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4) that <f>(ti) satisfies the differential equation :

-5-; = A — — when a > 0.

Hence determine the function /(a) defined by the integral.
46. Discuss the function defined by the integral :



y 6— •'•sir



•sin X dx,



47. Discuss the function

w

<f>(a) =r / tan«<^ d<f>,

plotting the graph and determining the essential chai-acteristics.

48. Show that the functions

/(a) = /^e-^cos(ax«)f/^',

<^(a) = / ^-^sin(a.r*)rfx,
satisfy the linear differential equations :

•^^*^ = 2(1"+ «^) [*^^*^ "^ *^*^]'

HARViOtD UNIVRBSITY,

CaMBRIDOR, MAH8AGHU8RTT8.



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NOTE ON THE PRODUCT OF LINEAR SUBSTITUTIONS.*

By H. B. Newson.

If two linear substitutions in n variables be compounded, the product is
also a linear substitution in n variables. The following method of expressing
the result in determinant fonn is believed to be new. The proof is given for
two substitutions in three variables, but the method and result are capable of
immediate generalization for n variables.

Let T and Tj be two substitutions as follows :



pxi = a^x -h bit/ -h c^z,
P!/i = ct^x -h h^y + v^z,
pz^ = a^x -h h^y -h c^z,



^1 : Pii/2 = o^^i + fii!/i + 7««i»
Pi^i = a^Xj -h fin!/i + 78^1-



The substitution T^ is obtained by elhuinating x'l, yi, Zi from the above equa-
tions. This may be done as follows : Find the inverse of T by solving the
three equations of Tfor x, //, z,' Thus we get

- x = AiXi -h A^yi + A^Zi,

A

9
where A is the detenninant of Tand Ay B^ et-c. have the usual meanings.

The three equations of T~^ and the first one of 7\ form a system of four
simultaneous linear equations ; hence



A
X

9

A

A

z

P






A^ A^



Ik Jh






ft



7i



= 0.



* Read before the Chicago Section of the American Mathematical Society at the Evanston
meeting, 2/8 January, 1902.

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148



NEWftON.



This equation expressen the i*elation between x^ y, z and x^.
equation for x, we get



Solving the last







X


Jxi Ji.^ -/Ij








y ih B^ B,




ppi^xt =


X C, C, C,
«, /8i 7,





like manner we get similar reuulto for y, and z, ; thus




X Ai Ai A^




X yi, ^, ^t




y A B, B,




// i/, A A


PPi^y» =


z Gi Cj Cj


: /»f»,A24 =


2 C\ Ct C,




a, A









«8 A 7.



When these three determinants are ex|)anded, A divides out of both sides of
the equation.

The general formula for 7) variables is

Ai A^ .... A„



/»/»,A— •x{" =



«i



^ Bi B,



X, N, JV.
m, /S,



B.






(t = 1 . . . n).



Thbobbm. • 7%e value of Xf" in the product of T and T,, two linear
aubstittUiona, m proportional to the determinant formed by bordering the deter-
minant of T~^, the inverse of 7\ vertically hi/ the variable* of Tand horizon-
tally by the coejfictente of the ith equation in Tj.



UmynnrrT or Kansas.

LAWBKNCa, Kamsah.



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CONTENTS.



PAOB



Space of Constant Curvature (Conclusion). By Professor Fred-
erick S. Woods, 93

Brilliant Points and Loci of Brilliant Points. By Mr. W. H. Roever, 113

Problems in Infinite Scries and Definite Integrals ; with a Statement
of Ceiiain Sufficient Conditions which are Fundamental in the
Theory of Definite Integrals. By Professor W. F. Osgood, 129

Note on the Product of Linear Substitutions. By Professor H. B.

Newson, .......... 147

ANNALS OF MATHEMATICS.

Published in October, January, April, and July, under the auspices of
Harvard University, Cambridge, Mass., U. S. A.



Gambridfje: Address The Annals of Mathematics^ 2 University Hall, Cam-
bridge, Mass., U. S. A. Subscription price, $2 a volume (four numbers)
in advance. Single numbers, 75c. All drafts *and money orders should be
made payable to Harv^ard University.

London: Longmans, Green & Co., 39 Paternoster Row. Price, 2 shil-
lings a number.

Leipzig: Otto Harrassowitz, Querstrasse 14. Price, 2 marks a number.

Pbtnted bt The Salbm Pbbss Co., Salem, Mass.. U. S. A.



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JULY 1902



Annals of Mathematics



(Founded by Obmond Stonk)



EDITED BY

Ormond Stone W. E. Byerlt H. S. White

W. F, Osgood F. S. Woods Maxime B6cher



PUBLISHED UNDER THE AUSPICES OF HARVARD UNIVFiRSITY



SECOND SERIHS VoL- 3 No. 4



FOB 8ALE BY

THE PUBLICATION OFFICE OF HARVARD UNIVERSITY
2 University Hall, Cambricls:e, Mass., U. S. A.



London : Longmans, Grbrn & Co. Lelpilg i Otto HARnAseowiTZ

89 Paternofiter Row Querstrasse 14






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NOTE ON A TWISTED CURVE CONNECTED WITH AN INVOLU-
TION OF PAIRS OF POINTS IN A PLANE.*

By H. S. White.

1. A Method for Gtonerating some Twisted Curves. The

points of a plane are said to be in involution when they are so related in sets of
n points that any one point of a set determines all the other n — 1 points of that
set. The simplest involutions, and the best known, are those where each set
contains only 2 points. Of involutions of higher order so much is known as
that they are all rational ; that is, that between the points of a plane and the
sets of points of any involution a one-to-one relation can be established.! But
for orders higher than 2 the classification and reduction to types have not yet
been given. Hence we shall understand for the present by "involution"
always an involution of order 2, one whose points are grouped in pairs.

Bertini showed that involutions of second order are of four distinct
types, every such involution falling into some one of these four classes. J Two
of the same type are usually not projectively equivalent, but each can be
transformed into the other by some Cremona transformation. As the repre-
sentative of each type he chooses the simplest.- The first type is that in
which collineation or projection effects an exchange of the points of each pair.
Such a collineation evidently must be a perspectivity, or central projection.
The second type includes as many distinct species as there are possible orders
of curves, one for every integer above 3. Among these we consider that of
lowest order, in which paired points lie on rays passing through a fixed point,
and are conjugate with respect to a fixed conic. § These two alone are to be
used in the present note.

The notion of an involution in a plane may be used to generate twisted
curves, just as the notion of involution on a line has been used (and may be

* Read before the Chicago Section of the American Mathematical Society at its meeting
January 8» 1902.

t G. CastelnuoYO, Mathematiache Annalent vol. 44 (1894), p. 125.

X Ricerche saUe trasformazioni aDivoche.inyolatorie nel piano. Aimali di mcUematica,
aer. 2, vol. 8 (1877), p. 264.

§ Compare the system of intersections of the circles of a net, nsed as an illustration, by
Professor B/^her in the Annals of McuherndtCcSf ser. 2, vol. 3 (1902), pp. 49-52, and especially
the first footnote on p. 62.

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150 WHITE.

used further) to generate plane curves. To make this explicit, recall the
method proposed by Steiner, and elaborated by Schroeter, for generating a
cubic curve in the plane. A quadric involution on a straight line, it will be
remembered, consists of pairs of points harmonic with respect to two fixed
points, either real or iniaginary, on the line. Two pairs of points determine
fully such an involution, since there is only a single pair of points separating
harmonically two given pairs. Any third pair of points, in order to belong
to the same involution, must satisfy a single condition. Hence if three pairs
of points are taken at random in a plane, and these are projected from a varia-
ble center in the plane upon any line, the requirement that their projections
shall form 3 pairs in quadric involution will subject the variable center to a
single condition ; i. e. the locus of that point is a definite curve. In this case
the locus is a general cubic curve.

In the same way, if a certain number of pairs of points in a plane suffice
to determine an involution of a specified type, then one additional pair of
points, in order to belong in the same involution, must satisfy two conditions.
Assume now, in space of 3 dimensions, pairs of points one more than suffi-
cient ; project them upon a plane from a variable center, and require the pro-
jections to form pairs in an involution of the specified type. This will be
equivalent to 2 conditions, restricting the variable center to motion along some
definite twisted curve.

In this way involutions of order 2 can give rise to an infinite variety of
algebraic twisted curves, and may be found to lead to important properties of
those curves. Indeed two species of involution in the plane which are equiv-
alent to each other by virtue of some Cremona transformation will usually be
connected with twisted curves which are not transformable into one another
by any Cremona transformation of three dimensional space.

To exhibit the method I have worked out the following two simplest
examples.

2. First Example : the Perspective Involutioii. In a perspec-
tive relation of the points in a plane, the axis o and the center may be any
line and any point. To find the conjugate to any point Pi (see Fig. 1), pro-
duce a line OPi to cut the axis o in a point P', and determine a fourth point
Pj on OPi harmonic to 0, Pi, and P'. All the pairs of points P1P2 consti-
tute then an involution of the first type. Otherwise, two such pairs may be
taken arbitrarily, and from them the center, the axis, and consequently all
other pairs can be found. Suppose Ai and A^, Pi and B^ to denote given



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ON A TWISTED CURVE. 151

pairs, then any third pair Pi P, must have their joining line pass through the
point where the line AiA^ meets B^B^. This is the first condition. Again,
this center determines two points A and B'
where the axis o must intersect A^A^ and BiB^ ;
the second condition is that the center and axis
o shall separate harmonically the points Pi and P^.

Assume in space three pairs of points A^A^y
BiB^y Ci O2 (no four in any plane) . If the pro-
jections of the 3 lines a^b^c which join these sev-
eral pairs are to meet in one point of a plane, the
center of projection X must lie on a line inter-
secting all three : a, 6, and c. That is, JT must lie ^^- ^'
on a ruled quadric having a, 6, c as directrices . This first condition restricts
JTto a quadric surface.

The second condition is satisfied if the six projecting rays XA^ XA^^
XBif . . . XC2^ lie on a quadric cone. For then the generator which passes
through JT, and its polar plane with respect to the cone, will project from X
into the required center and axis of perspectivity, inasmuch as their sections by
any plane will separate harmonically the projections of each pair of given points,
€, g. of Ai and A^. Conversely we see that this condition is not only suffi-
cient, but also necessary for the point X. We have to find therefore the locus
of the vertex of a cone passing through 6 given points. That this is a sur&ce
of the fourth order* can- be shown as follows.

A quadric surface is fixed by 9 points » Hence through the 6 given points
there pass all the quadrics of a linear system with 3 arbitrary parameters :

<^o + ^1 ^1 "i" ^ ^2 + ^8 4^9 = 0«
If ^ is to be a conical point upon one of these quadrics, 4 conditions must be
satisfied, viz, the vanishing of the 4 partial derivatives :t

<f>oi(x) + \4>n{x) 4- X2<^i(x) 4- X8<^8i(«) = 0,



<f>Qi(^) + \<f>u{^) + M<f>ti(^) + \<f>u{^) = 0.



♦ The ** Surface of Weddle." See full discussion in Cayley's Collected Works, vol. 7, p. 160
«qq. ; and a paper by Chasles, Comptes Bendus, vol. 62 (1861) pp. 1157-1162.

t The quadrics ^, ^, etc., are supposed to be homogeneous in the coordinates X], Xg, X3, X4.
Partial derivatives are then denoted by added subscripts. Thus the abbreviation 0si(x) means
d



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152 WHITE.

Eliminating from these 4 equations, linear in the (x), the parameters X,, X,, X^,
we find the equation of a quartic locus for A^ Notice that it is the Jacobian
of all quadric surfaces that pass through the 6 fixed points.

The intersection of a (luadric and a ({uartic is in general an octavic curve.
But this (]uartic contains entire lines, among them the 15 which join two and
two the 6 fixed points ; and 3 of these are the directrices a, 6, c of the quadric.
Hence the proper curve is a quintic, and cute every directrix in 4 points, but
every generator of the quadric (/. e. those of the system opposite to a, 6, c)
in only I point. This quintic belongs obviously to the class aoof Noether,* a
sub-species not particularly described by Halphen,t but included under his
second species of quintics. All directrices of the quadric surface are four-fold
secants of the cur\'e, so that its projection from any point of the curve upon a
plane is a plane cjuartic with a triple ix>int, proving that the quintic is of defi-
ciency zero, t. e. it is rational. Generators of the second system meet it in
only one point each, hence it can have no actual double point. Finally, a
counting of constants makes it probable that every quintic of Noether's class
ai can be generated by the above method ; if this is true, a geometric proof
would disclose an interesting property of the curve.

3. Second Example : Involutioii of Harmonic Conjugates on
Rays Cutting a Conic. Five pairs of conjugate points with respect to a
conic completely determine that conic. Hence 5 pairs of points suffice to

detennine an involution of the second type. But
these 5 pairs are not perfectly arbitrary ; they are
subject to the restriction that the 5 lines joining
the respective pairs must have a common point,
which we may call a radiant point. Any sixth
pair of points, in order to belong in the same in-
volution as the first five, must satisfy 2 conditions :
first, their joining line must pass through the radi-
ant {)oint; and second, the two points must be
*''°- *• conjugate with respect to the same conic which

separates harmonically the first 5 pairs. In Fig. 2 O is the radiant point,
Ai A^y BxB^y etc., the 5 pairs that determine the involution, o the conic which
separates each pair harmonically, and P| P^ any sixth pair in the involution.

* Zor Gmndlegang der Theorie d. algebraischen RaninciirFen. AhhmSlgn. der kgL Akad. d.
Wi8sen8ch€^ften xu Berlin, 1882, Anhang, p. 88.

t Sur la classiflcation dee coorbes gauches alg^briques. Jour, de VEeole polffteehniqne, vol.
62 (1882), p. 162.




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ON A TWISTED CURVE. 153

To secure a variable center of projection we have therefore to assume in
space, not 6 pairs of points taken at random, but one pair on each of 6 direc-
trices of a ruled quadric surfiice. Then a center of projection X anywhere
upon that quadric will give a set of projections satisfying the first condition,
for the 6 directrices will project into lines meeting upon the generator that
passes through X, As to the second condition, some quadric surfistce separa-
ting harmonically each of the 6 given pair^ of points must be a cone having X
for its vertex.* E^h pair of points, as being conjugates, supplies one linear
condition for the coeflScients of the equation of that quadric, leaving 3 free pa-
rameters (10 — 6 — 1 = 3). Here therefore the reasoning of §2 applies again,
and the locus of the vertex Xis a quartic surface, the Jacobian of 4 quadrics
that have generally no common point. This Jacobian must evidently contain
each one of the 12 given points, but need not contain any of their join lines.
The curve of intersection is accordingly of order 8 and is of the species a^ of
Noether {I. c, p. 96), or Halphen's 1st species {I, c, p. 165).

By counting constants we may judge that every curve of this species lies
on at least one Jacobian surface, and contains 24 pairs of points separated
harmonically by all cones of the system. But that these pairs, or any six
of them, will lie on as many of the quarti-secants of the curve is apparently
improbable. Hence the curve found from six arbitrary jmirs of points on six
generators of a quadric by the aid of the concept, involution of the second
type, must be of a very special kind under its species. To confirm this, it
would be needful to study minutely the system of lines joining corresponding
points on the general Jacobian surface.

Northwestern UmvERsmr,

EvANSTON, Illinois, February, 1902.

* See Cayley and Chasles, as cited above.



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ON SOME CURVES CONNECTED WITH A SYSTEM OF
SIMILAR CONICS.

Br R. E. AxLARDicE.

1. Introductioil. The problem has been propoesed by Steiner* of find-
ing the envelope of a system of similar conies circumscribed about a given
triangle, and of finding the loci of the centres and foci of the conies of the sys-
tem. He states that the envelope is a curve of the fourth order having three
double points, and gives some of its properties. The problem has been treated
by P. H. Schoute in a paper entitled Application de la transformation par
droites symMriques d un probUrrie de Steiner.^ In this paper the author dis-
cusses the problem of the envelope in detail by a geometrical method, to which
reference will be made hereafter, and gives the order of the locus of centres, of
the locus of foci, and of the locus of vertices, and the class of the envelope of
asymptotes, of the envelope of axes, and of the envelope of directrices. He
states, however, that the envelope of the asymptotes is a curve of the sixth
class, but does not point out that it breaks up into two curves, each of the third
class. I propose to determine analytically the equations of certain of these
curves.

2. Defliiition of Similar Conios by the Use of Asymptotes. It
may be taken as a definition or proved as a theorem, according to the point of
view, that two conies are similar when the angles between their asymptotes are
-equal. When rectangular coordinates are used, two conies with their centres
at the origin are similar and similarly situated if the terms of the second degree
be the same, or proportional, in the two equations. Now these terms of the
second degree represent the (real or imaginary) asymptotes, and the angle
between these asymptotes is not altered if the conic be displaced in any way.
It may easily be shown that the tangent of the angle between the asymptotes
of an ellipse is equal to 2abi/(a^ + 6*)> where a and b are the semi-axes, and
for a real ellipse this quantity is a pure imaginary whose modulus is less than

* Systematische Entwlckelung der Abhanglgkeit geometrlscher Gestalten 7011 einander
{problem 39 of the supplement), Gesammelte Werke, vol. 1, p. 446; Vermischte Satze and Auf-
gtkheu.ibid,, vol. 2, p. 676.

t Bulletin des sciences matkimatiques et astronomiques, ser. 2, vol. 7 (188S), pp. 814-324.

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CURVES CONNECTED WITH A SYSTEM OF CONICS. 155

unity. It is obvious that two conies for which this quantity is the same are
similar, as the quantity may be expressed in terms of 6/a, which in turn can
be expressed in terms of the eccentricity.

3. Laguerre's Projective Deflnition of Angle applied to the
Asymptotes- It has been shown by Laguerre and Cayley* that the angle
between two straight lines can be expressed in tenns of the cross-ratio of the
range of four points consisting of the points where the straight lines meet the
straight line at infinity and the circular points at infinity. More definitely, f
if a be this cross-ratio and the angle between the lines, then 6 = JHog a ;
and from this it may easily be shown that tan* ^ = — (1 — a)^ j (\ + a)*. Now
the infinitely distant points of the two asymptotes are the points where the
conic itself meets the line at infinity. We thus have to consider a variable conic
that circumscribes a given triangle and meets the straight line at infinity in two
points that form with the circular points at infinity a range of constant cross-
ratio. But instead of the line at infinity and the two circular points upon it we
may take any straight line with two fixed points on it ; and these fixed points
will be most conveniently assigned as the points of intersection with the given
straight line of a fixed conic circumscribing the given triangle. The special
case of the system of similar conies will then be obtained by taking the straight
line at infinity and the circumscribed circle for the fixed straight line and
fixed conic.

4. The Quadrie System of Similar Circumscribed Conies.
By means of the theory of invariants we may write down the condition that a
straight line meet two conies in pairs of points having a given cross-ratio a. %

Let

tZ-P = W, Xj + W2 ^2 + ^3 ^ =

be the given straight line ;

p% = Pix^x^ ^ Pi^xy^ -\-Pz^\^ =
be the fixed circumscribed conic ; and

Xj = XiXaXj + y^x^x^ 4- \zXiX^—

* Lagaerre, Note snr la th6orle des foyers, Nowoelles annalts de matJUmatiquea, vol. 12*
(1853), p. 64. Cayley, A sixth memoir upon qnantics, Philosophical Transactions, vol. 149
(1859), pp. 61-90, and Collected Mathematical Papers, vol. 2, pp. 561-592. See also Loria, Teorie
geometriche, where, however, the reference Is given to Gay ley's fifth. Instead of to his sixth
memoir.

t Compare Klein, Nicht-Euklidische Oeometrie, vol. 1, pp. 47-60.

X See Clebsch-Llndemann, Vorlesungen Uber Geometric , vol. 1, p. 281.



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156 ALLABDICE.

be the variable circumscribed conic. Tlie required condition is

/)'«(«- 1)» - DD'' (« + 1 )« = 0, (1)

where

- JZ) =Xlt4 + Xju|+ ^til- 2XiX4tiiU,- 2X,X,w,w,-2X,XiM,Wi;

- * ^" =i>iwj +I>tu5 +i>juj - 2/>i^,ttiU4 - 22),i),tt,ti,~ 2i>ti>i«,tii ;

- i Z)' = Xii), wj + X^i),t4 + Xsi),t4 - (X,^ + X,^,) u,tt, .

Since according to (1) the parameters X^, X^, X,, which occur linearly in
the equation of the conic, are connected by a quadric relation, the conies are
said to form a quadric system.

5. LoouB of Centres of Conios in the Quadric System. Cor-
responding to the problem of finding the locus of the centres of a system ot
similar conies we have to find the locus of the pole of u^ with respect to the
conic Xj..

I^ (yi> yt> ys) b® ^^ required pole, we have

"i = ^y« + ^y%^ etc.,

whence, omitting again a factor of proportionality, we have :

>'i = yi(wsyt + M,y8- wiyi); etc. (2)

Substituting these values of X^, X^, X, in (1) we get a curve of the fourth order
as the locus of the pole of u^.

If (Xi, X), Xs) be regarded as a parametric point, the locus of this point is
a system of conies (for varying values of «) of which the equation is given by
(1). This system of conies has double contact with the conic 2> = (an in-
scribed conic), the chord of contact being Z>' = 0. The relations (2) show
that this system of conies is transformed into the system of quartic curves by
means of a Cremona quadratic transformation. As such a transformation turns
a non-specialized conic into a trinodal quartic, we shall expect to find a system
of trinodal quartics, all having double contact with the quartic which replaces
the conic Z) = 0.

If we substitute the values of Xi, Xj, X, in Z), Z)', Z)", we find, putting
Pi = Wi {Pi'iH+PzU^ -i>i Wi), etc.,

I> =-(wiyi + ^y« + W8y8)(w2ya4- utyz-u^yi){u^y%-^ wiyi-ti,y,)

(^iyi + w,yt-««8y8)»
Z)' =PiWi3^ + P3^^yl4- Pansys- (P\^+ PsMyiys-etc.,

Z)" = -(Pi/>i + P,i>, + P,i>s).



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CURVES CONNECTED WITH A SYSTEM OF CONICS. 157

Hence the equation of the system of quartics is

(a-l)«[i>iWay?+ . . . - (P, % + i^^^i) yiy^ Y

. . . . = 0.

The fact that D = degenerates into four lines suggests a simplification of
the equation. Choosing the three components which form a symmetrical set,
we transform to a new triangle of reference by means of the equations

^2l/% 4- t^sys ~ wiyi = WiX^, etc.
The result is :

— (a 4- l)^kxiX^x^ (UiXi 4- u^x^ + u^x^) = 0,
where k = Pipi 4- P2JP2 + PzPs-

This equation represents a trinodal quartic having its double points at
the vertices of the new triangle of reference and having u^ for a double tan-
gent. By proper selection of the five constants involved in a, Wi, ?^2i '^s^PuPi^Pz^
the equation may be made to represent any trinodal quartic referred to the
triangle having its vertices at the double points. In other words, any quartic
with nodes at three distinct finite points and touching the line at infinity in
the two circular points is a locus of centres of similar conies through three


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