)
sin Ay
sin (B + B{)
sin Bi '
>J ^ »
sin((7+O0
sin Oi
The ratios of A are
1, 0, 0,
» ^ j>
0, 1, 0,
3J ^ J)
0, 0, 1.
Hence the equation of the line joining the two points A and
Ui is (Art. 7)
0.a + g.j3-h.yz=0]
[for BBi,
for GO,,
-/.a + 0./3 + A.7 = 0:
f.a-g. (^ + 0.y = 0.
By (Art. 9) the determinant of concurrence is formed from
■these three lines ; that is,
0 g -h
-f 0 h
f -9 0
= 0.
3. In the same manner, from the same figure, prove that
(BG, B,Gi), {GA, G,A,), (AB, A,B,) respectively meet in
; points which are collinear.
20
TRILINEAR COORDINATES.
15. A straigTit line through a given point and parallel to a
given straight line.
Let (I, m, n) be the given straight line, (/, y, h) the given
point, (Zi, mi, %) the required straight line.
The condition of parallelism of
Zia + mi/3 + niy = 0 (1),
and la-\-m jj-\-ny = 0 (2),
by (Art. 10), is
that is,
ll 'W2^ 7li
= 0
>
I m n
a h c
m n
+ Wll
n I
+ %
I m
b c
c a
a h
= 0
(3).
If the locus passes through /, g, h, we must have
lif+'mig + nih=0 (4).
We are now furnished with three equations to eliminate
ZiWi^i ; viz., (1), (3), and (4).
Hence
a /3 7
/ 9 1i
ABO
0
16. To show that
h c
B S
where B =
y a
7i "i
is the equation sought, where A, B, C stand for the minors
of (3).
= 2A(a-«0,
, ^= a /3
are the second and first determinants formed from the coordi-
nates of two points (a, /3, y), (aj, ft, yj),
h c = 0 —c b
B 8 a P y
"1 A 7i
0-10 =2A(a-ai).
a /3 2A
ai A 2A
Similarly, if
8 Q
TRILINEAR COORDINATES.
Q =
21
/3 y
A n
2A(/3-A), and
a b
Q B
2A(r-yi).
^ V?. Deduced coordinates of the triangle of reference.
1st. Of the angular points.
2A
WMA,
/3 = 0, y = 0
Hence
aa = 2a, a =
At B, similarly,
., 'f. 0.
At (7,
0. 0, 2A
c
2nd. 0/ ^^e middle point of JBG.
Evidently Z//3 = area of triangle = cy,
and a = 0,
Hence 0, --, — are the coordinates.
0 c
3rd, Of the foot of the perpendicular from, A upon BO.
2A
The perpendicular = — , by 1st case.
Hence
and
2A ^ p 2A ^
— cos u = p, — COS 5 = y ;
a Ob
2A
2A
0, — cos C, — cos B
are the required coordinates.
4th. Of the centre of the inscribed circle.
The point being equally distant from the three lines of
reference, we must have
n 2A
« = p = y = --- -.
a-\-b-\-c
22
TRTLINEAR COORDINATES.
Ex.— Prove that
a = r cos A, /3 = r cos B, y = r cos G,
r being radius of circumscribed circle.
18. Distance between two points.
Various expressions may be deduced. One only is here
given ; others will be given hereafter.
Let BiCi, BiAi, drawn parallel to the sides of the triangle
of reference BG, BA respectively, be two sides of a quadrilateral
BiGiPAi inscribed in a circle whose diameter is BiP ;
(a, /3, y), (ai, P^, yi) the coordinates of Bi, P ; r = the dis-
tance between them. Through Gi draw a diameter G^D.
Join AiGi, DAi.
A, Gl = PGl + PA\ - 2PGi . PA, cos G.PA,
= PGl + PAl + 2P(7i . PA, cos B (1).
The angle at D = the angle GiBiA, = B,
AiGi = GiD sin B = PB, sin B = r sin B,
PG, = a-a„ PA = y-ri.
Substituting these values in (1),
r" sin^ B = (a-ai)H (y-yi)' + 2 (a-n,) (y-yO cos B,
which is the required equation.
This, however, may be made symmetrical with the deter-
minants formed from the coordinates of the points B,, P.
These determinants are represented in (Art. 16) by Q, B^ 8;
also in the same Article it was shown that
h c
B 8
Hence
where
2A
4AV'sin2 5
X =
a — a,, and
a h
Q B
h c
B 8
2A
X'+Y' + 2XYcosB,
Y= a h \.
Q b\
y-yi.
TRILINEAR COORDINATES. 23
Developing the values of X and Y, we find
4AV sin^^ = h\Q' + B' + S'-2B8 coaA-2SQ coaB
-2QB cos G).
By (Art. 5), 2A sin B = Vh,
Therefore
T = y ^(Q'+B'-^ S'-2BS COS A-'28Q COS B-'2QB COS G).
19. The area of a triangle from the trilinear coordinates of the
angular points.
The area of a triangle expressed as a determinant in Car-
tesian coordinates, the axes being rectangular, is (Art. 10, D.)
Ill
Vi 2/2 2/3
M/i tVn tVo
rhich, referred to oblique axes, becomes
•^ cosec (o
1
1
1
Vi
2/2
2/3
X,
X,
a^3
F
F F
A
ft (h
"i
"2 "8
Let (oj, ag, Og), (/3i, /32, i^s) take the places of x and y, and
lultiply and divide by F, and we shall have
cosec^
2F
Multiply the last row by sin A and the second by sin B^ and
then take the sum of these new rows from the first.
Observing (Art. 5) that
a sin ^ + /3 sin ^ + y sin (7 = F,
C
24
TRILINEAE COORDINATES.
we are enabled to write
Ar
ea
cosec G
Yi sin 0
2V
A\
"i
1
71 72 73
2V
/^\ 1% f\
"i
«2 O3I
72 sin G yg sin (7
ft
A
or, more symmetrically,
= -L
2F
1%
It will be observed that the angle w, between the axes, is
changed to one of the angles of the triangle of reference in
passing from the Cartesian to the trilinear system.
20. Ferjjendicular distance of a point from a line.
Let the coordinates of the point be (a^, ft? 72) j ("5 /^j 7)5
(«i5 Pi^ 7i) ^^^ coordinates of two points in the line ; r the
distance between these last-named points ; and jp the perpen-
dicular sought.
Then pr = twice the area of the triangle of which the three
points are the vertices.
By the preceding Article,
F =
-1.1- D.
r Y
But, by (Art. 18), r =^ j^ Z,
where Z = the radical part in the final value of r.
«2 ft 72
a /3 y
Hence P = -77 =
«i ft 7i
Z
__ 0^0+ .
nH+Sy,
z
(1).
TRILINEAR COORDINATES.
25
We have already seen (Art. 7) that the equatioD to a line
joining two points is
la+mp-^-ny = 0.
But the numerator of (1) is, in fact, the same expression
under another form. As general equations of a straight line
joining two points they must be identical.
Hence we may write
la + mft + ny
* (l^+m^-\- n^ — 2mn cos A — 2nl cos B — 2lm cos 0)*
C2
26
CHAPTER 11.
THE EQUATION IN TERMS OF PERPENDICULAES— TANGEN-
TIAL AND TRIANGULAR COORDINATES— IMAGINARIES.
21. It is necessary, as we proceed, to introduce the equation
of the straight line under somewhat different forms. We have
considered a point as determined by its perpendicular distances
from the three sides of the triangle of reference. A line join-
ing two of these points has thus far occupied our attention.
Let now the perpendicular distances of the three angular
points A, B, C from a straight line be jp, q, r, and let it be
required to find the equation to this straight line in terms of
these quantities.
We will assume two points on this straight line, one upon
each side of the perpendicular ^ ; d = the distance between
them.
•^ = area of the triangle formed by these two points and
the point A.
For the coordinates of the point A we have (Art. 17)
(■
^, 0, 0.
a
1
Let (a, /3, y), (a^, /B^, yj be the coordinates of the two '
given points.
Hence (Art. 19) j
i'^^ 7
2^ 0 0
a
a ^ y
«i A 71
TRILINEAR COORDINATES.
27
Mand
qd =
rd =
1
V
0 \^ 0
0
a /3 y
«! /3i ri
1
F
0 0 ?^
c
a /3 y
"i /5i yi
Multiplying these equations by aa, &/3, cy respectively, and
Iding, we have (D. 7)
{ajpa -f hq^^ -f- cry) d =
2A
a /3 y
a /3 y
= 0,
«! /^i 7i
j that is, ajpa-\-'bc[^-\-cry =-0.
This is only another form, or a special case, of
la-\-mf^-\-ny = 0,
[in which Z, w, w are proportional to op, Sg^, cr, or to the de-
[terminants of (Art. 7, eq. 1).
22. The ]^erpendicular distance of a point from the line
wpa-\-hq(i-}-cry = 0 (1).
Let the given point be (/, g, h), through which a parallel is
[drawn ; d the perpendicular distance required.
Then the distances from A, B, 0 to this parallel will be
^represented by the perpendiculars dzhp, dzkq, dzkr^ which,
mbstituted in (1), give
a(d±p)a + b(d±q)fi + c(d±r)y = 0.
But, by hypothesis, this line passes through (/, g, h).
Hence a (d±p)f + b {d±q) g + c (d±r) h = 0,
(af+bg-\-ch)d = ^(apf-\rbqg-\-crh),
28
TEILINEAR COORDINATES.
which gives <i = ± <ml±^99+^\
an equation for the perpendicular distance.
23. The equation
2A
evidently gives the altitude of a triangle whose vertex is /, g, h,
and the equation of the base
ap/+ hqg -f crh = 0.
The equations of the sides will differ only in the perpen-
dicular ; hence these may be written
a:Pifi-hq^g + crih = 0,
apj-^lg^g + cr^h^ 0.
With these two equations, and
a/+5^ + c/i = 2A,
the values of /, g, h may be determined ; that is,
af : hg : ch : 2 A : :
qi n
:
npi
:
Pi <li
:
1 1 1
23 ^2
ni>^
P2 22
Pi 2i n
Pi 22 ^2
af __
92 r.
1
1
1
Pi
2l
n
P2
22
^2
2A
Multiplying this equality by j9, and the second and third
equations formed from the above proportion by q and r respec-
tively, and adding, we have
, _ apf-\rhqg-j~crh _
2A
p
2
r
Pi
2i
n
P2
22
^2
1
1
1
Pi
2i
n
Pi
22
n
TRILINEAR COORDINATES.
29
24. We will now show the method of expressing the posi-
tion of a right line by coordinates, and that of a point by an
equation.*
Let p, 2, ^ be the unknown, a, /3, y the known, coordinates ;
then, by the equation we have just considered, we are enabled to
determine a relation between p, q, r which will be true for any
right line drawn through the fixed point of which o, /3, y are
the coordinates ; that is,
aa2)-\rh(jq-\-cyr = 0,
which is called an equation, in tangential coordinates, of the
point whose trilinear coordinates are a, /3, y.
25. To find the tangential equation to the point of intersection
of two right lines.
Let (pi, ^1, r^), (2^25 5^2? ^2) ^^ ^^o tangential coordinates of
the two lines ; (a, /3, y) the trilinear coordinates of their point
of intersection. Evidently, then, (a, (3, y) is a point on each
of two lines whose perpendicular distances from A, B, C are
Pl» ^V n ; T'2^ ^2, ^2-
We first determine the ratios of the trilinear coordinates of
the point.
We have (Art. 21)
ap^a + hq^P-^-cr^y = 0,
ap^a-\-hq2(j + cr^y = 0;
and
hence aa : hjS
cy
^1 r.
n Pi
;
Pl^l
22 h
^2^2
P2 g.2
Multiply each of these ratios by p, g, r respectively, and
add ; then each of these ratios
aap + hftq + cyr
p q r
Pi 9.1 ^1
P'i ^2 ^2
* Salmon's Conies, p. 65.
30
TRILINEAR COORDINATES.
V
q r
Pi
g.1 n
P2
22 ^3
= 0
The numerator expresses a relation between p, q, r by the
preceding Article; but the denominator evidently expresses
the same relation.
Hence
is the equation required.
Otherwise, suppose
aap + hftq-^-cyr = 0 (1)
the equation of the point of intersection, which must be
satisfied by the coordinates of any line drawn through that
point ; but (pi, q^, rj, (jjg, g^i ^2) by hypothesis are perpen-
diculars from the points of reference upon lines drawn through
the point of intersection.
Hence aap^ + hPqi + cyr^ = 0 (2),
aap^-\-hPq^-\-cyr^ = 0 (3);
that is, (1), (2), (3) furnish the determinant
P
Pi
= 0,
P2 22 ^2
the same relation and equation as before.
26. Tangential equation of a point at infinity.
The point is clearly the intersection of two parallels.
Let (pi, 51, rj) and {pi + t, qi + t^ ^i + O t>6 ^^^ parallels.
But the condition of parallelism (Art. 10) is
P
Pi
Pi-^i
q r
qi n
qi + t n + t
which may be written
p q r
= p q r
Pi gi n
Pi ?i n
t t t
1 1 1
= 0,
= 0.
TRILINEAR COORDINATES. 31
The last determinant identically vanishes, as will be seen,
|if a common factor can be taken so as to make the first row
mity ; in other words, if jc>=zq=zr. That is, points at infinity
[•e comprised upon the line
p = q = r;
?and equation (1) of last Article reduces to
aa + i^/S + cy = 0,
relation which has already been interpreted (Art. 11).
27. Since we define the equation to a point in these coordi-
nates as an equation satisfied by the coordinates of all right
lines drawn through the point, it follows that, if
i = 0,
F=0,
^be two equations representing two points in tangential coordi-
lates, then the equation
L + hV=:0
'being satisfied, as it evidently is, by the coordinates of L and
F, must express a point on the line joining the given points.
28. Reserving for the present the further development and
the application of tangential coordinates, we will just mention
a system of coordinates known by the term triangular.
Instead of the trilinear equation
aa -f- fc/3 + cy = 2 A,
and, denoting the ratios of the left member by x, y, Zj we have
x-\-y + z = 1.
The ratios — -, -^, -^ evidently represent the ratios of the
triangles BPGy AFC^ APB to the triangle of reference.
32 TRILINEAR COORDINATES
It is clear how the trilinear coordinates ", /3, y are related
to x,y,z'j for, if we divide a; by a a, we have ^. In the same
manner, y divided by &/3 = — ; so that
^ _ y _ ^ _1_
aa 1(3 cy 2A'
29. The coordinates of the middle point of BO are, in tri-
angular coordinates, 0, ^, y.
This appears, since 5/3 = A.
But b(3 = 2Ay ;
hence 2/ = i ;
similarly, ^ = i ;
while X = 0.
30. We have seen (Art. 17) that the coordinates of the foot
of the perpendicular from A upon BG are
^ 2A ^ 2A -r.
0, — cos C, — cos B.
a a
These expressions, transformed as above, become
^ h cos G G cos B
a a
Referring again to (Art. 17), we find the coordinates of the
centre of the inscribed circle, which, transferred into the
triangular system, become
X _ y z _ 1
a b c a-^b-\-c
Transforming the area of a triangle (Art, 19), we have
JL .M
2V ' abc
x^ 2/i ^1
TRILINEAR COORDINATES.
33
= (Art. 5)
8A^ 1_
ahc 2(a8in J.-+-/3sin5 + y sinO)
The equation to a straight line joining two points is, in a
similar manner, found to be (Art. 7)
0.
x^ 2/i «i
= A
^1 2/1 2^1
^2 2/2 %
«2 2/2 ^2
^8 2/8 2^3
«8 2/8 ^3
a;
2/
z
«i
2/1
^1
«2
2/2
^2
The condition of concurrence is the same for both systems.
The equation to the right line at infinity (Art. 11) becomes
in triangular coordinates
x-\-y-\-z = 0\
and consequently the condition of parallelism (Art. 10) is
'easily transformed to
= 0.
I
m
n
k
^1
n.
1
1
1
The equation to the perpendicular (Art. 14, Ex. 1)
/3 cos B = y cos 0,
[transformed from trilinear to triangular coordinates, is
y cot B = z cot 0.
Thus a great number of similar transformations might be
written out.
These, however, must suffice for the present, and these are
probably sufficient to give the reader a correct idea of such
changes when they become necessary. The useful applica-
tions of these several systems of coordinates must be learned
chiefly from a study of lines of a higher order than the first.
31. The principle detailed by Dr. Salmon* is equally ap-
♦ Conies, p. 33.
34 TRILINEAR COORDINATES.
plicable to trilinear or triangular coordinates, or any system
in which a point is determined by coordinates ; that is, if
u = la-{- mf^ + ny =: Of
and V = Zia + m^/3+Wiy = 0,
then will u-\-hv = 0 (1)
represent a line passing through the intersection of u and v,
which line, it is evident, can be made to represent any parti-
cular line by giving particular values to the arbitrary con-
stant h.
Let us try a simple application.
Suppose the triangle of reference circumscribed by lines
whose equations are
u=Of v = 0, w = 0;
that is, representing A^Bi^ By^G^, C-^A^. Let A^B^ be produced
to some point JBg, and from B^ let B^ G^ be drawn, and let B^ G^
be the line whose equation is to be determined. Join B^ G^.
But Gi is the point of intersection of v=0 and w=0. Hence,
from what has just preceded, B2 G^ will be represented by
Also, since B^G^ and A-^B^ (produced) meet in B^^ the line
B^ O2, which is drawn through their intersection, will, by the
same considerations, be represented by
7c-^u-\-v-\-Jcw = 0,
which, written symmetrically and in the usual form, becomes
\u -\- fxv -\- yw = 0.
It is manifest that this proof is not restricted to lines form-
ing a triangle. It is equally plain that they should not be
parallel.
32. In order that a point may be determined upon the line
la -\- m/3 -j- ny = 0,
its coordinates must simultaneously satisfy the relation
TEILINEAR COORDINATES. 35
I
If such values prove to be irrational, they are, by conven-
tion, said to be the coordinates of an imaginary point. Since
quadratics involve two roots — sometimes imaginary — there
will be the same number of intersections, if the question is one
of intersection, real or imaginary, — or, more exactly, real, co-
incident, or imaginary. But as this truth is so well known
and so fully exhibited in Cartesian Geometry, we shall here
consider only what is peculiar to our subject.
33. It is evident that the imaginary roots of a trilinear
equation of the second degree must be of the form
a + a,y~l, /3+/3,x/^, y + y^Hl (a).
Suppose these roots to be the coordinates of an imaginary
point. Then, by the last Article, these must satisfy the relation
ao + &/3-fcy = 2A.
Making the substitution, we have
(aa + hfi + cy) + (aa^-{-h(i^ + cyO ^^ = 2A (1);
wherefore aa^ + &/3i + cyi = 0 (2),
and aa+hP + cy=z2\ (3).
From which we see that (1) is made up of both real and
imaginary parts ; the imaginary parts satisfying (2) the
equation to the line at infinity (Art. 11) ; while (3) is
of course satisfied by its own coordinates. The reader
will learn to distinguish between the coordinates of an
imaginary point and those of an imaginary point at infinity ;
that is, if the coordinates (a) had been regarded as the co-
ordinates of an imaginary point (or proportional to them) at
infinity, both (2) and (3) must have been written = 0.
34. The equation to an imaginary right line may be written
36 TRILINEAE COORDINATES.
35. Writers upon equations of the second degree represent-
ing right lines in Cartesian coordinates are accustomed to
dispose of the contingency of two imaginary roots by referring
both to two imaginary lines drawn through the origin, thus
determining a real point. So now we say that every imagi-
nary right line passes through one real point, and but one.
If we consider the equation of (Art. 34), we see that the
real and imaginary parts are not coincident, and consequently
j^the factor v — 1 does not divide out ; hence the equation may
be expressed w + vx/— 1=0 (1),
in which u and v are functions of the coordinates of the given
straight line. This equation is manifestly entirely similar to
equation (1), (Art. 31).
It is also to be observed that u and v are of the first degree,
and hence u ■= 0
and v = 0
are satisfied by real values, which values satisfy (1), which
passes through the point of intersection of w and v, and there-
fore each straight line passes through a real 'point.
36. Suppose the equation to a straight line
lf-\-mg + nh = 0
to pass through an imaginary point whose coordinates are
given in [Art. 33 (a)] ; then
la -\- mft + ny -f- (la^ -f- mft^ -\- yiy^ \/ — 1 = 0 ;
and consequently Za -|- mp -\-ny =0,
Zai + m/3i + ^yi = 0,
which equations determine the ratios of Z, m, n ; or we may
determine them fully by the determinant
= 0,
/
9
h
a
/3
7
"i
ft
7i
TRILINEAR COORDINATES.
37
rhich is the equation to the straight line drawn through the
^imaginary point whose coordinates are
a•^a,^/^ /3+A x/^, y + yi^^l
id consequently
a_a^yZa, p-p^^/^ y-y^^^.
Therefore, since imaginary roots enter by pairs into an
equation, the imaginary points of intersection of two lines
(curves) will bo found upon real straight lilies hij twos.
37. If we have an equation of the form
Z/3'-m/3y + %y' = 0 (1),
we can evidently subject it to the same reasoning which is
^ applied to the quadratic
x^ —px]) + qif = 0. *
Each equation is reducible to the form
(/3-5y)(/3-.,y) = 0;
that is, the two straight lines
^-sr = 0 (2),
/3-s,y = 0.. (3),
are real or imaginary according as we find, by the resolution
of (1) for the ratio /3 : y, that 4k is less or greater than m^.f
Examining (2) and (3) in the light of (Art. 31), we see that
these lines intersect in the point A of the triangle of refernece.
38. Excursus on imaginary right lines and points.
It is evident, from what has immediately preceded, that this
portion of the subject is capable of considerable expansion,
and that this system of coordinates is eminently fitted to deal
with the Infinite and Imaginary. From what has already
been said in reference to the adaptation of the reasoning
* Salmon's Conies, p. 69.
t Algebra, Bourdon, p. 159.
38 TBILINEAR COORDINATES.
employed in Cartesian methods to trilinear coordinates, the
views of high authorities upon these results are interesting.
Poncelet* has discovered and illustrated geometrically the
rationale of the principles which, upon purely analytical
grounds, we are enabled to re-discover, apply, and extend ; he
has pointed out the correspondence of points, some real and
some imaginary, and taught that theorems concerning imagi-
nary points and lines may be extended to real points and lines,
and hence shown how to indicate the properties of a figure
when some of the lines and points are real and some imagi-
nary. By the method of trilinear coordinates we are enabled
quickly to generalize all those theorems which are concerned
with the line at infinity. For example, if four points on a
conic, or four tangents to a conic, are given, and it is required
to find the locus of the centre of the conic, we proceed to find
the locus of the pole of the line
a sin J. -j- /3 sin 5 + 7 sin 0 = 0,
which also gives us, the conditions being the same, the locus
of the pole of any line
\a-f/i|(3-|-vy = 0.
In applying the method of projections, the analytic shows
its superiority over the synthetic method, by proving the
general theorem at once, rather than by inferring it by the
projection from a more elementary state of the figure.
As to the results reached in our discussion of parallelism,
and what we have said upon the theory and use of the line
a sin J. + /5 sin 5 + y sin 0 = 0,
nothing is affirmed beyond what has been received, almost
without dissent, from the first, both upon geometrical and
analytical considerations. See Chasles (Geom. Sup.), Town-
send (Vol. I., p. 16, Art. 136), Salmon (Conies, pp. 64, 318),
Poncelet (Proj. Persp., p. 53), Hamilton (Quaternions, p. 90).
* Traite des Proprietes Projectives des Figures.
TBILINEAR COORDINATES.
39
38. Tangent of angle betvwen two lines.
Let la + ml3 + ny = 0,
lia-\-m^P~\-njy = 0,
be the given lines.
P'' ! If 0, 01 be their respective inclinations to one of the lines of
reference, then, by the reasoning in Art. 14, we must have as
the tangent of the difference of the two angles, that is, the
tangent of the required angle,
tan (d-d,) = tan 0-tan d,
^ '^ 1 + tana tan(^i'
* which becomes, by a laborious reduction,
I
m
n
sin J.
sin 5
n,
sin G
P
where P is the sinister member of the equation of perpen-
dicularity given on page 17.
This is probably the simplest form possible in trilinear
coordinates.
Examples under Chapters I. and II.
1. Express the parallelism of
la-\-m(^-\-ny = 0,
with AG in the triangle of reference.
Ans.
a
h
c
I
m
n
0
1
0
0.
2. The same line with BG -j with AB.
D
40 TRILINEAR COORDINATES.
3. What relation of two lines is expressed by the determinant
= 0;
ah c
1 0 cos 5
0 1 cos J.
and what are the lines ?
4. What condition is expressed by
a b c
I m n
0-11
= 0?
5. Find the angle between the lines
^. a = 7 cos J?,
and j3 = y cos A.
6. If -zA+vv/— 1 = 0, it'-f ^^''Z— 1 = 0 are imaginary
straight lines having a real point of intersection, then the four
real straight lines u=-Of -17=0, -^.'=0, v=^0 are concurrent.
7. What is the determinant expressing the equation of the
right line drawn through the intersections of the pairs of lines
2au-\-'bv-\-cw :=0, &v + ci<; = 0;
26w-|-av + cw = 0, av— cm; = 0?
41
I
CHAPTER III.
THE TRILINEAE METHOD APPLIED TO CONICS.
39. We will now call attention to the fact, wliich may not
have escaped the notice of the reader, that trilinear equations
are always homogeneous. If not so in form, they can be made
so by a very simple process. Since
aa + 6/3 + cy = 2A,
aa-^hP + cy
we may write
1;
and therefore any term of an equation may be multiplied by
this fraction without affecting the pre-existing relation of
equality. Thus, if we have
a2-2a/3 + r=2,
we may proceed to raise each non-homogeneous term to the
i second order, as
40. Another consideration, which has been referred to, may
'be here emphasized ; viz., that we are not concerned with the
absolute values of the coordinates, but with their ratios ; and
this advantage we derive from the principle of homogeneity
'^which belongs to every trilinear equation ; thus,
a2-2a/3 + 7' = 0
is, in fact, (~)~^(~")
^ + 1=0,
d2
42
TEILINEAR COORDINATES.
in whicli only the ratios — and — appear. Beyond these ratios
it is not necessary for us to inquire.
41. It may be desirable to find tbe equation to the same
locus, but referred to another triangle of reference.
First Transformation^
when the equations of the sides of the new triangle are given.
These sides being represented by equations in terms of the
perpendiculars from the angular points of the original triangle,
we have (Art. 21)
coordinates of J., (^, jp-^, p^) ;
B, (q, q„ q,y ;
0, (r, r^, r^) ;
that is, ap f -{-hq g + cr h = 0 (1),
apJ+hq,g + cr^Ji = 0 (2),
apj+bq.2g + cr^h = 0 (3),
where (/, g, li) are the old coordinates of any point P.
To find the locus of the homogeneoas equation
When referred to (1), (2), (3), we observe that / represents
the perpendicular from (/j, g^, h^) — these being the new co-
ordinates of P — on the line joining P and G. Therefore
(/i, 9i^ K), (9^ 9i^ 22), (^, n, ^2)
indicate the angular points of a triangle whose area is found
by Art. 19,
double area
=./.i
/i 9i K
9 9x 92
r ri ra
'
Similarly, hg =
^1
/i 9i K
r r, r,
p p, p.
, 1
/i 9i h
P Pi V^
9 9i 9i
TRILINEAR COORDINATES. 43
from which the values of /", y, Ti are readily determined.
Hence, representing these determinants by Q, 22, 8 respec-
tively, we may write
F f ^, I-, -^) = 0
\ a 0 c /
as the equation with new lines of reference, the degree not
being changed by transformation.
42. Second Transformation,
coordinates of the new points of reference being given.
A triangle drawn within or without the original triangle
will sufficiently represent the construction.
Let the perpendiculars from A-^, B^, O^,, the new points of
reference, upon BC be denoted by p, pi, p^ ; on AG by q, q^, q^ ;
on AB by r, r^, r^ ; a^, &i, Cj the sides of the new triangle ;
/i, ^1, hi the new coordinates, and /, g, h the old coordinates,
of any point P. Then, by Art. 21, we find
a^pfi + hp^g^ + Cip.2\ = 0,
«i ^/i + K'^i9i + <^inh = 0.
Representing these equations by Q, B, S respectively, we
have, by Art. 22, the distance of P from each of the sides of
the original triangle expressed in a simple form ; that is,
/. Q B 7 t)
the old coordinates expressed in terms of the new.
43. We shall now pass on to the consideration of curves of
the second degree. An important property of these curves
was conceived by the early geometers ; viz., that every curve
of this degree might be regarded as a conic section. What
then can be easily shown may be stated here, that the section
of a right circular cone by any plane can be expressed by a
44
TRILINEAR COORDINATES.
homogeneous equation of the second degree in trilinear co-
ordinates. This can be readily proved by selecting particular
lines of reference ; and since, by the preceding Articles, we
may transform to any other lines without affecting the degree
of the equation, we may regard this as a general truth irre-
spective of the lines of reference.
Let us write
ua^ + i;/32 + wy^ + 2u, fty -f 2v^ ya + 2w^ a/3 = 0
as the general equation of the second degree in trilinear co-
ordinates.
This equation, it will be seen, contains six terms ; but as
the nature of the curve does not depend upon the independent
magnitude of these coefficients, we may simply regard their
mutual ratios, or, in other words, assign a particular value to
one of the coefficients, varying the Values of the others.
Here, then, as in the Cartesian coordinates, we can find the
equation to the conic described through five points. There
are, in other words, five constants to be determined whose
values substituted in the general equation will give the equa-
tion of the conic through five points ; that is,
a' {3' y' ^y ya a/3
«i R y\ ftiYx yi"! "A
ftl y\ Ays y5«8 "5/35
= 0.
44. Concurrence of the straight line and conic.
The well-known property that every right line meets a
curve of the second degree in two real, coincident, or imagi-
nary points* is readily exhibited.
Writing the general equations of the conic and straight line,
la + m./3 -{■ ny = 0,
W + ^;i8H^^yH2?^l/3y^-2v^ya + 2w;^a/3 = 0,
* Salmon's Conies, p. 132.
TRILINEAR COORDINATES.
45
and the simultaneous relation
aa + &/3-fcy = 2A,
we see from these three equations that we are enabled first to
express /3 and y as functions of a of the first degree, which
substituted gives us a quadratic, and this in turn furnishes
two roots, determining two points of intersection.
45. Excursus upon the fundamental form of the equation to a
conic section in trilinear coordinates.
Conceive the vertex of a right circular cone placed at the
origin 0 of x, y, z coordinates ; XYZ the plane of section, and
also the triangle of reference in trilinear coordinates ; Oj, 02, 9^
the angles which the perpendicular upon this plane makes
with OX, OY, OZ ; OX and OY supposed to be at right angles
to the axis of the cone OZ and to one another ; P a point on
the curve and the origin of a, /3, y.
Let a, h, c be the perpendicular distances of P from the
coordinate planes, and d the diagonal from 0 in the lower
face of the parallelepipedon.
Then the perpendiculars from P on XY = a, on XZ = /3,
on r^=y.
By the geometry of the figure, a (the perpendicular distance
of P from the plane OYZ) = a sin 6^, & = /3 sin 0^, and c (the
distance of P from the plane OXY) = y sin 6^,
d' = a'-{-b\
c = d tan 0,
where 6 ■=. semi- vertical angle of the cone.
Hence c^ = ((^H &') tan^ a ;
that is, y^ sin^ d^ = (a^ sin^ d^ + fi^ sin^ d,) tan^ 6,
or a^ sin^ 6^ tan^ 0 + /3^ sin^ 02 tan^ 0 - y^ ^^^2 q^ _ q^
which also may be written
W + mlj^ + ny^ = 0,
where it is understood that the signs are not all the same.
46 TRILINEAR COORDINATES.
We have, therefore, derived an equation to a conic section
homogeneous and of the second degree in trilinear coordinates ;
and in turn it may easily be shown that the general equation
ua^ H- 1'/32 + wy^ + 2u^Py + 2v^ya + 2w^al3 = 0
may be made to take the form
and hence every equation of the second degree may be said to
express some section of a right circular cone.
In tbe genesis of this equation it is evident how we might
proceed to make some applications in tri-dimensional Geo-
metry. For instance, let us take some function of x^ y, z as
an equation to a surface in three rectangular coordinates, as,
and let x cos d^-j-y cos 6^-\- z cos ^3=1?
be the equation to any plane ; also let the traces of the coordi-
nate planes upon the plane of section be the lines of reference ;
then, if x, y, z be the coordinates of any point P upon the
given surface, and if Q^, 6^, 6^ be the angles which the proposed
plane makes with the original plane, we must have
X •= a sinO^, y = p sin 6^^ z = y sin 6^ ;
and consequently, by substituting in the given surface, we
obtain the trilinear equation to the section, that is,
/(a sin d^, J3 sin Og, y sin 63) = 0.
In the same manner, the equation to the section of the
Q? y^ ^
ellipsoid ~2 + tT + "2 = -^
by the same plane would evidently become
g^ sin^ e, , (5' sin^2 . rl^n^ ^3 _ 1 r^^
a' "^ h' ^ c^ ~ ^ ^'
which is easily rendered homogeneous by first finding the
TRILINEAR COORDINATES. 47
identical relation among a, /3, y in the given plane ; that is,
by substituting the values of x^ y, z as above when
a sin 2^1 + /3 sin 20^+ 7 sin 208 _ -,
-' - 2p -''
and consequently (1) becomes
g^ sin^ e, fy" sin^ 0^ y^ sin^ ^3
a^ "^ &^ "'' c'
— r« sin 201 + /3 sin 2(92 + y sin 2a8T_ ^
46. By the last Article we are enabled at once to interpret
such an equation as
a/3-^ya; = 0 (1),
where, by ordinary abridged notation, a = 0, /3 = 0, y = 0,
ic = 0 are the equations to four straight lines, and k is any
constant.
In considering the given equation, we see that it is of the
second degree, and satisfied when a = 0 and y = 0 are at the
same time satisfied ; and hence we infer that the conic repre-
sented by the equation passes through the intersection of these
lines. In the same manner, another point is determined by
the intersection of /3 = 0 and ic = 0, and so on for the four
sets of lines determining four points through which the conic
must pass ; that is, (1) represents a conic circumscribing a
quadrilateral whose sides are a, /3, y, and x.
From this we readily pass to the interpretation of the similar
equation a/3 — Z;y^ = 0 (2),
which indicates that two of the opposite sides, y and x, are
coincident. And as each of the lines a = 0 and (3 = 0 can
meet the conic in but two points, they must be conceived as
drawn from a point without, and hence as tangents to the
conic at the points respectively where the coinciding lines
meet the conic*
* Salmon's Conies, p. 223.
48 TRILINEAR COORDINATES.
47. The triangle of reference self -conjugate with regard to the
conic.
B/eturning to tlie equation
la' + mfi'i-ny^ = 0,
we see tliat it expresses no possible locus while Z, m, n are
regarded as all positive or all negative ; but, as we saw in
Art. 45, these are not all of the same sign.
Let I, m be positive, n negative, and for tbem write n'^, v^, w^
respectively ; then
u'a^^v'p.^-m'y^ = 0 (1),
or y?a^-\-{v^ + wy) (yij — y) = 0.
After the analogy of equation (2) of Art. 46, the lines
vj3 + wy = 0,
v(i — ioy — 0,
must be tangents, and a their chord of contact ; in other
words, the line a = 0, which is the equation of BG, a side
of the triangle of reference, is the chord of contact of a pair
of tangents from the vertex A.
It is equally admissible to write (1)
whence we see, as before, /3 = 0, which is the equation of AG,
a side of the triangle of reference, is the chord of contact of the
lines ua + wy = 0,
and ua — wy = 0,
which are tangents from the vertex B ; or, still further, (1)
may take the form
or (ua-\-vl3\/'^) (ua — v(3\/ — l)—wy = 0;
whence 2fca + ?;/3 \/ — 1 =0,
ua—vjj \/— 1 = 0,
are the imaginary tangents from the vertex (7, and y = 0 their
chord of contact, which is also the equation of AB.
TRILINEAR COORDINATES. 49
Therefore, as we see, eacli side of the triangle of reference
becomes in turn the chord of contact of tangents from the
opposite angle, that is, the jpolar of that point with respect to
the conic ; and, conversely, each vertex is seen to be the pole
of the opposite side, or the triangle may be described as self-
conjugate with respect to the conic ; which was to be shown.*
48. Intercepts of a directed line upon the curve
Za2 + m/3Hn7^ = 0 (1).
Let («!, j.\, Yi) be the point from which the directed line h
is drawn to meet the conic ; s^, s,^, s^ sines of the given direc-
tion all measured in the same direction, the first from h to the
parallel of BG, the second measured from the same point to
the parallel of AG, and the third in the same direction round
to the parallel of AB, of the sides, respectively, of the triangle
of reference.
Then, evidently, a = a^-\-sJi,
^ ^ = Pi + s^h, *
These values substituted in (1) give a quadratic in h ; that is,
h^ (Isi + ms2 + nss ) + 2^ (Is-^a^ + rns^p^ + ns^y^)
+ (lal + mPl -\- nyl) = 0,
The two values of h obtained from this equation will be the
lengths of the intercepts from the given point.
Suppose this point to be on the curve, we shall then have
lal + mj^l + nyl = 0,
and consequently but one value to h (one intercept becoming
zero), which is manifestly the length of a chord in the given
direction.
49. Locus of middle points of parallel chords.
Let the curve be the same as in the last Article ; (aj, /3i, y^)
* Salmon's Conies, p. 227.
50 TRIUNE AR COORDINATES.
a point on tbe locus, and the cTiord from this point repre-
sented by h, whose direction is given by its sines, Sj, Sg? ^3-
Then, as before.
Hence the intercepts of the curve are given by
which, by the supposition, are equal ; that is, the two values of h
will appear with opposite signs ; and since they must be equal,
their sum, or the coefficient of Ji, will be equal to 0, and conse-
quently* ZsiaiH-mSgA + w^gyi = 0,
a straight line giving the relation, in fact, of any point on the
locus, and hence the equation required.
60. Tangent to a pomt on the conic.
Let now the point (a^, ft, yj be on the conic. We have
seen, by Art. 48, that when this point lies on the conic,
lal + m0l i- nyl = 0;
and therefore the quadratic in h of that Article reduces to
(Isi + ms2 -\- ns^) h + 2 (ls-^a^ + ms2fy^-\-ns^y^) = 0,
which gives the length of the chord.
When, now, the direction becomes that of the tangent, the
length of this chord, that is h, becomes zero, and we have
l\a^-^ms2fyi + 7is^y^ = 0 (1).
But if (a, jS, 7) be any point on this tangent, we must have
h ' ' h ' ' h '
which values substituted in (1) give
la^a-{-mj3^l3-\-ny^y = Za^ + m/3j + nyl = 0.
* Bourdon's Algebra, p. 160.
TRILINEAR COORDINATES. 51
Hence Z.aj a -|- m/3i (3 + ny^ y = 0
expresses the required relation, and is therefore the equation
sought.
51. Coordinates of centre.
The reasoning is similar to that of Art. 49. The direction
of a diameter being s-^, s^, s^, the lengths of intercepts by the
curve in this direction, measured from the centre (a^, /S^, yj),
will be given by the same equation as in that Article ; and
since the quadratic must, by the premises, give equal roots
with opposite signs, the coefficient of h will = 0 ; that is,
^^i^i + ^^a/^i + ^^syi == ^ (!)•
For the actual determination of a^, /3i, y^ we have
aa, + hl3, + cy, = 2A (2);
^ut since aa + hjj + cy = 2 A,
id a = a^ + s^7i, P = p^ + s^h, y = yi + s^hf
fe have as^-\-hs^-j-cs^ = 0 (3).
Comparing (1) and (3), we get
la^ ■m/3i nyj
a h c '
These ratios will enable us to find the values of a^, jSj, y^ ;
ins, by dividing (2) by — - or its equals, we have
CO
^ + ^' + ^' = — = ?^ = — •
I m n Zttj wz/3i ny^ '
and therefore the coordinates of the centre are determined.
It will be observed that these coordinates enable us to de-
termine the condition that the conic may be a parabola ; the
centre of a parabola being infinitely distant, its coordinates
mast satisfy the relation
aai + 6/3i + cyi = 0.
52 TRILINEAE COORDINATES.
Making the substitution, we have the required condition,
n^ h^ e^
that is, ^-i-JL +;^ = 0.
62. Equation of circle with respect to which the triangle of
reference is self-conjugate.
It may be inferred from Art. 47 that when an equation of
the second degree does not involve /Sy, ya, ajo, the conic, in
such case, is so related to the triangle of reference that each
side is the polar, with respect to this conic, of the opposite
vertex.
Let one side, as CJ., cut this conic in two points (/"i, 0, /i^),
(/a, 0, 7^2), and let this chord be bisected. Then the equation
of the straight line from the vertex to the point of bisection
is,evidently, /^^^ = ^ W.
which line passes through the centre of the conic.
We have seen (Art. 47) that the conic may be written
where, in this case, nothing is assumed as to which of the co-
efficients I, m, n should be attributed the negative sign.
Now /i and f^ are identical with the values of a given byl
this equation.
We can eliminate /3 and y by the relation
aa + 6/3 + cy = 2A,
remembering that /3 = 0 ; and we have the quadratic
2 _ 4:^10^ a _|_ _^^L__ _ 0 .
u^c^ 4- w^a^ uh^ + w^a^ ~ '
and therefore /i+/2 = -^2^T^^~^>
TRILINEAR COORDINATES. 53
since this coefficient is the sum of the roots of a ; by the same
reasoning we have
id consequently (1) becomes
line on which the centre lies, which may be written
u^a w^y
limilarly, ^ ^v^^
a h
Now it is a property of the circle that a line joining any
^point to the centre is perpendicular to the polar ; therefore the
line !!l?_!^ = 0,
a c
which is drawn from the centre to the vertex i?, is perpen-
dicular to /3 = 0.
But, by the figure, we have
a
=
y
cos
G
cos A '
u'
w"
a cos
A
c COS 0
u'
v'
therefore
similarly,
a cos A h cos B '
and the equation of the circle becomes
a co^Aa^-\-h cos5/3^ + c cos Oy^ = 0,
or sin 2 A a' + sin 2B (^' + sin 2(7 y' = 0.
The circle thus represented will be imaginary unless the
triangle of reference have an obtuse angle.
54 TRILINEAR COORDINATES.
53. The inscribed triangle.
Returning now to the general equation of the second degree,
^a2-|-v/3' + w;y' + 2wi/3y + 22;i7a + 2wia/3 = 0,
we see that if /3 = 0,
r = o,
the equation reduces to u = 0.
But this is the condition that the curve should pass through
the vertex A. In the same manner, it may be shown that when
i; = 0 and w = 0 it will pass through B and G.
Under these conditions the equation reduces to
u^(^y-\-v^ya-\-u\afi — 0 (1),
which also may now be written without the subscripts. We
may therefore write it
ul^y -h a (yy + wJd) = 0 ;
and since every straight line cuts the curve in two points,
the line vy-\-ivl3 = 0
must pass through the point where a = 0 and /3 = 0, since
these values alone will satisfy the equation ; but these points
are coincident, and determine the vertex A. This line could
not therefore be drawn within the curve, for it would then
meet it in three points ; it must be drawn without, and there-
fore is the tangent at A.
Equation (1) is an equation of the second degree, and re-
presents evidently, from what has been said, a curve circum-
scribing the triangle of reference, satisfied when any two co-
ordinates = 0, in which case each vertex lies upon the locus.
54 The conic ul3y-\-vya-\-wal3 = 0
will give values for the intercepts by the curve upon a straight
line from a given point ; the equation to the tangent at any
point ; the locus of middle points of parallel chords, in pre-
cisely the same manner as has already been shown in preced-
ing Articles.
TRILINEAR COORDINATES. 55
Let US here seek the condition that any straight line should
be a tangent to the conic. '
Since ufoy-^-vya + wap = 0
represents a conic described about the triangle of reference, it
passes through the point, as we have seen, where /3 = 0 and
y = 0.
Let /a-f^/5 + /iy = 0
be the straight line. If by means of this equation we eliminate
a from the equation to the given conic, we must have, evi-
dently, coincident values for /3 : y.
Now the quadratic which results,
y^ y \ gw I gw
will give equal values for — when the value of the radical is
zero ; that is, when
4ihgviv — {gv -f Jiw—fuY = 0,
or u^f+vY + w%'-2vwgh-2uwhf—2uvgf = 0,
This may also be written in the form
± \/uf dz "^vg ± vwh = 0,
which can be verified by clearing of radicals ; and this is the
condition that the straight line
fa+g/5 + hy = 0
may touch the curve
u(3y-\-vya-\-wa(^ = 0.
Dr. Salmon has called this the tangential equation of the
curve.
E
56 TRl LINEAR COORDINATES.
55. Pascal's hexagon : the opposite sides of a hexagon
inscribed in a conic meet, if produced, in collinear points.
Let the triangle of reference be inscribed in the curve,
and let Ax^, Bx^, Gx^ be three of the sides of the inscribed
hexagon .
Since, if (/, g^, h^), (f^, g^, hc^), (/g, g^, \) be the coordi-
nates of x^, x^, a?3 respectively, we shall have, by the figure,
/3 : y :: ^1 : Zz, ;
Ax^ will therefore be represented by
and x,G by g„ -f^, 0.
Hence (Art. 12) the point of intersection of these, sides is
9iA, 91921 K92-
The side Bx.2 will be subject to the coordinates f^ and h^ ;
the side x^A to li^ and g^ ; hence these sides will intersect in
the point
f^K h93i hK
The sides Cx^ and x^B will, in like manner, intersect in the
point /3/1, /i^3, hj^.
Hence we have, by the determinant of collinearity, the
condition
^1/2 9x92 K92
fih h9& hh
/3/1 /i 9s Kfz
0 (1).
which, as is evident, is also the condition ■ that the three
points cTj, a^a, x^ lie on one conic with the vertices of reference.
TRILINEAR COO"RDINATES.
57
For let (/i, g^y h^), (/g, g^, h), (/a, .^3» h) be the three given
points on the conic, and let the conic be represented by
ugh-\-vhf-\-wfg = 0.
Then will the three vertices of reference lie on this conic ;
and if the curve pass through the given points we must have
ug^h^ + vhj, ■j-wf^g^ = 0^
ug^\ + 'vlij^i + wf^ g^ = 0,
y'giK+'vhJ^ + wf^g^ = 0 ;
and the determinant by which u, v, and w are eliminated is
9x\ KA fi9i = 0,
g^K Kt\ hQ't
g^h hA A9z
which is identical in result with the condition given in (1).
Exercises.
1. A triangle being inscribed in a conic, are the points
coUinear in which each side intersects the tangents at the
opposite vertex ?
2. Prove the theorem of Hermes, that if (a^, p^, y^), (og, jSj, y^)
be two points on the conic
uPy-\-vya-^tvap = Of
then the equation to the straight line joining them is
"iQa PA 7x72
e2
68 TRILINEAR COORDINATES.
3. When does ul3y-\-v'ya-\-wa(3 = 0
represent an hyperbola ?
4. What is the chord of contact of the tangents
^ (/3+y) + (^/v± ywy a = 0 ?
5. What is'the condition of concurrence of the normals at
the vertices of the triangle of reference to the above conic ?
59
CHAPTER IV.
POLE AND POLAR— RECIPEOCATION.
56. Inscribed Conic.
Any conic inscribed in the triangle of reference may be
represented by _
via -|- \/m/3 + V7iy = 0,
which, cleared of radicals, is
l'a'-^m'l3'-h7iy-2mn(ir-2nlya-2lmaP = 0,
as we have seen (Art. 54), where it expressed a particular
condition.
If we examine this equation, we shall find that it may be
written in each of the three following forms :
4w%/3y— (m/3 + ny-Za)2 = 0 (1),
47ilay-(ny + la^mfDy = 0 (2),
4mZa/3-(Za + wi/3-n7)2 = 0 (3),
from which, as they differ only by a constant from the equa-
tion interpreted in Art. 54, we conclude from parallel reason-
ing that each represents a conic section in which the factors of
the first terms equated separately to zero are tangents to the
curve in whose equation they respectively appear, and the second
terms are the squares of their respective chords of contact.
Hence the lines of reference are tangents, and the conic is
an inscribed conic.
57. Conversely, every conic ivhose lines of reference are the sides
of a circumscribed triangle will have an equation of the form,
60 TRILINEAR COORDINATES.
since every conic may be represented by
If the triangle of reference be circamscribed, the side BC
will be a tangent and be represented by a = 0. This value
substituted in the general equation gives
which, from the nature of the case, must have equal roots,
that is, the left-hand member of the equation must be a perfect
square ; hence u^ =vw;
that is, ^1 = db \^vw ;
and similarly v^ = ^ s/wu,
w^ = ± vuv,
are the necessary and sufficient conditions that the conic should
touch the lines /3 = 0 and y = 0.
Substituting these values in the general equation, and re-
membering to write ?^, m^, 71^ for u, v, w, we have
db v"^ ± N/m/3 ± y 71^ = 0 (1),
which was to be proved.
68. Four conies may be inscribed in the triangle of reference
so related that the points of contact shall lie on the lines re-
presented by ± iJa ± m/3 ± wy = 0.
Eor it is evident that (1) of the last Article may be written
ZV + m2|32 + 7i2y2 ± 2mw/3y ± 2nlya ± 2Zwa|S = 0,
which, writing all the doubtful signs negative, or one negative
only at a time, breaks up into the equations to four conies, and
we are presented with four interpretations similar to (1), (2),
(8) of Art. 56. If the double signs be taken otherwise, the
locus will become simply two coincident straight lines.
These equations therefore, as representing conies, have
TRILINEAR COORDINATES.
61
twelve points of contact lying three and three on the above
four straight lines. It may be observed that the actual sign
of the quantities under the radicals in equation (1) of the last
Article depends upon which sign is taken with the coeflB.cients
of /3y, ya, a/3.
The process for finding tangent, intercepts, centre of conic,
&c. is similar to that already exhibited in the last Chapter,
and need not be repeated.
59. Brianclion' s Hexagon : the three opposite diagonals of
\every hexagon described about a conic concur.
The method of proof is quite similar to that already ex-
hibited. Let three sides be produced for the triangle of
^reference ; ABCDEF the hexagon ; AB, CD, EF the sides
■produced.
If \a-^m^Py + n^y:=0
be the equation io AF,
Zga + ma/B-frigy = 0
[to that of BC, and
^3a^-m3/3 + ^^37 = 0
to that of DE; then the diagonals AD and FC will be repre-
Jsented as follows : —
The point A,
» A
(AD),
(FG),
{BE\
y = 0, and Zja-f-mi/S = 0 ;
a = 0, and W3/3-r??3y = 0 ;
\m^a + m-^^m^(^-\-m-^n^y = 0 ;
l^n^a + n^m,^(^ -f- thn^y =. 0 ;
Hence, (Art. 8),
liTYi^ 'tn^^m^ m-^n^
n, m. n-, n.
= 0.
62
TEILINEAR COOEDINATES.
The condition that the three lines AF, BC, and DE shall
touch the conic
via -f Vmjj -\- Vny = 0
is found by first finding the condition of tangencj of each of
these lines, which is, for AF,
^
T-
m
= 0;
for BO,
i-
m
^2
for BE,
i-
W3
and therefore
1
h
^1
2
^1
= 0,
1
J_
1
k
nu^
W^2
1
1_
2.
k
m.
%
■which, it is seen, is the same condition as above.
From what follows on reciprocation it will be evident that,
by reciprocating Pascal's Theorem, Brianchon's Theorem may
be obtained.
60. It is proper here to notice a dijfferent form of notation
which is frequently employed in this subject.
Suppose / (a, /3, y) = 0 to represent the equation to the
curve, and s-^, s^, s^ the direction- sines of the tangent at the
point (ttj, /3j, y^) ; (a, /3, y) any point on the tangent ; and h
the distance between these two points. Then, following the
reasoning of Art. 48, the intercepts on h will be given by
substituting the new values,
a = Oj -f s^h, /3 = /3i + sji, y ^ yi + ^"^3^,
TRILINEAR COORDINATES. 63
in the above equation ; that is, by
f(a^ + sji, fi^ + s^h, Yi + sji) = 0,
which, when expanded as we have already seen in the Article
referred to, will consist of some function of (a^, /Sj, yj, a co-
efficient of hy and a coefficient of Jv^ which is some function of
the direction-sines, or
If now (aj, /Gj, y^) lie on the curve, we must have
Ik- /(«!, A, 70 = 0;
also one of the intercepts becomes zero, and since the line is a
tangent the length of the chord is zero, that is, the coefficient
of h vanishes, and we have
By substituting for s^, s^, s^ their values, we shall obtain
twice the function in (oj, p^, y^) which = 0, that is,
as may be shown by taking the differential coefficients of
ual + v(dI + wyl + 2ujD^y^-{-2v-^y^a^ + 2w^a^f^^ = 0,
in respect to a^, /3j, y^ respectively, multiplying the differential
coefficients by each of these coordinates and adding, when we
shall find that 2/ (a,, f3„ y,) = 0,
since (a^, /3j, y^) is supposed to be on the curve. There will
remain, therefore,
da^ d(iy tf,yi
64 TEILINEAR COORDINATES.
the equation to the tangent at (a^, (3^, yj, since it expresses a
relation among the coordinates of any point in the line.
61. Polar of a point in respect to the conic.
Let the fixed point be (a^, ft, y^) ; (og, ft, y^, (a^, ft, y^) the
coordinates of points of contact of tangents from the given
point. Then we can show, by an extension of the reasoning
of the last Article, that
«# + Af + 7.f = 0
da^ cla^ aaj
is the tangent from (a^, j3^, y^) to the point of contact (og, /Sg, y^) ;
and likewise «3 -/- + A -7^7- + 73 7 = ^
da^ dp^ dy^
is the equation to the tangent at (og, /Gg, y^. Therefore these
equations express the fact that the line joining these points of \
contact is a locus whose equation is
da^ fltpi dy^
that is, the polar with respect to the conic
/(a,fty)=0;
or we may proceed otherwise. Defining the polar of a given
point as the locus of the intersection of tangents drawn to the
points of section by a straight line through the given point,
we should have for the equation through the three points in
the same straight line
« (Ay2-ftyi) + /5(yi"2-y2«i) + y («ift-"2ft) = 0 ...(i),
where (a, /3, y) is the given point, (og, /S^, y^), (a^, ft, yj the
points of section in which any straight line cuts the conic.
TEILINEAE COORDINATES. 65
The intersection of tangents,
d£ 3f_ ^
will be .-^V = -^— = -P^ (2).
PiTa— P2T1 ri«2— y2«i «iP2— «2Pi
Equation (1) with (2) gives
i?/" 4. /3 ^ J- ^f _ 0
c?a <ij(3 c?y
This equation being independent of a^, /3j, yj ; Oj, ft, yg is
the relation at the intersection of the tangents ; it is therefore
the locus required, and, bj definition, the polar of (a, /3, y).
62. Coordinates of the pole of a straight line in respect to a
conic.
Let /a+^/3 + 7iy = 0
be the equation to the straight line, and
0 (a, A y) = 0
to that of the conic. If (a, (i, y) be the coordinates of the
required point, then its polar, by the last Article, is
da djj dy
and since this is the same as the given line, we have
d(p d0 d(f)
da ___ d(3 _ dy ^
7" 9 ^ ~f^''
that is,
ua-\-w^l3-\-v^y _ vl3-\-u^y + W-ia _ wy-[-v^a-\-u^l3
f ~ g ~ h •
6Q TRILINEAR COORDINATES.
Patting eacli member = —s, we have
ua'\-Wi(i-j-v^y-\-sf= 0,
and consequently
p
f
9
h
Vx
u^
w
w^
V
u^
f 9 h
V, u. w
y
f
9
h
^1
V
u^
u
w.
'^l
wbicli, with aa + &/3 + cy = 2 A,
determine the coordinates required.
63. Centre of conic.
If the equation be <p (a, /3, y) = 0,
and (oi, /3i, y^) be the centre ; then the roots of
0 (aj + Si/i, ft + V^ 71 + VO = ^
will be equal and opposite in sign.
Hence, since the coefficient of h must vanish in the quadratic,
therefore
da^ dpj^ dy^
Bearing in mind the proportionality of ^i, Sg?
relation as^ + hs^ -\- cs^ = 0,
that is, tbe
we have
c?0 d(^ dip
da^ __ dfD^ dy^
which, fully written out, will give determinants similar in
form to those of the last Article, with «, &, c in place of
/, Qj hj and a^, /3i, y^ in place of a, ft y.
TRILINEAR COORDINATES. 67
64, The conic will hreah wp into two right lines wlien we
have the condition
u W^ Vj
v. u, w
0.
Eor suppose the two lines into which
0 («, ^, y) = 0
breaks np to be represented by
/a + ^/3 + ^y =0,
Then will
0 («,ft y) = {fa + gP + hy) (Aa+g,P + \y),
and g =f(Aa+g,fi + Ky) + f^(fa + g[i + hy),
with corresponding values for -^ and -^. Hence, reverting
to a principle already explained (Art. 31),
da ^ d(3 ^ dy
are straight lines which pass through the intersection of the
given lines ; that is, the lines
u a -{■ w^fi + v-^y =: 0,
w^a -\- V (3 -\- u^y = 0,
v^a + u^P -\- wy == 0,
concur, and give the above determinant.
65. When some of the four points of intersection of two
conies become coincidejit, some of the common chords will
68
TRILINEAR COORDINATES.
coincide ; others will toach. at a common point, that is, be-
come tangents. There will, in general, be three pairs of
common chords ; if two points coincide, the conies touch ; if
the two remaining points also coincide, the conies have double
contact and a chord of contact.
66. Equation to the asymjptoies.
From Art. 64 we can easily form the equation to any pair
of common chords. Thus, if
0 («,ft y) = Qa2 + E/3H^y' + 2Qi/3y + 2E,ya-f 2;Sia/3 = 0
and
0i(a,/3,y) = ua^ ^viD'^wy^-\-2u^(Dy + 2v^ya-\-2w^ai^ = 0
represent the two conies, the locus in question will have the
equation (Art. 31),
^(«,Ay) + %(a,/3,y) =0 (1),
which must be so conditioned in h as to represent two straight
lines, hence (Art. 64) *
Rx + kv^ Qi + hu^ 8 + hw
= 0.
If now 0 (a, /3, y) = 0 breaks up into two coincident
straight lines, as,
(fa + gft + hyy = 0,
we shall find
k =
u w^
f
^1
f
u.
9
w
h
h
0
(=U),
W
i=W),
* Ferrers, p. 85.
TRILINEAR COORDINATES. 69
which, substituted in (1), gives
^(a,/3,y)Tr+ Cr<^i(a,/3,y)=0,
or (fa + gP + hyyW+ Ucl>,(a, fi,y)==0 (2).
This equation now represents, under the above condition,
not a pair of common chords, but a pair of common tangents
whose chord of contact is
fa + gl3 + hy=zO.
We have now only to introduce the condition that the chord
of contact is at infinity ; that is, that
aa+fe/3 + cy = 0;
; wherefore (2) becomes
Jaai-hP + cyy
+ fi (a, /3, y)
u w^ v^ a
u\ V i»i h
v^ U-^ w c
a h c 0
= 0,
rhich is the equation of the asymptotes.
I
^B Cor. 1. — Since every parabola has one tangent altogether at
an infinite distance, the vanishing of the second determinant
in the above equation expresses the condition that the conic
may be a parabola.*
Cor. 2. — The conic will be a rectangular hyperbola when
the asymptotes are at right angles to one another ; that is,
when (Art. 14) the two straight lines into which the conic
breaks are subject to the condition of perpendicularity,
11^ — (mn^ -{- m^n) cos A + mm-i^—(nli + n^l) cos B
+ ^Wj— (Zm^ + Zim) cos G.
In other words, if the conic be
01 («, ft y) = 0,
* Salmon's Conies, p. 224.
70 TRILINEAR COORDINATES.
the required condition becomes
u-\-v-\-w—2ui cos A — 2vi cos B—^w^ cos (7 = 0.
Def. — TF is called tlie discriminant of the function
^j (a, /3, y), and U the bordered discriminant of the same
function (D. 43), where f, g, h, as the coefficients of a, /3, y,
are = tt--, -— -, r— - respectively. For the conic,
2A 2A 2A ^ -^
W = ?m?^,
Z7 = — (a^mw + &^wZ + c^ Zm) -— ^ .
4A^
Numerous other functions may be determined.
67. Space does not permit extended illustration of the use
of the abridged notation thus far exhibited. The reader can
easily apply it. For instance, if the function be (p^ (a, |(3, y),
and we wish to express the equation of the straight line at
infinity in terms of the derived functions, the required equa-
tion might be written
and since aa + h^ + cy = 0 (1)
represents the straight line at infinity, we have
iif+ w^g-^- vji _ Wif-\- vg + '^Ji '^i/'+ % 9 + ^^^
a , b c *
These equivalents represented hj —Tc give us
uf-^- w^g + v-^Ji -\- ah =■ 0 (2),j
w^f ■{■ V g ■\- u^h + hh = 0 (3),
■^iZ+^i^ + wh-\-ch = 0 (4)^
TEILINEAR COORDINATES.
71
Eliminating now between (1), (2), (3), (4), we obtain
the condition that the minors of the bordered discriminant in
respect to its f^ g, Ji are proportional to jT, ^, Ti of the given
equation, which minors being represented by A^ JB, 0, the
equation becomes
da djo dy
as the straight line at infinity.
68. The equation of the nine-point circle.
We first find the condition that the conic
ly represent a circle. If the conic be a circle, / (^j, ^g, 53)
constant, that is, all diameters will be equal ; and since, in
le equation for finding the lengths of the intercepts,
/(«, A r) + ^ («i f + «. J + h J) + V/(«i, h, s,) = 0,
le coefficient of h vanishes, we have
rhich gives the radius in the given direction. To reduce this,
re may express the condition that diameters in three direc-
ions (that is, directions of the lines of reference) are equal.
We have, therefore, to express this,
^, ^/(«.fty) _/(«,fty)_/(«,/3,y)
/(<») f(y) fi") '
'herefore /(*) =/(y) =/(«) ;
)r, for direction of BO,
«, = 0, 82 = 81110, 85 = — sin B.
Lence /W =/(0, c, -i).
72 TRILINEAR COORDINATES.
Similarly, for OA and AB^
f(y)=f(.-c, 0, a),
from the proportionality of sin A, sin B, sin G.
Hence we have the two conditions,
v('^-\-wh'^—2u^bc = wa? + uc^—2vj^ca = uh'^+va^^2w^ah.
In the second place, we see that, if the curve pass through
the middle points of the sides of reference, a, ^, y must in
succession be taken = 0 ; whence
VG'^+wh^ + 2u^hcz= 0 ^
wa?-\-uc^ + 2v^ca = 0 \ (1),
uh^-]-va^-\-2w^ah = 0 J
which follows from the condition involved, that
ip = cy = aa.
Comparing the two sets of equations, we find
w\ab =:ViGa =■ u-J}c.
Hence, if equations (1) are true, they will hold whatever the
value of -Wj. Let u-^=l —a.
The resolution of these equations gives
w = 2a cos A,
V = 2b cos J5,
w = 2c cos C.
Hence the circle which passes through the middle points of
the sides of reference (tJie nine-point circle) becomes
a^ sin A cos A + /3^ sin B cosB -{- y ■ sin C cos G
— (jy sin ^ — ya sin 5 — a/3 sin (7 = 0,
or a^ sin 2A + (S^ sin 2B + y^ sin 2(7
- 2i3y sin J. ~ 2ya sin ^ - 2a/3 sin (7 = 0. ■
TRILINEAR COORPINATES. 73
COE. 1. — If now a = 0,
(P sin 2B + y' sin 2(7 - 2/3y sin ^ = 0.
But since 2 sin J. = 2 sin (1?-|- (7),
/32 sin 25 + y2 sin 2(7 - 2/3y (sin 5 cos (7 + sin 0 cos 5) = 0.
This breaks up into tlie factors
(/3 sin 5 — y sin (7) (/3 cos 5 — y cos (7) = 0.
The circle therefore meets BG in two points.
The one when, by the hypothesis,
a = 0, biJ =■ cy, i. e., /3 sin 5 = y sin G,
which determines the middle of BG.
The other is evidently when
a = 0, /3 cos B = y cos (7,
the foot of the perpendicular from A. Similarly for the other
sides.
Cor. 2. — The last equation of this Article shows that the
nine-point circle passes through the points of intersection of
the circumscribed circle and the circle in respect to which the
triangle of reference is self-conjugate.
Cor. 3. — The difference between the equations of the cir-
cumscribed circle and the circle through the middle points of
the sides of the triangle is
a cos A -\- 3 cos J5 + y cos (7 = 0 multiplied by a constant,
since a^ sin 2A + /3^ sin 2B + y" sin 2(7
= (a COS A-\- p> cos -B -f y COS (7) (a sin J. + /3 sin 5 + y sin (7).
But a sin A ■\- pt sin -B + y sin (7 is a constant, and there-
fore a cos J + /3 cos B •\- y cos (7 = 0 is their radical axis,
or the homological axis of the triangle of reference and that
formed by joining the feet of the perpendiculars.
Cor. 4. — By similar reasoning we find that the same
circle passes through the middle points of the sides of the
74
TRILINEAR COORDINATES.
triangles of which the point of intersection of perpendictdars
is the vertex. Nine points are therefore determined.
POLAR RECIPROCALS.
69. Reciprocation — the principle of duality, or that analysis
(or synthesis) which, while determining the distribution of
points, coordinately fixes the position of lines — though of great
interest, is altogether too large a subject for this Tract. Some
theorems may be introduced. In general, we may say that
to reciprocate involves interchanging "angular points" for
"sides," "inscribing" for "circumscribing," "join" for "in-
tersect," &G. &G.
For instance, if the well-known theorem, that " If two
triangles be inscribed in a conic, their sides will be tangent to
a conic," be reciprocated, we may write, " If two triangles cir-
cumscribe one conic, their vertices will lie on a conic."
This proof and its reciprocal may be exhibited by a common
process in triangular and tangential coordinates (Arts. 24, 27).
Let vertices of one triangle (sides of the same) be represented
Ijy (Pi^ ?i» n), (P25 ^2, ^2): (P&, ^35 ^'3) ; let the other be the
triangle of reference, and suppose
/(_p, q,r)=0
the tangential equation of the conic passing through the points
of reference; or the equation may be represented in both
systems by Iqr -\- mpr -\- npq = 0.
Then the equations to the one triangle will be
Ip mq . nr __ ^
r^Ps g'2 23 ''2 ^^3
Ip ^ mq ^ nr ^ ^^
g'3?i
mq
ly> ^ mq ^ nr _ ^^
TRILINEAR COORDINATES.
75
By comparing (Arts. 56, 57), we see that by this form of
representation the inscribed conic (circumscribing) may be
expressed by
\/^+ v/%+ ^Nr = 0 ;
that is, this conic will be inscribed in (circumscribe) both
triangles provided the conditions of tangency be satisfied,
Lp^^ + -^^2^3 _|_ NTYh
0
I m n
^VxVi ^ -^^19^2 ^_ -^^n^2 _ Q
I m n
(!)•
But since the given points (pi, q-^^ r^, &c.), (vertices),
lie by hypothesis on the conic, the condition must be ex-
pressed by
1 1 1 =0,
I
Pi
^1
n
1
1
1
B
^2
ra
1
1
1
B
^3
^3
which condition satisfies equations (1), and proves the theorem
and its reciprocal. The determinant follows, it is evident, as
the eliminant of the equations.
l^
+
in
+
n
.—
o>
Vx
^1
r.
^
+
m
+
n
—
0,
P2
^2
'^2
l_
+
m
+
n
0.
B
^3
n
76 TRILINEAR COORDINATES.
70. If m, w, p, q are the poles of the sides of a polygon
abed, then the points a, 6, c, d are the poles of the sides of
the polygon mnpq.
The conic with respect to which the poles and polars are
taken is the auxiliary conic.
TJie 7'eciprocal of a conic is a conic.
By Art. 60, the polar is given by
da d(i ay
If therefore (/, g, h) be any point on the reciprocal curve,
its polar with respect to the auxiliary conic,
Ua'+Vfi'+Wy' = 0 (1),
will be given by the equation
Ufa + VgP-^Why = 0 (2).
Let the conic to be reciprocated be
la^ + ml^' + ny^ = 0 (3).
To find the condition that (2) may touch (3), we eliminate
a between the equation of the conic and the line ; and if the
line be a tangent, the values of /3 : y must be equal (Art. 57),
and we obtain
I m n '
This being of the second degree, giving two points of intersec-
tion of the straight line, is a conic, and is the reciprocal of (3)
with respect to (1).
71. Two straight lines are conjugate when each passes
through the pole of the other. Required to express this con-
dition. Let fi^ + g3 + Ky = ^,
f^a-^g^fj + h^y = 0,
TRILINEAB COORDINATES.
11
the two lines. Then (Art. 62) we may express the condi-
)n by the equation
da^
d^
dud(3
d^(l>
dady
d\
dp da
dF(f>
d\
dpdy
d'(p
d^<p
d^
/l
91
dyda dydft dy^
it 92 K 0
= 0,
^here
0 («j A y) = ua^+v^^ + ivy^-^ &c.
London : Printed by C. F. Hodgson & Son, Gough Square, Fleet Street B.C.
ICAL TEACTS,
No. III.
INVARIANTS.
4:*
'-^^t-i^ '7^ C
^.
!%n.
^ C-
TEAOTS
Al^.
/<P3^^
RELATING TO THE
MODEM HIGHER MATHEMATICS.
TRACT No, 3.
INVARIANTS.
BY
Rev. W. J. WRIGHT, Ph.D.,
MEMBER OF THE LONDON MATHEMATICAL SOCIETY.
Plato, Rep. VII., 527, *.
LONDON:
0. F. HODGSON & SON, GOUGH SQUARE,
FLEET STREET.
1879.
My acknowledgments are due to R. Tucker, Esq., M.A., Honorary
Secretary of the London Mathematical Society, for valuable assistance
rendered in passing these sheets through the press. — W. J. W.
CONTENTS.
CHAPTER I.
Symmetric Functions of the Differences of Eoots
Eliminant by Symmetric Functions
Discriminants
PAGB
7
12
16.
CHAPTER II.
Invariant of the Binary Cubic
covariants -
Emanants
contravariants
CHAPTER III.
Canonical Forms
Canonizants
Combinants
Tact-Invariants
Absolute Invariants
Series of Covariants
CHAPTER IV.
Computation of Invariants
Self-Conjugate Triangle
Locus of the Intersection of Normals....
Equation of the Four Common Tangents
Theory of Foci
21
24
25
28
39
41
44
46
48
50
56
57
60
67
74
PREFACE TO TEACT NO. III.
This Tract takes up the general Theory of Invariants.
It is published in pursuance of a purpose, announced in
the first number of this series, to give an account of the
principal new methods, processes, and extensions which,
since 1841, have been introduced into the study of Ma-
thematics. The chief requisite to this undertaking, which
undoubtedly is one of considerable magnitude, is evidently
a sufficiently comprehensive reading upon these various
subjects. •
The English, German, French, and Italian Mathema-
ticians have contributed to their journals and learned
societies innumerable memoirs and treatises, whose value
and bearing upon the matter in hand the reader cannot
determine without some degree of careful examination.
It also frequently happens that the time consumed in
tracing a fugitive paper is in inverse ratio to its impor-
tance.
The reader who wishes to read fully upon this Theory
may adopt one of two methods. He may begin at the be-
ginning, reading in order of time the papers of its chief
authors and expounders, commencing with the essay of
the late Dr. Boole in the Cambridge Mathematical Jour-
VI PREFACE.
nal for 18il, and follow this with the numerous papers of
living authors — Sylvester, Cayley, Hermite^ and Salmon —
papers extending through the subsequent volumes of the
Cambridge and the Cambridge and Dublin Mathematical
Journals, and the Philosophical Magazine, together with
the various contributions of Clebsch and Aronhold, and
others, in Crelle, from Vol. 39 to Vol. 69. Or he may take
a reverse course^ beginning with the Lessons of Salmon^
and those of Serret, on Modern Higher Algebra, which,
as compends of this and connecting Theories, are in the
main works of great excellence, though oftentimes not as
clear and satisfactory as could be desired, or as full and
explicit as may be found elsewhere ; and then he may
extend his reading to the Journals above mentioned,
together with the proceedings of the contemporaneous
societies — as the Philosophical Transactions, Comptes
Rendus, &c. But, whatever course he may take, he will
doubtless never be able clearly to determine to what
authorship he is to ascribe some parts and illustrations of
the Theory.
The best reading-room for this work, so far as I can
judge, after an experience of nearly two years in European
libraries, is that of the British Museum.
In view of the extensive literature upon this subject,
it may be asked, what can be accomplished by a work of
the size of this Tract ? Its actual value, evidently, remains
to be seen; but I believe that within these pages the
reader wiU find such an account of the Theory as will
enable him to gain a knowledge of its principal proposi-
tions, and also to judge, from the explained applications.
PfiXFAd. yn
of its real valae in Greometry. The compntalions of In-
variants, Chapter IV., will afford snch a gnide in the
arioos applications that he will probably be at litlle loss
in extending them at his pleasure. I have been desiroos
of making these calculations so foUy^ that no one wiUi a
fair geometrical knowledge need fail of understanding
how each result was obtained. Nowhere else can sndi
work be found in so elementary a form, and for this
reason I hope it may proYe acceptable to those persons
whose time and opporkmities <^ stud^ are somewh^
limited, and to those also who are unwilling to obtain and
"0 read the larger works.
In reviewing the notes whidi I had taken of the prin-
cipal contributions to this Theoiy, I found that I had fre-
quently omitted the proper credits, either through sheer
n^l»;t^ or want of sufficient knowledge; and hence, with-
out attempting to supply these omissions^ as could not
weQ be done in the absence of the books and journals, it
was concluded to omit them nearly altogether.
The number of persons who have obtained the preced-
ing Tracts of this series, and who have expressed them-
s^es in terms highty &yorable to thdr publication, is
deemed sufficient evidence that they are meeting a public
want. One thing which was e^qiected has certainly fol-
lowed,— a goodly number of my countiymen have been
awakfflied to look, for the first time, upon a Tsst on-
traveraed domain of mathematical knowledge. To tiiese
persons, at leasts tiieare can be no doubt as to tlie direction
of thegoaL It is now deartyand definitely fixed that
maUiematical researdies wiD^ fiur a k»g time to come;, be
Vlll
PREFACE.
mainly conducted through the media of methods and
processes, to whose exposition these Tracts are devoted.
The time is not far distant, if it has nob already arrived,
when a knowledge of these subjects will be considered as
necessary to the equipment of a mathematician as the
Calculus. It is not meant by this, that it is the duty of
every mathematician to make a specialty of algebraic
forms, either with or without their geometrical interpre-
tation. But it is meant that the modern treatment of the
Higher Geometry should be studied as a part of the general
preparation necessary to a student of Physical Science.
Cape May Point, N.J.:
April, 1879.
W. J. w.
INVARIANTS.
CHAPTER I.
PROLEGOMENA.
1. The Theory of Invariants, as will appear, is based upon a
knowledge of the General Theory of Equations and several of
its later important extensions. Some of these extensions must
be stated, because, although perhaps familiar to the reader as
commonly or formerly expressed, they may not be easily
recognised in their modern dress or terminology; others,
because they have no existence outside of their present form.
2. Symmetric Functions, — If the general equation be
the Newtonian formulas give us
Si = —cLi, S2=al — 2ai,
8^ = — % + Ba^a^ — Sag, &c.,
or, as they are written by Hirsch, Cayley, and others,
Sa =-«!,
2a2 = a? -2%; 2a/3 = a^,
Sa' =— «! +3aia2— Saj,
Sa^/3 =—^1^2 + 3%,
Sa/3y=— ag, &c. ;
in which we have expressed the sum of the roots and the sum
of their products by twos, by threes, &c.
3. If we consider any one of these products, as a-^a^, we say
that its weight is 1 + 2, or, in general, that the weight of any
term is the sum of the suffixes. Looking at these functions,
B
8 INVARIANTS.
however far we may extend them, we see that they are sym-
metrical as to weight. The order is estimated by the number
of factors in each term. Hence a^a^a^ is of the third order,
and its weight is 1 + 2 + 3. This being stated, it is easy to
see, by inspection of the several functions written above, that
the weight of Sa^/3" is t + u, and the order the greater of t, u.
The order of Sa/3y (being the sum of the products in threes)
can evidently be, so far as the coefficients of the given equa-
tion are concerned, only unity. If, therefore, we regard a as
the leading root, appearing in every function, we might predi-
cate the degree of the function upon the degree of a. In this
case any symmetric function of the p**^ order must contain
more or less terms involving a^. There will then be p factors
each including a. Hence 2a^, Sa^/^y are each of the third
order in the coefficients of the given equation; that is, the
highest order in any term is three. In general, then, the
order of any symmetric function is determined by the highest
degree in any one root, while the weight is estimated by the
total degree of the roots as factors. The literal part, then, of
any symmetric function can thus be at once written out. For
the sake of clearness, it is necessary to notice that the functions
of roots in this manner may be expressed in terms of the
coefficients of the given equation, as will be seen by solving
the linear equations just written for /S^, 8^, &c. ; and, con-
sequently, any function of the differences of roots can be>
expressed in the same terms.
3. 8ym7netric functions of the differences of roots. — These we
shall see are invariants. For the present let us consider what
relation such functions ought to satisfy. We begin by observ-
ing the effect upon the coefficients of the given equation of
increasing or diminishing all the roots by the same quantity.
There will plainly be no change in the resulting functions of
the differences of roots. Let then x + l he substituted for a?,
and we^have
«'> + («i + wO ^'*'^
tla,+ (n-l)la, + ^n(n-l)l']x''-' + &c. = 0.
INVARIANTS. 9
Next, observe the form of any function/ of the coefficients
«i, ^2, ^3, &c., when a^, cig, &c. are changed into a^ + da^j
This form will be
But, by the substitution of a; + Z for x, a^ becomes a^ + nl^ and
^2 becomes a^^+in—V) la^-\-\n (w— 1) 1?.
Clearly, then, if this substitution were made in any function
of the coefficients %, a^, &c., and the result arranged with
reference to Z, we must have, by (1),
/+Z L J£-4- (^_1) a, J/L +(^_2) a2#1 +&C. = 0.
L da^ da^ da^J
This is true whatever I may be.
Let 1 = 0, and we have, as the condition which any function
of the differences will satisfy,
da^ da^ da^
This relation is both necessary and sufficient in order that the
given function of the coefficients should remain unchanged by
the substitution of aj + Z for x in the given equation.
We can now write, not only the literal part, but the coeffi-
cients of any symmetric function. For instance, if we are to
form 2 (i3— y)^ we see that its order is 2 and its weight 2.
There can be no more than two factors in any term, while the
weight for each term must be 2. It must be of the form
Aa<^-{-Ba\ . By the above differential equation,
lA{n-l)-\-2nB']a^ = 0.
n — \
This ffives B •= -r — , when J. = 1 ; or the function can
differ by only a factor from {n — 1) a^ — 2na^.
B 2
10 INVARIANTS.
We may see fhat this factor is unity by supposing y = 1
and the other roots 0 ; then flg = 0 ; and a^ = 1, since
a + /3-f &c. = %, and a {p-\-y-]-&c.)+l3y + &c.= a^,
4. The homogeneous equation
(%, (^i a,,) (a;, 2/)"
or aoaj** + ?iaiaj'*~^2/ + Jw(7i— 1) a2aj""y+ + «„2/'' = 0
reduces to the general equation of Art. 2 by dividing by a^y"*.
And it is plain that the differential equation of the last Article
will undergo a corresponding change. Hence, for the substi-
tution of x + l for X, we must write
fto #- + 2^1 -^ +3a2 -^ +&C. = 0,
da^ da^ da^
while for the substitution y-\-l for y it must be written in a
reverse order, that is,
na^-^-^{n-l) a,-^ + (n-2) a,-^ +&c. = 0.
da^ rfaj da^
5. The symmetric function of the homogeneous equation in — . —
Suppose a one of the roots, then — = a. That is, any system
of values, as — ^ = a, in other words, any ratio which is = a,
will satisfy the homogeneous equation. Or, we may state it
thus : any symmetric function expressed in terms of its roots,
as a?i, x^, cBg, &c., may be reduced to the corresponding func-
tion of a homogeneous equation of the same degree, by dividing
each a?!, X2, &c. by y^, y^, &c., and then multiplying this
result by any power of y-^y^^ &c. that will clear it of fractions.
Hence we may write any function of the differences as the
sum of products of determinants
«i 2/1 X ^i Vi &c. X (yiyi&c.y,
«2 2/2 ^3 2/3
INVARIANTS. 11
where n = the variable power necessary to clear of fractions.
Thus, to form for (a, b, c, d) (x, yf the sum of the products
of the squares of the differences of the roots, we have the
ratios, or roots,
^1 ^2 ^3
2/l' 2/2' 2/3'
that is,
aji
2/1
^X
X2
2/2
'X
x^
2/3
«2
2/2
X,
2/3
x^
2/1
-i' In this case the order is 4 and weight 6.
i The form is therefore
^ Aa^ao -{-Ba^a^a^aQ + Ga^ai + D^ 0^0 + -^^2 «i (!)•
f' Operating with a.— \-2a-, — l-Sa, -; — > ^^ g^t
f c^cti c2a-2 da^
(B + 6A) a^a^al + (3(7+25) a^al a^
^' +(2^+6D + 3J5)a^aiao+ (4:^7 + 30) a^ctj = 0,
J which gives us, taking J.=: 1, and equating each term to 0,
M 5 = -6, (7=4, &c. ;
or^ since ^q = a and a^ = d, we have
] a'd:'-6ahcd-^Wd-^4iac'-Wc\
< The result would be the same had we required the product
of the squares of the differences
I The order being the same, 4, and the weight also 6, the form
I ^ would be
I — Aa^ + Ba^a^a-^ + Ga^a^ + Ba^, + Ea2 % .
[ ' To render this homogeneous, as if derived from a homo-
5 geneous equation in x^ y, each factor must be divided by a^,,
and the whole multiplied by the highest power of % in any
denominator. It would then be identical with (1).
12 INVARUNTS.
6. The eliminant* or resultant of a system of equations is
that function of the coefficients whose vanishing expresses
that the equations are simultaneous. If we have as many
independent equations as we have variables, we can ordin-
arily, by direct elimination, arrive at such a function freed
from any of the assumed variables. This function is generally
indicated by A.
7. Eliminant by symmetric functions. — The product of the
several roots of an equation is a symmetric function, as Sa/3y
or Sa^/3. If we have a, /3, y as the roots of the equation
f(x) =0, and a, (j^, y^ as the roots of /^ (a;) = 0, then, since
they have a common root a, the eliminant condition is involved.
If the first set of roots be substituted in the second equation,
fi{x), the result for the value a will vanish; therefore the
continued product
/.(a)X/.(«X/.(r)
will vanish ; and consequently will conform to the definition of
an eliminant, since it is plain, being a symmetric function of
the roots of f(x), it can be expressed in terms of the given
coefficients of f(x) = 0 and /^ (x) = 0, however they may be
written.
From this it is seen that the eliminant is a function of the
difierences of the roots of the two or more equations.
If the equations are homogeneous, f(x, y) = 0, f^ (x, y) = 0,
they may be treated as non-homogeneous by dividing each
equation by the coefficient of the highest power of x and the
highest power of y. To illustrate this form of operation, let
us find the eliminant of
aa3'+ ^hxy-Y cy"" = 0 (1),
a,x^ + 2b,xy-{-cy = 0 (2),
* Thus U * I = 0, \a c \^-\b c\x\a *| = 0
are determinant expressions for the eliminants of
ax +b = 0 . ax^+bx +c =0 j.psT)Potivplv
a,x + b, = 0 ^"""^ a,x + b,x + ci = 0 respectively.
INVARIANTS. 13
or, written in the non-homogeneous form,
0^ + 72^ + % =0, z^-\-mz-{-mi = 0.
The symmetric function is then
(u^ + ma + mj) (fP + ml3 + m{) = 0,
or
aV^ + mal3 (a + /3) + m^ (a^ + i3') + m^afi + mm^ (.ci-\-(i)'i- m\ = 0,
But a^^-\Z^ = n^-2n^ (Art. 2), a;3 = w^, a+/3=:— w.
Hence (%— mi)^ + (??2— w-)(nim— ^imj) = 0.
Giving m, n^ m^, w^ their values, we have
or (^ca-^—c^ay + 4i(h-^a — ha-^(h^c — hc-^) = ^j
the eliminant. This method is useful in this place simply as
an exercise in symmetric functions. In practice, it would be
far easier to eliminate directly.
8. The order. — By inspecting this example and others, we
are enabled to determine inductively the order of the eliminant
in the coefficients. The symmetric function consists of as many
factors as there are units in the degree of the first equation,
but each of these factors involves the coefficients of the second
in the first degree. On the other hand, the entire product
consists of the several symmetric functions of the roots of the
first equation, and the highest degree of these is the same as
that of the second equation ; hence it is evident that the orders
of the coefficients in the eliminant are the same as those of the
^ven equations, but taken in an inverse order ; that is, the co-
efficients of the first equation have the order of the second, and
le contrary.
If, for instance, there were three homogeneous equations in
bhree variables of the 2nd, ord, and 4th orders, then the eliminant
^ould be a homogeneous function of the 12th order in the co-
14 INVARIANTS.
efficients of the first equation, of the 8tli in those of the second,
and of the 6th in those of the third.
9. The weight. — It is not so easy to determine the weight. But
we may begin by considering that the elirainant is a symmetric
function of the differences between the roots of the first and
second equations expressed in terms of their cofficients, and
then the number of these diSerences is equal to the product of
the orders of the equations. If we multiply each root by any
factor as Tc, we do, in efiect, multiply each difierence by h ; and
consequently, the eliminant, which is the product of these
differences, is multiplied by A; to a power equal to the product
of the degrees of the equations. Now, each root in the
equations
a^x'' + na^x''-^y + \n{n—l) a^x''-^y^ + &c. = 0 (1),
h^x"^ -\-mh^x'"-'^ y -^^m (7n-l) h^x'^-^if + ^c. = 0... (2),
will, it is evident, be multiplied by Ic when we multiply a-^, \ ;
ftg. ^2 5 ^y ^) ^^» <^c. ; and therefore each term of the eliminant
would involve k to the mn^^ degree. In this manner we can
readily determine the weight of each term, which we shall find
to be constant, that is, m7i for each term.
10. It is easy to see, from the definition of an eliminant and
from the results of (Art. 3), that the eliminant must satisfy
the difi'erential equations there given ; or, if referred to equa-
tions (1) and (2) of the last Article, must be of the form
«o -^ •" 2^1-5 1- 3^2— h&c. + &o-T7— +<^c. = 0,
aaj da^ da^ db^
where A represents the eliminant of equations (1) and (2).
11. The eliminant of three equations in three variables. — The
eliminant vanishing, the equations are simultaneous. This can
be brought under the system of two equations. For, solving
between any two equations, and substituting these values in
the third, the product of these substitutions must vanish, since,
by hypothesis, there is a community of values between the
INVARIANTS. 16
different sets, the number of which must equal the weight of
the eliminant of those two equations, that is, the product of
their degrees. These substituted successively in the remaining
equation, and multiplied together, will furnish the requisite
symmetric functions by which the coefficients of the solved
equations may be expressed, which gives the eliminant whose
weight is equal to the product of the degrees of the three
equations. For four equations we proceed in the same manner,
solving for three and substituting these values in the fourth.
12. In reviewing this method of elimination, it will be seen
to be of the widest generality, and all its results susceptible of
very satisfactory proof. It is not introduced for any use in
actual elimination, but that the reader may here avail himself of
important assistance in the study of the Theory of Invariants.
IB. The reader interested in determinants will naturally
seek some form for elimination by this method. That of Euler
leading in this direction is of high theoretical value. Two
equations, homogeneous or otherwise, are supposed to be satis-
fied by a common root of the first degree ; then the first, multi-
plied by all the remaining factors of the second, is evidently
equal to the second multiplied by all the remaining factors of
the first ; as, if we have
x^—(a + h) x-\-ah — 0,
x^—(a-\-c) x-\-aG ■= 0,
then (x — c){x^—(a-\-lj)x-\-ab}=. {x^b) {x^—{a-\-c) x + ac]',
or, in general, if we multiply the homogeneous equation
f{x^ 2/) = 0 by any arbitrary function of a degree one less
than /i (aj, y) = 0, and the latter by any arbitrary function
with a degree one less than the former equation, and then
equate term to term, we shall have a number of equations
equal to the sum of the degrees of the two given equations,
and the eliminant will of course appear in the form of a deter-
minant.
16
INVARIANTS.
To eliminate between
ax^-\-2bxy + cif,
(hx^ + Uxy + hf) (ax^ + 2hxy + cy^)
= (Jc,x-\-l,y)(a,x^-\-Sh,x^y + Sc^xy^ + d2/) ;
equating like terms,
ha—\a^ = 0,
2nc + Ua-2>W—\a^ = 0,
Tcc + la + ^Uh-Zh^c^ — ZW = 0,
2lb + Jdc-]c^d—Sl^r.^ = 0,
lc-\d = 0.
Eliminating fc and Z, we have
aOO— CTi 0=0
26 0 a -36i -«!
c a 26 -3ci -36i
0 2& c - c? -3ci
0 c 0 0 — d
as the eliminant.
14. The various other methods, such as Bezout's method,*
Sylvester's dialytic process, the uses of the Jacobian in elim-
ination, explained in (D. 39), f since they do not illustrate the
Theory of Invariants, may be omitted.
We will now pass at once to a subject which is intimately
connected with that theory.
15. Discriminants. — If an equation, or quantic, as it is
called when it is not equated to 0, be differentiated with
* I must qualify this statement, so far as it relates to Bezout's method.
It is well known by those acquainted with Dr. Sylvester's researches, that
what he calls a Bezoutiant is the discriminant of a quadratic function in
any number of variables, and is expressible as a symmetrical determinant
which is written, as in (D. 22), with a double suffix. The eliminant of two
equations of the w*^ degree may be similarly expressed. The use of the
Bezoutiant in the theory of equations is exhibited in a Memoir by Syl-
vester, Phil. Trans., 1853, p. 513.
t Tract No. 1, Determinants.
INVAKIANTS.
17
respect to its variables, the eliminant of these several
differentials is the discriminant. As the quantic is under-
stood to be homogeneous, it is evident that the discriminant
must be homogeneous also. The order of the discriminant
is clearly the product of the degrees of the differentials of
which it is the eliminant.
Observing the same order of the suffixes a^^ %, &c., the
weight of the discriminant will depend upon the number of
differentials and the order of the quantic. Thus, for a binary
quadratic the weight must be 2 ; for a ternary cubic, that is, a
quantic containing three variables, 3 (3 — 1)^. This arises from
a slight modification of the reasoning in Art. 9. The weight
would be evidently (w — 1), taken as many times as a factor as
there are variables, were it not for the consideration that all
but one of these differentials begin with a coefficient whose
relation to the leading variable is the same as in the original
quantic ; in other words, with a suffix one greater than the first
differential which begins with % ; hence the number of suffixes
must be increased in this proportion. If j) = the number of
differentials, (t^ — l)^ must be increased by (%— 1)^~\ that is,
{n—\y + {n-iy-^ = n {n-\y-\
which is the sum of the suffixes for each term of the dis-
criminant.
16. If we divide the homogeneous equation by y'', the result
is reducible to a product of factors, as
X y X X y X X y X &c. = 0.
^i yi ^2 2/2 • ^s 2/3
Comparing this product with
we see that 2/i!/2 2/3 ^^' — ^oj
since the product of
(xy^—x^y) (xy^ - x^y) (xy^ -x^y)&c.= Q (1)
^ives 2/i 2/2 &c. for the coefficient of x^.
18 INVARIANTS.
17. The discriminant is equal to the Jcontinned product of
the squares of the differences of the roots of the given quantic
taken two and two.
Suppose a3i?/i, x^y<^^ x^7j^, &c. are the^rootsjof (1) above,
then -j^ = 2/i {xy^—yx^){x])^-yx^ &c. + ?/2 (xy^-x^y) &c.
ax
fid
Observing the effect of substituting a^ij/i in - , which is
ctx
2/i (^i2/2"~yi^2) <^C' j substituting in the same manner x^y^^
ajg^/j, &c. in the same equation, and taking the continued pro-
duct, we must have
2/i2/2 &c. fe2/2 -2/1^2)' fe2/3-2/i^3)^&c. = 0 (1),
which, as we have seen, is the eliminant (Art. 7) of Q and -— ^.
ax
The same product, divided by y-^ y^ &c. = a^, will give the dis-
criminant.
This will more fully appear when we consider that
then, when we have substituted successively all the roots of
— — = 0 in Q, we shall have for the continued product
ax
2/12/22/8 ^^- niultiplied by a similar result of substituting the
same roots in —P^- But this latter result is evidently the
discriminant. Hence, if (1) be divided by a^,, that is, if the
eliminant of the quantic and its first differential with reference
to X be divided by the product of the ?/'s, we shall obtain the
same result as if we had found the eliminant of the first differ-
entials with reference to x and y.
18. Enough preliminary matter has now been introduced to
enable the reader to follow with profit all that will follow.
INVAEIANTS. 19
To those who wish to pursue the theory of discriminants
further, and desire to study an interesting geometrical applica-
tion, the theorem of Joachimsthal, taken as the basis of an
investigation on the nature of cones circumscribing surfaces
having multiple lines, by Dr. Salmon (" Cambridge and Dublin
Math. Journal," 1847 and 1849) would probably prove as
fruitful in this direction as any that could be mentioned.*
* The theorem above alluded to is included in the following statement.
If we have the quantic {aQ, a-^ ... ««_i, an'^x, y)", and a^ contain a factor ty
and if ^o contain t" as a factor, the discriminant will be divisible by t^ ; also,
if a^ contain ^ as a factor, and if a^ and aQ contain t^ and t^ respectively,
then the discriminant will be divisible by t^, and so on. The application
by Dr. Salmon was that, if Uq + a^x + a^'^ + &;c. be the equation to a sur-
face, and if xy be a double line, a^ will contain y in the second, and a^ in
the first degree. The discriminant in respect to x is divisible by y\ and
the locus is a tangent cone.
20
CHAPTER II.
FORMATION OF INVARIANT FUNCTIONS.
19. The definition of an invariant and covariant of a
single quantic has already been given (D. 42). In pur-
suance of this, we might proceed at once to show how in
general such functions can be formed, and then give some
explanation of the geometrical importance of the theory;
but, for the sake of clearness, we will commence with one of
the simplest examples of an invariant function.
20. The determinant of a system of linear equations is an
invariant of that system, because, as it will be remembered,
when the variables are all transformed by the same linear
substitution, the determinant of the transformed equations
is equal to the determinant of the given equations mul-
tiplied by the modulus of transformation (D. 42). In
other words, the determinant (function of the coefficients) of
the given equations, which remains unaltered by the trans-
formation, is called an invariant. The equations of (D. 17)
will exactly illustrate this.
When the linear equations
dv + ev^ +/v2 = 0 ^
^l^ + ^l^l+/l^2=0i (1)
are transformed by the substitutions
ax -\-by +CZ = v ^
a^x + h^y + c^z = v^ I (2),
then the determinant of the transformed system will be equal
INVARIANTS. / 21
to (dcif^) X {ab^c^, which may be written
/(transformed) = A'* /(given),
where A = the modulus = the determinant formed from the
sinister members of (2).
The /, which is also a determinant expressing the coexistence
of equations (1), is in this case called an invariant.
21. It is not difficult to see from the above that a somewhat
complicated problem is now presented to us. We are to trace
the effect of linear transformation upon the same functions of
the coefficients of an equation, to determine the number of
the functions which remain unaltered by such transformation,
and to deduce convenient rules for their formation. It was
seen, for instance (D. 41), that when the binary quadratic
(ahc^Xj yY was linearly transformed by the substitution of
X = lx-\-my^
y = \x-\-m{y,
we wrote Ax'^ + 2Bxy + Gy^ as the transformed quadratic,
in which
A= a?-Y2Ul^-Vcl\,
C = am? -\-2hmm-^-\- cnii ,
JB = aim + h (Im^ + Z^m) + cl{in-^,
and from which we obtained
AC-B'' = {ac-h^){lm^-\my.
The invariant in this case, ac— h^, is no other than the dis-
criminant of the given quadratic ax^ + 2hxy-^cy^.
22. Proceeding now to the binary cubic (a, h, c, d'^x, y)*,
we obtain its discriminant ; that is, we find its two differentials,
and by direct elimination their eliminant, which is the dis-
criminant, viz.,
4>(bd-c')(h'-ac)-\-(ad-hcy,
which, in form, is the same as that obtained in Art. 7. And
here, again, we say that the invariant in this case is no
other than the discriminant.
22 INVARIANTS.
23. Since ac—h^ is an invariant of the quadratic
we can, by the introduction of a constant, derive not only two
invariants after the analogy of ac — W, but one other whose
constituents are derived from the coefficients of the trans-
formed system.
Thus, if we have the two quadratics,
s^x^-\-2t^xy+uy,
we may multiply the second by k, an arbitrary constant, and
obtain by addition
(s + Jvs^) x^-\-2(t + U^) xy + (u + Jcu^) y\
which, by a transformation identical with that in Art. 21,
becomes
(8+hS,) X' + 2 (T+JcT,) XY+(U+hU,) Y^
and the invariant, consequently, by symmetry, is
i8+JcS,)(U+hU0-{T+hT,y
= A^\_(s + hs,)(u-\-7cu^)-(ti-M,y].
Since this equation is satisfied by any value of Tc, and therefore
identical, the coefficients of the like powers of h must be equal,
and therefore
8U-T' = A''(su-f),
8,U-Tl = A'(s,u,-f,),
8U,'j-8,U-2TT^ = A\su,+s,u-2tt,).
The last of these is therefore an invariant of a system of two
quantics.
And here it would be well to observe that, if we had operated
upon the given invariant su—f with ^i "T" + ^i -jT + ^i -t->
the result would have been the same as that actually obtained
by the substitution of s-\-ks^ for 5, &c. ; and thus, in general, if
we have an invariant of any known quantic, we may find the
INVARIANTS. 23
invariant of a system of two or more simultaneous quantics of
the same degree, either by substitution or by the use of the
operator, as above.
24. As we have stated, the binary quadratic and cubic have
no other invariant than their respective determinants ; but, as
we shall see, binaries of a higher degi'ee may have two or more
functions unaltered by transformation. For instance, if we
take the binary quartic
ax'^ + 4ibx^y + Qcx^y^ + 4dxi/^ + ey*,
and operate upon it with the symbols* -— and -, that is,
ay dx
substitute _ - for x and — - for y ; we shall find that the
dy dx '^
result ae—4<hd-\-2c^
will conform to the definition of an invariant. The same v^ill
be true if we expand the determinant formed from the fourth
differentials of the quartic, viz..
a h c
bed
c d e
ace-{-2hcd-ad'-eh'-G' (1).
That these two invariants may be derived from the binary
quartic, may be shown by actually transforming the given
* The theory of these symbols must be reserved for another part of the
subject ; see Arts. 28, 33. The actual process is to introduce the symbols
into the quantic, thus obtaining a differential symbol. Thus, by substi-
tuting for v, and — for a; in the given quantic, we have
dx dy
a 4o + 6c — - — — Ad + e — -.
di/ dfdx dy^dx^ dydx^ dx"^
Operating upon the quartic with this symbol, we get
A8ae~192bd+lUa^
Hence the relation is . ae — ibd + Sc^.
If the quantic had been of odd degree, the result would have vanished.
24 INVARIANTS.
quartic, equating the values of -4, B, C, &c., and we should find
and also another, (1), which entering into the discriminant
forms still a third. These latter, however, do not at present
concern us beyond the assurance that they exist, the theory of
their formation being reserved for future consideration.
25. The Theory of Covariants will be found to be immedi-
ately connected with the Theory of Invariants. This follows
from the fact that the covariant is a function not only of the
coelEcients, but also of the variables of the given quantic ;
that is,
/(iS, Z7, &c. X, Y, &c.) = A'"/ (5, u, &c. x, y, &c.)
To illustrate this, let us take the Hessian (D. 40) of the
quartic
ax*+Ux^y + 6cxY + Mxif-\-ei/ (1),
and we have
ax^-{-2hxy + c7/, hx^ + 2cxy-{-dy^
tx^ + 2cxy + dy"^^ cic^ -|- 2dxy + ey^
which, expanded, gives the invariant form mn—l^, and differs
in form only from the invariant of the quadratic (Art. 23)
ac — lr' by the variables of the given quantic.
Looking at this example a little further, we see that (1) and
(2) contain the same powers of the variables, and equally the
same coefficients. Hence the invariant of the covariant in
this case can be no other than the invariant of (1), and this
conclusion may easily be seen to be general.
26. Covariants may be formed by substituting in the given
quanbic x-\-hx^ and y + hy^ for x and y. The coefficients of
the several powers of h form covariants, and, taken in order,
are called emanants of the quantic. Thus, if we take the
binary cubic
ax^-\-2>l)xhj + Zcxy'' + dy\
* To transform this quartic, ax'^ &c., the reader has only to repeat the
process of Art. 21 on a larger scale.
(2),
INVARIANTS. 25
and substitute as proposed, we shall find the several emanants
^1 — + 2/1 , j , in which form the coefficients of the
ascending powers of h appear. -
The emanants of any quantic can in general be expressed in
the form
(^v^+2/.^) (1)
as first, second, and n^^ emanants.
If we take the second power, to get the second emanant,
we may write
Xi rr^ + Ix^y^mn + \j^ n? (2)
as the result, where w and n represent the differentials ■ , -— ,
^ ax dy
and a?i, ij^ are regarded as cogredient to,* or vary as, the given
variables. Now it is easy to see that, if we regard (1) or (2)
as a function of x^, y^, and the original variables as constants,
and proceed to form the invariants, these invariants will iu
turn represent covariants of the given quantic, if we then con-
ceive 03, y as variables.
This will appear at once by reference to Art. 23, where
we were enabled, after transforming the quadratic, to write
Transform now (2), and write its invariant, and we shall have
dX' ' dY' KdXdYJ Idx' ' dif \dxdyl A '"^ ^'
where V = the transformed, and v = the original quantic.
* To exhibit this, let (abc^x, t/Y be the given quantic.
Making the substitutions, we have
k^ (axi^ + 2bxit/i + cy^) and 2k { axx-^ + b [xy^ + x^tf) + cyy-^ }
as first and second emanants ; but, by hypothesis, X\, y-^ are cogredient to
xy ; hence each of the coefficients above resume the quadratic form, in
other words, they become identical. Hence there are not, as we shall see,
two covariants to the quadratic (^abc^x, y)^, but one (as there is no func-
tion of the differences of the roots), and that one must be the quantic itself.
26 INVARIANTS.
This invariant is now the covariant of the given quantic, as is
evident algebraically if we compare (2) with the form for the
second emanant,
/ dv , dv\^
in which x, y are first treated as constants, by which supposi-
tion we form the invariant, and then as variables, so that the
result conforms to the definition of an invariant. It is plain
also that (3) is the expanded determinant formed from the
quadratic emanant, and in this sense may be regarded as a
discHminant of the quadratic function, and is therefore an
invariant with the limitation that its variables are regarded as
constants. The first emanants of a system of linear equations
yield a determinant. Hence in general we may say that the
Jacobian (D. 39) of the first emanants of a system of linear
equations — that is, the first difi'erentials of these equations
regarded as functions of a^i, yi, ^j^ — will form a determinant
which is a covariant of the system.
27. Inverse Suhstitution. — Tn Trilinear Coordinates a, (3, y
are used ordinarily to express the coordinates of the point.
Let now a3a + ?//3 + ^y = 0
be the equation of a straight line in which a, /3, y are the
tangential coordinates of the line, that is, its perpendicular
distances from the three points of reference ; and ^, y, z the
perpendicular distances of any point in the line from the three
lines of reference, that is, its trilinear coordinates (T. 2, 25).*
By transforming this equation to new axes by linear substi-
tution, it will be seen that, while the trilinear coordinates are
transformed by direct substitution, the tangential coordinates
are transformed at the same time by the inverse substitution.
Let X = l-^X+m^Y+n-^Z^
y = Zg^+^c.,
z = Z3X+&C.
* Tract No. IL, Art. 25.
INTARIANTS. 27
Then the new equation of the line may be written
AX+Br-f r^= 0 (1),
where A = l^a -\- 1^(d + l^y,
B = m^a + rn^lj + m^y,
r = n^a -^-n^i^+n^y.
The two sets of coordinates in this case are said to be
contragredient to each other ; and in general it may be stated
that the tangential coordinates, whether of a line or a plane,
will be transformed by a different — that is, inverse — substitu-
tion, from the coordinates representing different points. The
latter are said to be cogredient, as x, x-^, y, ^/i, &c., because
transformed by the same substitution ; while the former are
said to be contragredient, because tranformed by an inverse
substitution.
28. This may be stated in another form ; for, since the equa-
tion which was a function of x, y, z has been transformed to a
function of X, Y, Z, the total differential coefficients with
respect to the latter are functions of those with respect to the
former.
We have, from (1) of the last Article,
a {l^X-^-m^Y+n^Z) + /3 {l^X+m^Y+n^Z)
-\-y{l,X+m^Y^n,Z)^0',
and therefore
d _ 1 ^ y 1 ^ \ 1 d^
dX ^ dx ^ dy ^ dz^
dx , «
smce -—. = fci, <x;c.
clX
Comparing oc, y, z with ., — , — -, we see that the substi-
ct^ cty ctz
tution which linearly transforms the one will linearly trans-
form tlie other, but by a reciprocal relation as expressed by
28
INVARIANTS.
the determinants of the coefficients
h h 's
m-i Wg Wj
If now , , ~z~, — vanish, as will happen only when the dis-
bar di/ dz J
criminant of the quantic or system vanishes, then — - will
necessarily vanish also.
29. The consideration of inverse substitution leads directly
to a function well known in geometry as the contravariant. For
it will be seen at once that, if a quantic which is a function of
two sets of variables x, y, z; a, /3, y, be linearly transformed,
the function involving the coefficients and the variables, re-
garded as transformed by the inverse substitution, must be
similar to the covariant, but which is called the contravariant ;
that is,
/(A, A,, &c. A, B, r) = A-/(ao, «!, &c. «, /3, y).
It is evident also, from what has preceded, that the contra-
variant may be deduced in a manner similar to that exhibited
in Art. 28. If we take the binary quadratic,
ax^-^2hxij + cy^ (1),
and combine it with Jc(xa-\-yl3y (2),
we shall have
{a-\-1ca')x' -\-2(h + lcaf5)xy + (c + hP')y' (3).
If now (1) becomes, by linear transformation,
and (2) becomes h {XA + ^B)^
then (3) becomes
INVARIANTS. 29
and the invariant form gives
(A + JcA') (C + JcB') - ( 5 -f JcABy
And since we may equate the coefficients of the like powers
of k, we have
AB2-2i?AB-f-CA2 = A\al3^^2hal3 + ca^) (1),
that is, a/3^— 2&op + co^, differing only by a power of the
modulus from the corresponding function of the transformed
coefficients and variables o, (j, is a contra variant.
In reviewing now three functions thus considered, it will
be seen that they all equally possess the property of in-
variance.
30. In general, when a, fy,y are regarded as contragredient
to X, 2/, ^, the contravariant may be expressed, by application
of the preceding Article,
^0^" + &c. + A: (XA + YB + zry
and the invariant would be
/(^ + M", A,-^hA^'-'B, &c.) = A"7K + ^-"«", a, + Jca»-^l3, &c.)
We have thus to develop the sum of two functions, which,
by Taylor's Theorem, gives us for the coefficients of the con-
stant /j,
\ claQ da^ db^ I
If r=l, this formula gives us what has been called the first
evectant.
* P = the invariant of the quantic.
80 INVARIANTS.
To apply this, we know that ac — \p- is the invariant of the
quadratic. We have then
which is identical with (1) of the last Article.
If the quantic be a ternary quadratic, as
we shall have for an invariant
a^ ^6 ^'4 ■= %a^a^-\~^a^a^^^ — a^a.^ — a^(Xi^ — a^<i^ = F,
ag «! ttg
^4 «3 «2
which is the discriminant of the quantic ; whence
\ aa-0 aaj, aaj actg aa^ daj
+ 2 (rt^ag— ao%) /^y + 2 {a^a^ - a^a^) ay + 2 (dga^ - aa^Tg) a/3
is a contravariant, and in geometry expresses the condition
that a given line represented by a trilinear equation shall
touch the given conic, or, in other words, is the tangential
equation of the conic*
It is to be observed that the discriminant of the first evec-
tant of the second degree can be written as a determinant :
dE
da
= 0
Salmon's '* Conies," p. 249.
INVARIANTS. 31
and may be regai'ded as the invariant of a system, or of the
given contravariant.
It is also to be observed that there are instances of functions
involving both x, y, z and o, /3, y, and which do not change
by transformation of the quantic, that is,
/(Jo, A - X, Y... J,5...) = A-/K% ••• ^-.y ■■■ «,/5...),
which have received the general name of mixed concomitants.
That such functions may easily be formed may be seen by
examining the covariant (1) of Art. 26.
If we subject it to the operation of finding the coefficients
by Taylor's Theorem, we shall have ( «^ y +"/^ 77 "^^^^T ) ^>
where C = the covariant in question.
31. We may here perhaps interest the reader by introducing
an. illustration of the geometrical application of invariants. It
is well known that when we transform from one rectangular
system to another, that a+h and ah — Ji^ remain unaltered by
the transformation. Suppose it were inquired as to the form
these quantities take when the transformation is made from
rectangular (or oblique) to oblique axes, where a, h, h are con-
stants in the quadratic
ax^ + 2hxij + bi/ (1).
Let the transformation be from axes inclined at the angle to
to axes of any other inclination, as O. Then, by making the
proper substitutions, (1) becomes
AX' + 'IHXY+BT' (Art. 21).
By symmetry, cc^ -\- 2oi^i/ cos w+y would become
X^ -1-2X^008 12+1^,
as either expresses the square of any point from the origin.
Adopting now a method with which we are familiar, we say •
ax"^ -f 2](Xi/ + hf -h k {x^ + 2x'ij cos w -fy')
= JXH2IfXr-i-5r^ + X;(XH2Xrcosl2+ Y^).
32 INVARIANTS.
If we determine k so that the first side of the equation may
become a perfect square, the second will become a perfect
square also, that is, h must be one of the roots of
^2sin'^w + (a + &-2/tcos w) Z; + a& - /.'^ = 0.
This value of h will make the left member a perfect square.
A similar quadratic will be found in the right-hand member,
which will make it also a perfect square. Both members
become perfect squares for the same value of h^ and are there-
fore equal.
Equating coefficients of corresponding terms, we have, what
we already knew (Art. 5) in form,
a-^-h — 211 cos M _ A-\-B — 2Il co^ £1
sin^ u) sin^ il '
sin^ (1) sin^ li
(This elegant demonstration is due to the late Dr. Geo. Boole.
See Camhridge Math. Jour., N. S., VI. 87.)
32. From Art. 28, we learn that x, y, n; a, (3, y sustain a
reciprocal relation to each other. The same is to be observed
of Xy y, z and —, — , - . The transformation of the former
ax dy dz
transforms the latter, but by an inverse method. In this way
the contra variant is obtained, Vihich, as has been remarked,
possesses the property of invariance. Know in the contra-
variant we substitute --, &c., we shall obtain a function which,
dx
containing signs of operation, and being itself unchanged by
transformation, may be called an operating symbol — a type form
which, if applied to the quantic or to its covariants, must give
either an invariant or a covariant according as the variables
disappear or remain after differentiation.
ca^— 26a/3-f-a/3* being a contravariant of ax^-^ll/xy-^-ci/y
INVARIANTS. 38
we obtain, by applying to the quadratic the operating symbol
(f cP d?
c -— ^ — 2h " — — + a -— the invariant ac — h^.
dx* ax ay ay"
38. Proceeding upon the principle now before ns, we are
enabled to generate, as will be seen, the three functions con-
sidered, by means of simple substitution. Since /(/?, —a)
becomes, by a linear transformation, ' > that is, a
contravariant, we have then, in a binary quadratic, only to
substitute /3 and —a for x and ij to obtain the contravariant.
If, therefore, we write a with the negative sign, there is no
reason why we should not say that — a, /3 are transformed by
the same rules as 03, y. The symbols — , &c., which we re-
garded as contragredient to ce, ij^ may be with equal reason
called cogredient to — a?, Xj ; and, conversely, — , — y- may
be taken as cogredient to a?, y. Hence, if we substitute these
symbols in either the quantic or its covariants, we obtain a
new set of functions of the same form. The exception is seen
in the binary quartics, where, for instance in the quadratic,
the substitution gives 4 {ao—h^), an invariant.
34. A fuller investigation of the quadratic, in the general
theory, will lead to what is perhaps already sufficiently evident,
that the quadratic (a, &, c'^x, yY has no covariant but the
quantic itself.* We have seen that its discriminant is the in-
variant ac—P, and its contravariant ca^ — 2&o/3-|-«/3^ ; and
since ac — h^ is an invariant, we learn, from Art. 26, that the
second emanant is a quadratic in x-^, ?/i, and its discriminant
is a covariant, for a quantic higher than the second degree.
We know (Art. 22) that the invariant of
(a, h, c,d^x,yf (1)
is a'd'-\-4^ac'-6ahcd-\-Mb'-Sb'c' (2).
* An invariant being a function of the differences of roots, there can be
no such function formed other than the given quantic. See Note, p. 24.
84
INVARIANTS.
Hence for every quantic higher than the third we have the
covariant
L \ dx^ dy I dx^ \ dec du I dx^
d^ d'
dyl dx^ Xdx dy J dx^ dx^ dij dx ny^ dy^
dy^ dx^ dy \dx^ dy dx dy'^l \
The covariant of (1) may be found by forming the evectant
(Art. 30)
where P = (2).
Then, by substituting a;, y for a, /3, we have
03^ (acZ2-3&c(^ + 2c«) + Zxhj {-acd-^'lhH-hc^)
+ 'Sxy' {-aU^'lao''-V"c) + y^ (a^d-3abc-h2h^).
And thus generally for binaries, when any invariant is known.
35. If we take any quantic, and observe the effect of any
linear substitution, it is easy to see that its invariant will
remain unchanged if for x we substitute y or Ix, and y for x.
It will be seen that the order or degree of the invariant is
still constant, and also that the weight, which is estimated by
taking the sum of the suffixes of the factors of the several
terms, is constant for each invariant.
If s, Si, s.^, &c. represent the suffixes before transformation,
n — s, n—s-^, n—s^ &c. will represent the suffixes of the same
coefficients after transformation, and we shall have
s + Sj 4- cs*2 &c. = n — s-\-n — 8^ -\- n—s^ &c,,
or 2m; = nt^
where w = the weight of the suffixes for each terra of the
coefficients, and t =. the degree or order of the invariant. In
other words, the weight is = \ut.
INVARIANTS. 35
In this way the invariant of any quantic may be written at
once, the required degree being known.
If, for instance, an invariant of a binary quartic of the
second degree in the coefficients is required, we have
w = \nt = 4.
There will be as many terms of the proposed invariant as the
sum of two numbers 0 ... 4 inclusive can be written ; hence
is the required invariant. The values of ^q, &c. will depend
upon other considerations. The first is, that an invariant must
be a function of the differences of the roots ; for it is to be
unchanged when we effect the transformation by substituting
x + l for X ; it must therefore satisfy a differential equation for
the function of the differences of the roots, as
«o^- + 2«,^ + 3^,§^+4«3^ + &c.= 0 (2).
da^ da^ da^ da^
The second consideration is, that the coefficients thus ob-
tained are clearly proportional.
Applying then (2) to (1), we have
(.42 + 4^o) a^a^ + (4^^ + 8^2) «ia^2 = 0.
Taking ^^ = 1, we find the invariant to be
a^a^— 4aiCt3 + 3(X2 ;
or, using the coefficients of the quartic,
Thus the differential equation furnishes the conditions to
determine the values of A^, &c.
If the number of conditions is greater than these coeffi-
cients, there is no additional invariant ; if the same, one more,
or one alone ; if less, more than one. If we wished to obtain
the discriminant of the quartic, which is also an invariant, by
this method, or rather if we wished to obtain an invariant of
36
INVARIANTS.
the sixth order in the coefficients, we should find the number
of ways in which 12 can be written as the sum of 6 numbers
from 0 ... 4, and we should have as many conditions as
\nt—l or 11 can be written as the sum of 6 numbers from
0 ... 4.
36. We may arrive at the covarianfc in the same manner.
If n represent the degree of the quantic, n^ the degree of
the covariant, in x and y, and m the degree of x in any term,
we have
m-\-s-\-s^-{-s^ &c. = n^—ni + n — s -{-n — s^ -j- n—s.^ &c.
Calling m + s+Sj &c. the weight, the equation gives
w ■= \ (nt + n^.
If we wished to form, for instance, the quartic covariant to
the quartic of the second degree in the coefficients, we could,
instead of taking the Hessian of the quantic, which would give
the required covariant, estimate the terms multiplying each
variable, since ^ = 2, ^ = 4, m = 4, and, if we are concerned
with the coefficient of ic'*, n^ = 4.
The weight would then be 6, and hence
4 + s + 5i = 6,
5 + 51 = 2.
There are therefore two terms multiplying x^ each of the
second degree, that is, a^^a^ and a-^a-^, or ac and IP'.
In the same manner we find, for the terms which multiply aj',
3 + 5 + Si = 6.
Hence the terms are a^^a^ and a^cig, or ad and hc^ &c. &c.
Now it will be perceived we do not know how by this pro-
cess to connect ac and V', ad and he.
To ascertain this relation, let us suppose that
^;fc"^ + Mi^"^~V+'^^-%^^2-^"'"'y + &c (1)
represents the covariant.
INVAR IA^'TS. 37
Suppose also
f^Q-j — \-a^-Y- &G. and na,-rr--\-(n—V) a.—~ &g.
to be represented by a and /3. If now, in (1), we suppose the
same substitution as was made in the original quantic, and
that
da -^' da -""'' J^-^^^' ^-^'^^'
then f=a,f+2a,f+Sa,^ + &c.;
da da-^ da^ da^
and we can write, on the supposition that these changes are
identical,
-- ^ = 0, &c. as above,
da
and, for the same reason.
Thus, when A^ is a function of the differences, we can find
all the other terms of the covariant ; that is, we can, by suc-
cessive differentiation, pass from one term to the other, and
thus, by the use of these two operators, determine the exact
form of the coefficients of the covariant. Thus, in the case of
the quadratic covariant to the quartic, we found A^ to be of the
dA
form Aaffi^-\-Ba^a^^ which, operated upon by -r-^, becomes
da
(J.H-25)«o«i = 0. If ^.=1, then ^= — 1, and Af^-=. a^a^— a^a^.
dA
Operate upon this latter with —r^^ which in this case is
da^ da^ da^ da^
and we get 2 («5o^3'~"^2^i) = -^i*
dA
Again, operating with -j^ upon a^^a^—a^a^, and we have
4«ifl'3+a^«0— 2^3«l — S^oflTg = ^2 = flr4«'o+2«ifl'3— 3«2^2-
38
INVARIANTS.
dA
Operating then upon this latter with -—, we obtain
2(a^a^-a^a^) = A^
And finally we have
dA,
di3
= a^^a^—a^a, = A^.
The covariant then, written fnlly, is
(ac- W) x' + 2 {ad- he) x^i/ + (ae + 2hd-Sc'') xY
+ 2 (he-cd) xy^ + (ce-d') y\
We see, therefore, that A^ is the source of the covariant, and
we can readily write
as the law of derivation.
That Jo is appropriately called the source is evident from
its repeated use, being, in fact, operated upon by each succes-
sive differential symbol, as is seen on p. 36.
39
CHAPTER III.
THEOEY OF LEAST OR CANONICAL FORMS.
37. When a quantic has been reduced to the least form in
which it can be written, and yet retain its generality, it is said
to be reduced to its canonical form. The theory has been
presented by Dr. Sylvester (see Philosophical Magazine, Nov.
1851). The name canonical seems to have been first applied
by Hermite. The number of constants remains in most cases
implicitly the same.
Since lx-\-my may be represented by X, and I'x + m'y by Y,
a cubic in two variables may be represented by X^-\-Y^. This
is evident, as the entire number of constants is implied in
this form.
The quadratic (a, h, c'^x, yf can be reduced with four con-
stants* to the form x^ + y^, or to a similar form Az^-\-Bi^ con-
taining the original number. But the binary quadratic in
geometrical investigations is so completely manageable in its
* To reduce 2x"+ lix + 29 to the sum of two squares.
We have {Ix + myf + {}'x + m'yf
as the transformed quadratic, or
{x + ty- + (a; + t'Y = 2a;2 + 14a; + 29,
where t = — , and if = %,
I i
whence f^ + f^ = 29,
and t + t'=^7 or t = 2, t'=5,
while the coefficient of x is plainly 1, therefore {x + 2f + (a? + o)- is the
expression.
D
40 INVARIANTS.
original form that its reduction to a sum of squares is not a
matter of mucli interest.
But the reduction of the cubic is of more practical impor-
tance, since, independent of geometrical considerations, the
reduction to the sum of two cubes furnishes a method of
solution of numerical equations. The cubic
becomes, we will suppose, by transformation
and, remembering that the Hessian
^ ^_ I ^^^^ ^ ^
dx^ dy^ \dxdyl
gives a covariant which may be transformed in the same
manner and into a function of the same constants as before,
that is,
Idx' ' df [dxdyj J dX^ ' dY' [dXdY/ ^ ^'
we see that the transformed becomes ADXY when B and G
vanish.
Or, since we are simply seeking the factors into which the
Hessian may break up when B and G vanish, we may omit the
factor A^ (being composed of the constants of transformation),
and examine the left-hand member of (1) for the required
factors X, Y.
With these conditions, the Hessian cannot differ by more
than a factor from XY.
As an illustration, let us take
4a3H30a32+ 78» + 70 = 0 = w.
The Hessian is
2x-\-5 5x-\-lZ I = x^+bx-\-6.
5aj + 13 13a3 + 35
INVARIANTS. 41*
Taking the factors of this, x + 2 and a? + 3, we have
A(x + 2y-{-D(x + Sf
for the determination of A and D by comparison with the
given quantic
^+ D=4,
SA + 27D = 70,
or A=2, D = 2.
Hence 2 (a! + 2)» + 2 (x + Sy = u,
that is, (a; + 2)^ + (a;4-3)^ diifers by only a factor from it,
and therefore
(aj + 2) + (a3 + 3) = 0
gives a; = — f as a root of the given cubic. The other roots
of this cubic being imaginary, it is evident that not every
cubic can be reduced to this form, since it must differ from
one which contains three real factors, or one containing a
square factor.
In the latter case, we could evidently express the canonical
form of the given cubic by (Ix -\- myf (I'x + m'y) or (x + tf (x + f)
or x^y.
To reduce x^+7x^-{-16x + 12 to the form x'^y.
We have x'+(t'-\- 2t) x^ + {2tt' +f)x-\- tH\
whence i'+ 2^ = 7,
m+ f = 16,
tH'= 12,
^ = 2, r=3.
38. The canonizant. — This is a name given by Dr. Sylvester
to a determinant which is used in the extension of the method
of the last Article. The theory assumes that a quantic of the
D 2
42
INVARIANTS.
fifth degree can be reduced to a sum of three terms of the
fifth degree, one of the seventh degree to the sum of four
terms of the seventh degree, and thus for every odd degree ;
and then proceeds to make the assumption good in the
following manner. The transformation is supposed to be
efiecfced, as before, by letting
s = Ix+my, t = l'x-\-my, v = V'x + m'y.
The theorem then requires that
(a, &, c, d, e,f^x, yf = s^ + f^-v\
Since the right-hand member of this equation contains implicitly
as many constants as the given quantic, it must be capable of
expressing that quantic when s, tj v have been properly
determined.
Jjetu = the left-hand member, and CTthe right-hand member
of the above equation ; then, by successive differentiation, we
shall have
U
d^ d'
dx* dx^dy
d* d'
dx^dy dx^dy'
d' d'
dx^dy^ dxdy^
d'
dx^dy^
d"-
dxdy^
d^
dy'
= TJ
d^
dx''
dx^dy^
d^
dy'
or the symmetrical determinants,
ax + hy hx + cy cx + dy
hx + cy cx-\-dy dx + ey
cx + dy dx + ey ex^-fy
ih rt rv
Ims I'mt rm'v
I'
P
r
= s.t.v
I r
2
r r
2 ^
r
I
Im
I'm'
l"m"
m m'
m' w!'
VYl'
m
m'
m'
m"
INVARIANTS.
That is, if the expansion of
ex + dy
43
ex+fy
yields the factors s, t^ v, then these factors will differ from the
factors {x + ty), (x + t'y) , {x + fy) by only nnmerical coefficients ;
and, consequently,
(a, &, c, d, e, fix, yy = Tix + tyy+Tix + fyY + r (x + fyf.
39. In the same manner, to find the condition that a quantic
of even degree can be reduced to the sum of — n^ powers,
where n is even.
The nature of this condition is seen from the last
Article. The determinant formed from the n differentials
will, on the supposition that the quantic can be reduced to the
sum of — n^^ powers, vanish by the same process whicli
proved that a quantic of odd degree, as for instance the fifth,
could be reduced to a sum of three powers of the same degree.
The determinant formed from the n differentials in the latter
case being a covariant, gave the necessary factors s, t, v, while,
in the case now under consideration, the proposed determinant,
it will be seen, gives an invariant whose vanishing proves that
the quantic can be reduced to a sum of powers each of the
n^ degree.
To see if 2a;* + 12ajH30«H36a;-}-17 = 8 can be reduced to
a sum of two fourth powers, we take the fourth differentials
as in the last Article and we obtain the determinant
2 3 5
3 6 9
5 9 17
= 0.
The vanishing of this determinant shows that in this case
44 INVARIANTS.
the reduction is possible. To obtain the binomials, we equate
like powers of S and s, v^in 8 = s*+v^, and we find
s = :r + !I, V = a;-\-2.
40. It is hardly necessary to carry this proof into the higher
powers. But it may be said, in general, that if the quantic
does not break up into sums of powers of binomials, it will be
sufficient to add to these powers some multiple of their product
or product of their powers, as
(a, h, c, d, ej^, yy= s' + t' + 6DsH\
and (a, h, c, d, e,f, gjx, yf = s^^-f+u^-\-Estu.
That these are the least or canonical forms may be seen
by extending the proof ah'eady given. The subject in such
form as developed by Sylvester and others would not be
necessary here.
41. Comhinants. — We have seen (Art. 20) that the eliminant
of a system of linear equations is an invariant. An invariant
or eliminant of a system of equations or quantics of a uniform
degree higher than the first is called a comhinant. One
peculiarity of the combinant is that it satisfies the equation
da db
where 0 is the combinant of
ax''-\-nhx''-''-j-&c. = 0 (1),
a,x''-\-nh,x''-'-\-&c. = 0 (2).
&c. &c.
42. Another peculiarity to be observed is, that if a pair of
quantics have a common factor, their Jacobian will contain
this factor in the second degree.
Take the equations as above, and form their Jacobian, and
INVARIANTS.
45
the truth of this will be evident ; or, let a be a common factor in
u=. ax^ + Zay'^,
then
du
du
dx
dy
dv
dx
djo_
dy
= - 20a'xy.
If, in (1) and (2), (Art. 41), w = 2, we shall have, for /,
ax +hy hx +cy \ = /,
a^x-i-\y \x-^c^y\
whose discriminant we find to be
4 (a&i — a^ h) (Jbc^ —h^c) — (ac^ — a^ cy.
This, as we have seen, is the eliminant of u and v as quad-
ratics; or, in other words, we find that in this case, at least, the
J of u and v contains their eliminant as a factor : and this is a
truth to be observed when ?i = 3, 4, &c., in which cases the
discriminant of J, as is evident, will be composed of the
eliminant of the quantics and some other factor whose form
may be determined.
43. If (1) and (2) above be represented by u and v, then
u -f hv represents a locus common to u and v ; and, by assigning
varying values to h, we shall obtain a system of quantics some
of which will contain square factors ; and in the involution of
points formed by these quantics there will be as many double
points as there are quantics which contain square factors.
The number of these is seen to be 2 (n—1), or is the same as
the order in the coefficients of the discriminant. The number
of double points may then be determined by the Jacobian of
w and V. If u-\-hv has a factor (x—af, then a will satisfy
^ + 7,^^1 = 0, and^ + ^^=0. It will evidently satisfy the
ax ax dy dy
4^6
INVARIANTS.
equation obtained by eliminating h, and therefore the Jacobian
of (1) and (2). We thus have an easy method of determining
the number of double points resulting from the involution of
these quantics. In this form we see that h can be so determined
that u-\-hv shall contain the square factor (aj — a)^ ; and, by
adding another condition, we may determine the valae of con-
stants so that the quantic shall contain {x — af. The coefficient
in (1) and (2) will then be of the degree 3 (?? — 2). Conversely,
if (x — aY exists as a factor in u-\-liv-Ymt, this factor in the
first degree will exist in the three second differential coeffi-
cients, and consequently in their eliminant with respect to li
0,
and m ; that is
d'^u d'v d^t
dx" dx^ dx^
d'u dh dH
dxdy dxdy dxdy
d^u d\' dH
df df df
which gives the number of triple points in the above system
u-\-hv + mt ; and, if 2/ = l? it expresses the number of these
points on the axis of x, or 3 {n — 2), which fulfils the condition
of a combinant.
44. Tad-invariant. — When we find the eliminant of (1) and
(2), and equate it to zero, we express the condition that the two
curves may be tangent to each other. If we express also the
existence of a cubic factor in any quantic of the series u -}- hvj
that is, a cuspidal curve, by ^"=0 and by V=0, one having
two double points, or two square factors, and by TF=0, what
has been described above as the tact-invariant ; then the dis-
criminant of u-\-hv with respect to h will contain Z7, F, W as
factors.
If u and V are tangent to each other, then the discriminant
of u+hv will, as a function of h, have a square factor ; in other
words, when expressed geometrically, it is the condition that a
curve has a double point.
INVARIANTS. 47
45. The tact-invariant is of the order Sn (n—1) in the
coefficients. Hence, if we have three surfaces L, P, Q (of Z, m,
n degrees), the condition that two of the Imn points of inter-
section will coincide is called, in this case, the tact-invariant,
and the coefficients of 1/ are in the degree mn (2l+n-\-m—4<)f
and so of P and Q.
The tact-invariant of two surfaces aL and P have the co-
efficients of X in the degree m(V + 2lm + Sm^ — 4Z— 8m + 6).*
These results are obtained from quantics of four variables.
The geometrical importance of these results will be further
seen.
46. As to the numher of invariants of a binary quantic, we
have already seen that a quadratic has one, that a cubic has
one, each of these being the discriminant of the given quantic.
If we take the next in order, the quantic
(a, b, c, d, e^x, yY,
we can easily determine the number of ordinary invariants,
omitting from our enumeration those which are expressible as
rational and integral functions of the same or lower degrees.
Remembering that the invariant must satisfy the differential
equation
ao-^+2ai--— +3a2-_— +&c. = 0,
rtaj da^ da^
and that the last invariant must be of the order 2 in the
coefficients, it must therefore be of the weight 4 in the
coefficients, that is,
Aa^a^^ + Ba^a-^ + Ga^a,^.
Operating upon this with the differential equation, we have
AiAa^a^ + SBa^a^ + Ba^a^ + 4^Ga^a^,
* Terquem's Annales, Vol. XIX., and Quarterly Journal, Vol. I.
48
INVARIANTS.
which, taking A as 1, gives for B, —4, and for C, 3 ; and we
have by substitution
or ae—4>hd-\-dc^,
as the invariant fun'ction, which is the same as would have
resulted by actual transformation. Had we followed the
latter method, we should have found that the function of the
new would be equal to the old when multiplied by the fourth
power of the modulus, (Im—TmY or A^, or, written fully,
AE-4.BD + W^ = ^' {ae-4;bd + 'dc^) (1).
Proceeding now to the invariant of the third order in the
coefficients, we see that the weight would be 6, and must be
of the general form
which embraces all possible forms.
By applying the differential equations as before, we have
ace + 2hcd-~ad^—eh'^ — c^j
or AGE+2BCD-AD'-EB'-G'
= A^ (ace + 2hcd-ad''~Gh'-c^) (2).
If the A^ does not follow clearly by symmetry, the actual
transformation will make it evident. If we proceed to the
fourth order in the coefficients of another invariant, we shall
find only a function of those already found, which therefore is
not to be counted in the enumeration.
47. Absolute Invariants. — If we eliminate A between (1) and
(2) in the above, we shall obtain what has been called an
absolute invariant, that is,
{ACI]-\-2BGB--ATy-BB''-Gy {ae-4;bd-^Zcy
= (^AE-WB-^^Cy {ace + 2bcd-ad'-eh'-cy.
INVARUNTS. 49
And if J and T represent the invariants (1) and (2), their
ratio P : T^, as is seen, is unchanged by transformation.
48. As to the discriminant of the quartic which is the elimi-
nant of its two first differentials, we shall see that we can
arrive at a method of derivation by means of (1) and (2). We
have only to remember that the eliminant vanishes if the
differentials have a common factor, and that this factor will
exist if the binary quantic contains a square factor. We have
only, to arrive at this condition, to suppose the first two coeflB.-
cients to vanish ; the quantic then has a square factor, since it
is divisible by y^. It is clear, also, that the invariant of such
a quantic must vanish. The one contains the other as a
factor when the two first coefficients a and h vanish. Or we
may state it thus : — The invariant is a symmetric function of
the differences of the roots, and the discriminant is the product
of the squares of the differences between any two roots (Art.
17) ; that is, the invariant, on the above supposition that the
roots are equal, as expressed in the terms of the roots, must
contain the difference between the roots taken two and two.
Now, since the ratio ot P : T^ is unchanged by transformation,
a new invariant may be constructed from them, and we see
that P — 27T^ will vanish when a and h are each 0; that is,
I becomes 3c^, and T, — c^ on that supposition. And since we
know (Art. 15) that this form gives us the required order in
the coefficients, we conclude it to be the discriminant, that is,
(ae—Ud + dc'y-27(ace-\-2hcd-ad''-eh'-c')\
which, being of the form of PzkJcT^, is not commonly reckoned
as distinct from I and T ; and thus generally when, as in this
case, I and T are expressible as an invariant, a function both
rational and integral of I and T, such function is not counted
as a new invariant. We would infer also, in the same manner,
that, if I and T are invariants of the same degree, then IdikT
need not be counted.
60 INVARIANTS.
To sum up our number of invariants thus far, we have
ac—V' the invariant of the quadratic (a, &, cja;, y)^,
which is the discriminant.
Next, o?d? - 6a bed + Wd + 4ac« - Sh'c'
is the discriminant of the cubic (a, h, c, d'^x, yY (Art. 5); that
is, it is the eliminant of its two first differentials.
This is its only invariant (Art. 22).
And, lastly, the I and T of the quarfcic (a, h, c, d, e^x^ y)^
just considered, which are two ordinary invariants.
49. The Series of Covariants. — It follows from the definitions
of invariants and covariants, and may easily be verified, that
every invariant of a covariant is an invariant of the original
quantic, and the contrary ; consequently the quadratic can have
no other covariant than the quadratic itself; or we say that this
fact follows immediately from the consideration that there are
no difiierences of roots — there being in this case but one differ-
ence— and that there can be no function of the difierences of
the roots. But in the cubic, since a symmetric function of
differences of roots, and differences between x and one or more
of the roots, is a covariant, we can form a covariant distinct
from the cubic. The form of this covariant,
(aH-Sahc + 2h\ ahd-2ac' + h\
-acd + 2h'd-hc\ Zlcd-ad^-2c'-\x, y)\
has been investigated in Art. 34, and the process need not
be repeated here. We have also the Hessian which Dr.
Salmon writes
B =
a b c
bed
y^ ^xy x^
= {ac-b')x^ + {ad-bc)xy + (J}d-c^)y'
These two covariants examined in connection with the quantic
itself, which is also a covariant, show at once that the list for
INVARIANTS. 51
the cubic is complete. For we see tliat the coefficient of o?
in each case is a, ac—lcP'^ o?d — ^abc-\-W^ which are called the
leaders. Kecurring now to the discussion (Art. 35), we find
that whatever analytical relation exists between the leaders of
covariants, that same or similar relation will hold with the
covariants as a whole. This being the case, we need only
operate upon these leaders in order to discover the successive
covariants.
Thus B. above is the Jacobian of the first covariant
(a, 5, c, c^Jaj, ?/)', or F, and the original quantic (a, &, c^a?, ijfi
say ; so also the third covariant in the above series, whose
leader is o?d — 3a6c4-26', is the Jacobian of the above Hessian,
and the original quantic, which in this case, the cubic, is F,
and thus each succeeding covariant, is found by taking the
Jacobian of the last covariant of the series and the original
quantic, whatever that may be. For the cubic this last
covariant is indicated by /.
50. The question whether any other covariants may be
properly added to this list, as regards the cubic, may be
examined as follows. We see that a-, ac, o^d^ &c. are divisible
by a. We find then what new functions, rational and integral,
of these leaders may be formed whicb contain a. In this case,
the leaders of H, /, ac — V"^ aH ■^dahc + 21^^ become, on the
supposition that a = 0, 4B'*+ J'^ = 0. It therefore contains
some power of a. Performing the operation indicated by
4iH^-\-J^ = 0 and dividing by a^, we obtain the discriminant
of the cubic aH^-6ahcd-Sh'c'-\-4<ac^+Mh\
Now it must be remembered that a covariant, as also an
invariant, is by definition a function of diJfferences of the roots,
and that a covariant is known when its source or leading co-
efficient is known (Art. 36) ; hence these leaders, as well as
resulting invariants, will satisfy concurrently the difierential
equation F ( «o -r— + 2«i -; h Sa. h &c. ) = 0,
\ da^ da^ da^ I
where F is any leader or invariant.
52 INVARIANTS.
From this fact, and in conformity with tbe definition, we
might, for the purposes of this classification, include the
invariants with the covariant of a quantic. The above dis-
criminant, then, may be classed with the coefficients of the
CO variants.
Regarded in this light, we shall find that a quantic of the
^th degree will have n covariants, including the quantic itself,
so that each other covariant, multiplied by some power of the
quantic, will be equal to a rational and integral function of the
n covariants. Thus, at once, if we represent the discriminant
(invariant) by A, we shall have
A7^ = /2 + 4B■^*
or, using the canonical forms,
a^dP {ax^+dyy = a'd' (ax' - dfy -\- 4^ {adxyf.
51. If in A we let « = 0, we have left a quantity containing
coefficients which cannot be eliminated by combining with
—h^ or 25^. In other words, no new functions of ac — W,
a^d—Sahc + 2h' can be formed divisible hj a. Hence we may
say for the cubic the list is complete.
52. The covariants of the quartic are first the Hessian,t and
then the Jacobian of this Hessian and the quartic itself must
be taken. We find H to be
{ac-h\ 2(ad-hc), ae + 2bd-3c\ 2(Jbe-cd), ce-d^Jx,yY.
The Jacobian has its first term, or leader, a^d^dahc + 2b' &c.,
which, by Prof. Cayley's symbolical representation (where the
Hessian of every binary quantic is written 12^, and the
Jacobian ofH and the quantic 12^, 13), is easily distinguished,
and indicates a basis of calculation.
* Prof. Cayley, "Phil. Trans.," 1854.
t Known in geometry as the Harmonic Conic.
INVARIANTS. , 53
53. We might state here more fully the principle of this
symbolic representation.
In Arts. 30 and 34, it was shown that — — , — — , &c., regarded
ax ay
as operating symbols contragredient to a?, y, &c., while trans-
formed by a direct substitution x, y, &c., will be transformed
by an inverse substitution, and the contrary ; and that, repre-
senting — — , — — , &c., by a, /3, &c., operating symbols could be
ax ay
formed which, substituted in the quartic, a covariant or invari-
ant could be formed according as the variables were or were
not removed by differentiation. We can thus form an opera-
tive symbol for a system of quartics by a system of determi-
nants formed of a, (3, &c. Thus a-^f^^—a^^^-^, represented by 12, is
an invariant symbol of operation. If we operate on two quan-
tics 8 and V, the result of the operation upon their product
^Fby 12 is the Jacobian.
If these are quadratics,
then the result of the operative symbol 12^, or
a\ /32 — 2ai ft agft + «2 /3? ,
on iSFwill be an invariant, i.e., ac^-\-ca^—2hi^.
In the same manner, 12^ 13 expresses the operative symbol
(or its effect upon a binary quantic)
(aift-«2A)' («i/53-/5s«i).
54. We have then, as the effect of 12 on SV, the Jacobian
d8 dV _ dS dV
dx dy dy dx^
and the application to any two quantics may be expressed by
Id8 dV dS dVy
\ dx dy dy dx I ^
or 12".
54 • INVARIANTS.
In the former case, the exponent of the power does not apply
to 8 and V, but only to the symbols of differentiation. The
result is, of course, the same in both cases — an invariant if
n = the degree of the quantic, since all the variables are removed
by differentiation, or a covariant if n is less than the degree of
the quantic. From this it will immediately appear that, if by
this process, we wish to form the covariant of a single quantic,
we have only to make S = V. Thus, if we desired to form the
covariant of a single quantic with the symbol 12^, or
fds dv _ ds dvy
\ da: dy dy dx /
we have only to make iS = F, and the latter symbol becomes
g r d-'s d's ( d's \n
Ida;' dy' \da;dy)y
which, applied to two quadratics 8 and F, would in this case
give 2 (ac — h'). Hence, in general, the quantic to be operated
upon may be conceived to be the product of two or more quantics
8, F, T, &c., whose variables are distinguished by subscripts,
as ^1, yi, 0^2, y2> <^^'j ^^^ when the differentiation is complete
the variables are written solely ^r, y. Since 32 and 23 are
clearly the same with opposite signs, as also 12 and 21, it
will appear that either of these symbols with odd powers will,
when applied to any single function as 8Vf cause it to vanish.
Following this analogy, we can easily write the symbol for a
system of ternary quadratics. If, for ^i, y^^ &c., we write
J J &c. (in which the cogredient variables can be written
dx^ dy^
as a determinant
VX 2/2 2/3
= VL6\
INVARIANTS.
55
we shall have, when the symbol 123^ is applied to the ternary
quadratics
a^ + ly^ + cz^ + 2/?/;3 + 2gzx + tlixy
= 6 (abc-\'2fgh-af''hg^-ch^),
a
h
9
h
b
f
9
f
c
i.e., six times the discriminant of the ternary quadratic when
a = (Xj = ag &c.
66
CHAPTER lY.
COMPUTATION AND GEOMETRICAL APPLICATION OF
INVARIANTS.
55. The attentive reader of tlie preceding pages will have
now no great difl&cnlty in making a variety of important ap-
plications of the Invariant Theory.
It is shown in works on the Conic Sections, that if Fand Fj
represent two conies, there are three values of h for which
IcVzkV-^ represents a pair of right lines.
We take
ax^ + ly'^ + cz^ + 2/2/» + 2^«aj + 2'hxy = 0
as the general homogeneous equation of the second degree in
three variables ; and this is intimately connected with
ax^-\-'by^-\-2hxy + 2gx + 2fy-^c = 0 (1) ;
the latter being derived from the former by making z-=\.
This latter may represent two right lines, and does in ge-
neral, when its coefficients fulfil the relation
a
h
9
h
h
f
9
f
c
= ahc-\-2fgh—af—hg^—ch^ = 0,
which is obtained by the resolution of (1) as a quadratic ; the
above determinant being the condition necessary to make the
quantity under the radical a perfect square. If we call
then
Fi = OiX^ + 6i2/' + Cjg' + 2f^yz + 2g^zx + 2hiXy,
Aj = ajfejCi -j- 2/1^1^1 — ctifi — &i^i - Ci^i .
INVARIANTS. 57
It is not difficult to see that the three values of h^ for which
fcFzb Fi represents a pair of right lines, is obtained by substi-
tuting ka + a-^i hh-\-h^, &c., for a, &, c, &c., in A. Writing
this result in full, we shall find that h^ will have A for its co-
efficient; W and h will have functions for their coefficients,
which may be represented by d and d^ ; and lastly, that A^
appears as the absolute term ; that is,
A^H0/^' + 0iA; + Ai = O.
The value of
0 = {hc-f) a, + (ca -/) h, + (ah - h') c^
+ 2 (y/.~a/)/, + 2 Qif-lg)g, + 2 (fy-ch) h, ... (2),
and 01 = (&iCi —fi) a + &c.,
the same as 0, the accents being interchanged.
Now between A]c^+dh^ + djc + \z= 0 (3),
and ^7+ Fi = 0,
we may eliminate ^, which gives
AF'--0F'F+0iFiF='-F'Ai = O,
denoting the three pairs of lines which join the four points of
intersection of F and Fj.
56. Since any two conies have a common self-conjugate
triangle, and since they may be written
V =ax^+ hif + cz" = 0,
(see T., Arts. 45, 47, 56,) or
Fi = i6^ + 2/' + «' = 0,
where x is written for ic-v/^i, &c., we obtain, by Invariants,
the three values for which hV-^ + V represents right lines.
£ 2
68 INVARIANTS.
Then A reduces to ahc,
6= ah-\-'bc + aCj 6^^ = a + h + c, Aj = 1 ;
or, were we to substitute Jca+a-^, &c., in ahc, we must have,
for the required condition,
h^ + Jc^(a + h+c)-\-k{a'b-\-ac-\-hc)-\-ahc = 0,
which is satisfied by —a, — &, — c.
For another example, let us take the ellipse*
and the circle (x—x^y+ (y—yiY—'^'^ = 0 = Fj.
In forming A from F, we must remember to affect the result
by the negative sign, since c or the coefficient of z^, as well as
z^ itself, is reduced to unity with the minus sign. Hence
'^^"■^'
To obtain 6 we must recur to the general equation of the circle
x' + y' + 2gx + 2fy + c = 0.
From which we find, by comparing the values of the coeffi-
cients with those in the preceding Article, that
-h-
Sfl = »!.
/i = Vi,
'=^.
c, = x]
+ y\-
r\
c=-l
From these values we
find d to be
_ i _ i + '^'i + yl-^
* Students who are familiar with Salmon's " Conic Sections," will at
once recognize these examples. It is believed that the treatment here
given them will completely remove the difficulties which hitherto have
been experienced by many in their solution.
INVARIANTS. 59
^.« + yl-^-'^'-n
In the same manner, by interchanging the accents, we find
+ (l-0)(-l)
0^ = (^\+y\-''-yl)~, + (.^\+y\-^-''])i
~ J '^ h' ^ ''Aa' "^feO'
and Aj = ajj + 2/J — r^ — 2/J — »5 = — »•' ;
from which the equation in Jc is formed.
If, instead of the ellipse, we had taken the circle
»^ -h 2/' - r' = 0,
Fj remaining as before, accenting the r, we should have had
A = - r^
since a = 1, 6 = 1, c = — r^\
e:=(-r'-0)l + (-7^-0)l + (l-0)(xl+y\-r\)
= xl + yl- 2r' ^r], by (2) of Art. 55 ;
»2
A, = — r^
as in the previous case.
57. Since A, A^, 6, 6^ are invariants of the system of conies
under consideration, their computation should be carefully
studied, because in solid, as we shall see, as well as in plan©
geometry, these functions are fundamental.
Take the parabola y^ = 2px, and Fj as before, the circle
€0 INVARIANTS.
Here 5 = 1, while p corresponds to y in the more general
equation, as is evident from (1), Art. 65, the other coeJSicients
reducing to zero.
We have then A = — 5^^ = — p',
d = (0-/) l + (0 + 2p) (-C.0 = -p (2x,-\-pl
A = — r^ , as before.
If x^ + y"" = r" and {^^-x;f-\-{y-y^^ = rj
represent two circles, and d the distance between their centres,
we have, as before,
A = -r^ 0 = cZ^-2r2-rf, Q^ = d^-r'-^r^, Ai = -rJ.
58. If we turn to equation (3), Art, 55, and observe its degree,
and remember that two conies always intersect in four points,
and that four points may be connected by six lines, viz., 12, 13,
14, 23, 24, 34, we may conclude that this equation is that of
the three pairs of chords of intersection of the two conies.
An easy application of this equation is found in the problem,
to find the locus of the intersection of normals to a conic from
the ends of a chord which passes through a given point.
The equation of the normal to an ellipse is
d^xy^ — h^x^y = c^x^y^.
If we interchange the accents, the right line becomes a curve, \
in fact, an hyperbola a^x-^y—V^xy-^ = c^xy,
expressing that the point on the normal is known, and that
the point on the curve is sought ; consequently, we see that
the intersections of the given ellipse and the equation last
written are points whose normals will pass through the given
point ; that is, x^ y^
INVARIANTS. 61
Let 4+?^-l=0 = 7,
a ' 2,3
2 {a^xy^-V^x^y—c\y^) = Vy
This latter equation, it is evident, should be, as has been
done, multiplied by 2 in order to sustain the fixed numerical
relation expressed in the corresponding coefficients of the
general equation. The equation of the six chords joining the
feet of normals through xy, the locus required when satisfying
the given point, is readily formed by substituting the requisite
invariants in equation (3), referred to above.
We have then
since \ = c^,
9i = ^Yf and ai = h^ = e^=z 0,
and therefore 6^ = -(aV-c*+&Vi^)>
Ai= ^2a'b'c\yy
Hence, if a/3 represent the given point, we have
~ (a'ISx-h'ay-c'al3y + &c. = 0,
an equation of the third degree, reducing to a conic when
the axis is a part of the locus.
59. In the cubic for k, its values, for which hVzk Fj repre-
sents right lines, remain the same without reference to the
coordinates in which V and Fj are taken. In other words, the
relation between the coefficients A, 0, &c., remains unaltered
by a change of coordinates, and these coefficients for the new
system are equal to those of the old, multiplied by the square
of the modulus of transformation, or in general by some power
of that modulus. (Art. 20.)
62 INVARIANTS.
60. If 1 and 2 of the foar points of intersection of two
conies coincide, then 13 and 23 will coincide with 14 and 24.
In this case the cubic in h will have two equal roots. Let us
take the differential coefficient of this equation, and proceed as
if to find their greatest common divisor. This condition may-
be expressed as
(00i-9AAiy-4(0'-3A0O(9i2-3Ai0) = 0.*
In this case the conies are said to touch each other, though it
miist not be supposed that there are not also two other real
or imaginary points in which the conies meet. A great vai-iety
of examples will at once occur to the reader which will illus-
trate the foregoing. We might exhibit an application of the
last example. Art. 58. Expressing that the two curves touch,
we must have, since 0 = 0,
27AAiH40i' = 0.
Now that this equation will apply to the finding of the evolute
of the given curve — that is, the ellipse — we have only to remem-
ber that the coordinates of the centre of the osculatory circle
and those of the evolute coincide, that two of the normals
coincide which can be drawn through each point of the evo-
lute J and we have
as the required equation.
61. Before passing to other applications, we may discuss
the conditions under which Aj, 0, and 0^ vanish.
If Ai = 0, how shall we interpret Q and B^ ? Since Fj breaks
np into two right lines when \ = 0, we may represent these
lines by a and /3, and then instead of V-\-hV^ we may write
7^+2A;a/3, whose discriminant maybe found by substituting
A + fc for A in A, from which we obtain
A + 2h(fg-ch')-cJc^ (1).
* This condition may be found by equating the discriminant of the
given cubic in k (Article 55) to zero.
INVARIANTS. 63
But when the coefficient of h vanishes, that is, when fg = ch,
we have the condition that the pole of the axis of x in the
general equation should lie on the axis of ?/ ; in other words, in
this case, that the lines a and /3 are conjugate with respect to V.
Now the vanishing of the discriminant indicates, as we know,
a double point in the curve, and hence the vanishing of (1)
shows us that the point aj^ lies on the curve V\ that is, the
coefficient of A;^ vanishes when, in this case, c = 0 ; and conse-
quently that, when ^1=0, the intersection of the two lines is
on r.
More generally, the geometrical interpretation of 0=0 may
be shown if we take the trilinear equation (T. 47) of the
general form ax^ -f iy"^ + cz^ = 0,
in which the triangle of reference is self-conjugate in respect
to F^. We have then
Again, from (T. 53), we see that
fiy^-\-gi^^'+K^y = ^ (2)
represents a curve circumscribing the triangle of reference.
Hence we say that, if V-^ has the form of (2), d will vanish,
since, in that case, a^=L'b-^-=c^=-0', that is, d will vanish when
the triangle of reference inscribed in F^ is self- conjugate in
respect to V. If we reverse this relation, taking the triangle
of reference as self-conjugate in respect to F^,
d=^(bc-f)a,+ {ca-g'')\^{ah-h')c^ (3),
since in this case yj = ^^ = /^^ = 0.
We see that (3) will vanish if we impose the condition of
equal roots in the general equation ; that is, if he =/^, &c.,
which is the condition of coincident tangents, or that » as a
line should touch F; that is, that the triangle should circum-
scribe V while self-conjugate in respect to F„ in which case
« = 0.
64 INVARUNTS.
62. Since V= ha^ represents a conic having double contact
with F, a being the chord of contact, if now V represent
the general equation in a;, y, z, and lx-\-my-\-nz the equation
of a line in trilinear coordinates, the equation of any conic
having double contact with V on the points of intersection
with the given conic, can be written
W+{lx-\-my + nzf = 0 (1) ;
and suppose it were required to so determine h that this equa-
tion may represent two right lines. In this case A remains
unaffected, but 6 evidently becomes
(hec-f) 1^+ (ca-g') m^+ {ah-h^) n"^^ (gh-af) mn
+ 2 Qif-hg) nl + 2(fg-ch) Im = 0.
But since, by hypothesis, Fj breaks up into two right lines,
Aj = 0, and 0 also vanishes, since there is double contact, or
the intersection of the two lines is on F ; hence the cubic in k
reduces to Ah^+dk^ = 0.
In other words, there are two roots = 0, and we have
kA-\-e = 0 (2)
to determine the other. When there are two equal roots, the
conies touch each other (Art. 59). Hence, finding the value
of A; in (2), and substituting it in (1), we have
eV= A(lx+my + nzy,
which is the equation of the pair of tangents at the points
where the conic is cut by the given line. Where 0 = 0, re-
presenting its new value as above, we have the condition that
the line touches the conic, and the tangents coincide with the
line.
63. It may be well here to remind the beginner, that
by a tangent is understood, analytically, in general, a line
meeting the curve in two coincident points, and that when
INVARIANTS. 65
the curve breaks up, as we have supposed, into two right lines,
the only tangent which can meet such a locus must be on the
intersection of these right lines ; and since a curve of the
second degree may always have two tangents, both tangents
must coincide with the line at the point of intersection.
We know that, when Fand Fj represent conies, V+hVi = 0
represents a conic passing through their points of intersection.
If now the condition were sought that the line lx + my-{-7iz = 0
should pass through one of these points, we may equate z in
F to 0 and in the equation of line = 1 ; and then, substituting
the value of y found from the equation of the line in F= 0,
we have a quadratic in x whose condition of equal roots we
wrote in the last Article, viz.,
e = {hc-f) V-\- (ca-g') m^-^ {ah-h') n' + 2 (gh-af) mn
-f 2 Qif~ ly) nl+2{fg- cJi) Im.
Let this right member now be represented by S, the condition
that the given line touches F. If in this expression we write
a-\-Jca-^ for a, Z) + A:&i for h and c, we shall manifestly have the
same condition for V-\-JcVi, or any conic of the system, which
we had for F. Hence, multiplying out, we have, for the co-
efficient of &,
ihc,-\-h^c-2ff^) l'' + (ca^ + c,a'-2gg;) m'^-\-(a\ + a^h-2]i\)n'^
+ 2 {Q\^-gi^—af^—aJ) mn + 2 Qif^-^liJ-hg^—^g) nl
+ 2 (fgi +fig-c\ - c^h) Im.
Representing this by O,* and the coefficient of k^ by S^ we have
The condition that this equation should have equal roots is
$^ = 4SSi; or is the condition that the given line should pass
through one of the four points ; and as the " envelope of this
* Wlien * = 0 we have tlie condition that the given line shall be cut
harmonically by V and Fj. It is also to be observed that this condition
is a contravariant of the system of conies V and Fj.
66 INVARIANTS.
system is clearly only these four points, tlie equation last
written may be regarded as the envelope of the system. It is
to be remembered that we are here really discussing functions
which remain unaltered by change of axis, because, if V and V^
by transformation to a new set of co-ordinates become V and
Fj, then V+JcV-^ becomes V+kVi, Jc still remaining constant.
Now
= 0 is the determinant whose vanishing
a h ff
h b f
9 f c
is the condition that the general equation may represent right
lines. DiflPerentiating this function with reference to each of
its letters, we have the coefficients of S above. Also both ^
and Dj are functions of A, in such manner as to possess the
character of invariance.
64. If we seek the condition that
shall touch the conic represented by the general trilinear
equation (T. 43)
aa2-|-&/32 + C7H2//37 + 2^ya + 2/ia/3 =0 (1),
we shall have the condition represented by S, as above. For
the coefficients there given, hc—p, &c., we may write A, B,
&c., or
Al?-\-Bm' + W + 2Fmn-\-2Qnl + 2mm = 0 (2),
which is sometimes called the tangential equation of the conic.
If between this equation and the equation of the line we elimi-
. . f
nate n, we shall have a quadratic in — r, and the condition of
m
two equal roots, or that it breaks up into straight lines ; or,
which in this case is the same thing, the envelope of the line is
{BG-F^)a? + {GA-G^)^^+&.Q.=:0 (3),
an equation symmetrical with S, — the latter in Z, w, n and
INVARIANTS. Q7
its coefficients in small letters, the former in a, jS, y and its
coefficients in large letters. We may see, then, that the enve-
lope of a line whose coefficients fulfil the condition S is the
conic (1), for we have only to substitute for A, JB, &c., their
values bc—f^, &c., and (2) becomes AF= 0 when 7"= (1).
Consequently, if we write the trilinear equation corresponding to
we have AV+JcD+k^A^Vi = 0 (4),
in which D is symmetrical with $ ; that is,
an equation in x, y, z when Fand Fj have the meaning we have
heretofore assigned them.
The envelope of the system (3) is
but the envelope in this case is the four common tangents.
Hence this is the equation of the four common tangents to the
two conies.
To illustrate this, take the two conies
2aj*+4/+62^= 0.
A =15, Aj = 48, ^ = 15, J5 = 5, (7 = 3,
A^ = 24, B^ = 12, (7i = 8 ;
I>=2(18 + 20)a;'^ + 12(10+6)^H30(4+6)g'.
Hence
(7603^+192^'^+ 3002^^)2 = 2880 (a?H32/' + 62')(2aj2+4/+6a«)
is the equation of the four common tangents to the two conies.
If 3^'-2a;''-4a;y = 0= F,
and J + ^_1=0=F„
what is D ?
68 IXYARIANTS.
65. As has been before intimated, an invariant is a function
whose vanishing indicates some property of the curve inde-
pendent of the axis to which it is referred. In the same
manner, as we know, covarianta are particular loci whose
relation to the equations whence they were derived is inde-
pendent of the axes of these given equations. In other words,
the two functions agree so far as axes are concerned.
Turning our attention now to covariants, which, as we have
seen, contain the given variables, we may find our illustration
in the system of conies we have been considering, V and Fj,
which we will again refer (Article 56) to their self- conjugate
triangle, that is.
If we proceed now as in the last Article, we find
A = bCf B = ca, C = ab^
A, = B, = C, = 1;
consequently
D=:(ah + hc)x^ + (bc-\-ba)f+(ac-{-cb)z^ (1) ;
equation (2) of the preceding Article becomes
Al'-^Bm'+ 071^=0,
or the condition that a line should touch V. Hence the locus
of the poles with regard to Tj of the tangents to V is
Ax' + By^+Cz^ = 0 (2).
Adding (1) and (2), we have
(A+B + G)(x'+f+z') = D.
Or, since (Art. 56) e = A + B + G,
we have GFj = D as the equation of the polar conic of V with
respect to Vi in terms of the conies of the system and the
conic D. The locns OFi = D is therefore a covariant of Fand
Fi, and this relation will not be altered when V and Fj are
INVARIANTS. ' 69
transformed to othes axes. Similarly, 6j^V=D is a locns, a
covariant, the polar conic of F^ in regard to F".
Returning to ^ = 0 (see note, Art. 63), we see that it
becomes, retaining the same expressions for V and Fj,
(6 + c)P+(c + a)m2 + (a + 5)^' = 0,
which may be called the tangential equation of the conic
enveloped by a line cut harmonically by V and Vy Now the
trilinear equation, as found from this, is of the form of
equation (3) of the last Article, that is,
or
(c-{-a)(a + h)x'+(a+hXb + c)y' + (c + a)(b + c)z^ = 0...(1),
since in this case A = (&4-c), &c.
Adding the value of D to (1), and reducing, we have
as the equation, a locus, covariant with F^and F^, expressing
in terms of these conies a conic enveloped by a line cut har-
monically by the conies in question. If D breaks up into two
right lines, we have simply A = 0 in equation (1),
or {a'b + ac)(hc+ha)(ac + ah) = 0.
66. It would be a profitable exercise for the reader, at this
stage, to reduce a few conies to the forms
a^-\-y' + z' = 0,
This can be done with the help of
Aic''+ejc^+ejc-\-\ = 0 (1).
That is, the roots of this equation will give us the new a, 6, c ;
then we shall have
x'+y' + z' = F, ax' + by' + cz' = V„
when V and F^ are the given conios.
70 INVARIANTS.
We shall still need one more equation, and for this we can
conveniently nse equation (1) of the preceding Article,
(ab + ac) x^+(hc + ah) 7/ + (ac + ch) z^ = D,
with this caution, that, as the discriminant of F^is 1, D must be
divided by A to put the three equations upon the same relation.
Thus, if Fand F^ are
x^-2xy-\-2i/-4x + 6y = 0,
Sx^-6xy-\-5y''-2x-l = 0,
we see these are of the general form
ax^+2hxy + hy^+2gx + 2fy + c = 0.
The A of the first is -5 (Art. 55),
0 = -14, 01= -9, Ai=-ll;
and since (Art. 56) the roots of (1), when the conies are re-
ferred to their self-conjugate triangle, are —a, —6, — c, the
actual form of (1) for numerical use must be
or in this case
-bJc^+Uh'-9h-\-ll = 0 (1).
In order to calculate the co variant D, we must first know
A,B,C, A, B,, Oi, &c.
These may be computed by equation (2), Art. 64.
gives us
Vi gives
a = l,
a, = 3,
h=-l,
ft. = -3,
i = 2,
6. = 5,
? = -2,
S> = -1,
/=3.
/. = o,
c = 0;
c, = -1.
As given in Art. 63,
A = hc-A
B = ca-g\
C=ab-h',
F =:gh-af,
G=}
'f~hff,
H=zfy-ch.
INVARIANTS. 71
In the same manner ^dj = h^c^—ff^ &c.
The value of D must be computed from the general equation,
which we now write in full,
+ (AB^ + A^B^2HE,)z'-{-2 (GH,+ G,H-^AF^'-A^F)yz
+ 2 (HF^ + E,F- BG, - B,G) xz
+2 (FG,+F,G-GH,-G,E) = D.
Now suppose the roots of (1) to be represented by a, h, c
(the new a, h, c), and we shall have
From which we can obtain the values of X, Y, Z, which
were required. The reader can complete this example. These
computations are important on account of their frequent
occurrence in geometrical investigations, as will be seen in a
succeeding Tract.
67. Another of a large class of examples will show how in-
variants determine the situation of a conic, as for example a
fixed locus.
Let us take F, a curve circumscribing the triangle of
reference (T., Art. 53),
that is, 2 (u(3y-{-vya + waP) = 0.
Let Vi be touched by two sides of the triangle. This can be
represented by the tangential equation, in this case (T., Art.
54), by aH/32 + y^-2/3y-27a-2a/3 (l+wh\
since a=0, /3=0, in each case, satisfies the equation, giving
perfect squares. Then will hV+ Fi, a conic passing through
their intersections, be touched by the third side of the triangle.
Computing the invariants as before, we have
72 INVARIANTS.
A = 2uVWj
:= —(u-\-v-{-wy—2uvwhf
0j = 2 (u-\-v-\-w) (2-\-w]c), Aj = —(2 + hJc)\
From which we obtain
0i«=4AAiJfc+40A,
and, eliminating the parameter k between tbis last equation
and 7cV+Vt„ we have the envelope of the third side of the
triangle of reference; or — which, in this ease, is the sametbing —
by substituting the value of Jc^ derived from that equation, in
the latter, we obtain, plainly, a fixed conic touched by the
third side, that is.
When 0j' = 40A, h = 0^ and is simply the condition that the
three sides of th^ triangle are touched by V^.
68. If I and m are any lines at right angles to each other
through a focus, we can construct an equation, a particular
form of u^a^ + v'(3' = wY, (T., Art. 47)
that is, P + m' = eY,
where y, the polar of the focus, is the directrix. If e = 0, as
in the circle, we have the equation which determines the
direction of the points at infinity on any circle ; or, in other
words, P-^m^=:0
is the tangential equation of these points, or the condition that
the line lx-\-my + n = 0
should pass through one of them.
Now the necessary relation between these constants, in order
INVARIANTS. 73
that this line may touch the curve represented by the general
equation, sometimes called the tangential equation of the
curve, is given Art. 63, equation (2). Distinguishing this by
Si let us proceed to examine the discriminant formed from
which is A^+TcA (a + h) + J(^ {db-h%
Form also the discriminant of
8 + h8^i
which is A^ + hAJd^ + Ic'A^d + h^A\ ,
and we see that a + & corresponds to 6^ and ab—h^ to d. Hence
we say that, the invariants of any conic and a pair of points
at infinity being formed, we can express the condition, by
placing ^1 = 0, that the curve is an equilateral hyperbola, and
by 0 = 0, that it is a parabola. This result follows from the
theory of invariants, — viz., that whatever homogeneous relation
is seen to exist in the one case will also exist in the other, ir-
respective of the coordinates in which the curves are expressed
or the axes to which they are referred.
We now seek for the corresponding expression in Trilinear
Coordinates. The length of the perpendicular on one of these
four imaginary common tangents from any point must be in-
finite. Hence the denominator of p (T., 20) must be put = 0,
that is,
l^-\-m^ + n^—2mn cosA — 2nl cos B—2lm cos 0 = 0
must be the general tangential equation of the points in ques-
tion in trilinear coordinates. Combining this with Sj as before,
we find that ©i corresponds to
a + h + c—2f cos A — 2g cos B^2h cos (7,
which, equated to 0, is the condition that the conic S-\-7c8i
shall represent an equilateral hyperbola.
In this computation the coefficient of 7c only, it is evident,
need be formed, which divided by A must give the condition
74 INVARIANTS.
sought. To find tlie condition that the curve shall represent
a parabola, it will be necessary to form the coefficient of Tt^
and then divide this result by A^.
68. By the theory of foci, the four tangents drawn through
the two imaginary points at infinity on any circle form a quad-
rilateral, in which two of these vertices are real and the foci
of the conic. Now, since ^-f^^i touches the four tangents
common to 8 and 8^^ it will represent these two vertices or
foci in question, when Tc has been so determined that the
conic (8+lc8i) reduces to a pair of points, with the condition
that 8i represents the two points at infinity.
To find these foci, we proceed to find the value of k in
which, substituted in S + k (l^-\-m^), gives two factors, viz.,
(l^ + m^-^ + n) (l^+m^+n),
in which — ^, ^ and — , ^ are the coordinates of the foci, one
Zi Zi z^ z^
value of k giving the real and the other the imaginary foci.
As a simple illustration, let us seek the coordinates of the focus
of a;2 + 2a;^ + i/2-2a;--2y + 2 = 0.
Here ah— I? = 0, and consequently
reduces to 2k^-\-^'' = 0.
But A = 2 + 2— 1-1-2 =0.
Hence 8y or
AV + Bm^ + W + 2Fmn + 20^1^ 2Slm + k{f + m»),
reduces to l^ + 'ni? — 2lm or Q — m^il — m).
INVARIANTS. 75
Therefore the cordinates of the focus are 1, 1, if we regard
Zi as the linear unit in the equation of the line
Ixi-^myi + nz^.
But if these variables are conceived of as functions of one
another, or the line as a function of the variables, then, as z^=0
and the coordinates are represented by -^, ^, these become
infinite, which result is still consistent with the geometrical
conception of the foci of the parabola, where one focus is re-
garded as at infinity.
London : C. F. Hodgson & Son, Printers, Gough Square, Fleet Street.
VOLUMES ALREADY PUBLISHED.
Tract No. 1.— DETERMINANTS.
„ No. 2.— TRILINEAR COORDINATES.
,, No. 3;— INVARIANTS.
UNIVERSITY OF CALIFORNIA LIBRARY
BERKELEY
Return to desk from which borrowed.
This book is DUE on the last date stamped below.
'?o.
"%^
i4faai52HB
*i^
]3Nov'55fi|jF
T 3 0 1955 m
V}^
-ii
,-,t^r""n
«rS T:f>
vw
-^^^
«^1"1"S^
auH9
CD
«65
LD 21-100m-9,'48XB399sl6)476
M5a:S4;2i
I
Live Music Archive
Librivox Free Audio
Metropolitan Museum
Cleveland Museum of Art
Internet Arcade
Console Living Room
Books to Borrow
Open Library
TV News
Understanding 9/11