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a

are conjugate points with respect to tlie conicax? +By2 + yz?=0, is

1 1 1
+ + =0.

B 7 Or writing this equation under the form By+ya+=0, and substituting for a, b, y their values, we have the equation already found, as the condition in order that it may be possible in the conic x + y2 +z=0 to inscribe an infinity of triangles the sides of which touch ax? + by2 + czo=0. Theorem. Let

ax2 + by2 + cz?=0 be the equation of a spherical conic, and let (5:9:8), a point on the conic, be the pole of a great circle cutting the conic in two points; the conic intersects upon the great circle an arc given by the equation cos =

(a+b+c) vĘ? + m2 + 42

w (a +b+c)?(&? +m2 +82) 4(bce? + cana + abg?) Hence if a +b+c=0, d=90°; or there may be inscribed in the conic an infinity of triangles having each of their sides equal to 90°.

It is worth while, in connexion with the subject, and for the sake of a remark to which they give rise, to reproduce in a short compass some results long ago obtained by Jacobi and Richelot. The following are Jacobi's formulæ for the chords of a circle, subjected to the condition of touching another circle ; viz. if in the figure we put

2162

a

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Then it is clear from geometrical considerations that

do "

MA MA'
We have
MA’=cA_cM=q? + R2 + 2aR cos 20-22

= (a + R)---4aR sin?
= {(a + R) 2 — 12}(l – ko sin 20),

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=

or

2

do
V1-ksin? ✓1-k2 sin’¢'
where

ᏎaᎡ
k? =
(a + R)? — p2*

2Whence also

(R-a)-72 k2=

(R+ a)2 — 229 It will be convenient for comparison with the formulæ of Richelot to write Z ACQ=24; this gives

2y=-20. And the differential equation is df

df va? +R? - p2 - 2aRcos24 va? +R? — 2 — 2aR cos2yar

i. e.

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Or if

tan y=g tano,

1-g2 tano 0 cos 24=

1+ go tan

(n) cos? @ + (m + n)ge sino m-ncos 2y=

cos? 0 +g2 sin0

2

2

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m

C

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i. e.
del

d
- n cos 24 Nm+nv1- ka sin? Ó'

2n where k?= has the same value as before. Hence the rela

m+n tion between is

d

d-k2 sin ✓1-ksin? 01 which is identical with that between $ and $'; and in fact the equation between 0, is

1 tan 0 tan ø=

R which, if b=amu, gives O=am(K-u).

The differential equation contains only a single arbitrary parameter; hence the same differential equation might have been obtained from different values of a, R, the parameters which determine the circle enveloped by the moveable chord. The

m' condition for this of course is

n
a? +R?—72 d' + RP — pol2
2aR

2a'R i. e.

(aa' – R?)(a-a') -R(a're - arl2)=0, which implies that the enveloped circles intersect the other circle in the same two points, or that all the circles have a common chord.

Suppose for t=0, W=B, then it is easy to see geometrically that

--p2 ? ß

m

i. e.

n'

n

tau?B= (R—a)e—r?

tan?a=

2

Let the corresponding value of Ol be 0=a, i. e. suppose that for 0=0, we have 0'=u, then

(R+ a)--p2 (R-a-72
(R-a)? — 72 g2

(R+ a) ?-
tan?a=

72 or, what is the same thing,

R

i, e.

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a

seca=

r

And a having this value, we have for the finite relation between 0,8',

FO=FO + Fa.

Richelot has shown, by precisely similar reasoning, that for circles of the sphere we have

dfs
V cos?—(cos R cosa + sin R sin a cos24)?

dy
✓ cosir - (cos R cosa + sin R sin acos 24)
which is of the form
dyl

dy
'1-(4+u cos 24)
4+

✓1-(1 + u cos 24')? wtere

cos Rcos a λ:

=

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And it is very important to remark, that this equation contains the two parameters , M, so that the same equation cannot be obtained with any new values of the parameters a, r; or the formulæ in plano for three or more circles do not apply to circles of the sphere: the geometrical reason for this is as follows, viz. in the plane a circle is a conic passing through two fixed points (the circular points at oo), and consequently any number of circles having a common chord are in fact to be considered as conics, each of which passes through the same four points. But circles of the sphere are not spherical conics passing through two fixed points, but are merely spherical conics having a double contact with an imaginary spherical conic (viz. the curve of intersection of the sphere with a sphere radius zero); hence circles of the sphere having a common spherical chord are not spherical conics passing through the same four points. I am not sure whether this remark as to the ground of the distinction between the theory of circles in plano and that of circles on the sphere has been explicitly made in any of the treatises on spherical geometry. To reduce the equation, write

1-2+u) tan y=A

(

tan 0; 1ー(シール)

-(1-4 then after a simple reduction, dy

sin?8. V1-(4+ucos 24)?

4u

(1+u-19 Or the relation between the two values of 0 is

d

de 71-k? sino ✓1-2 sin

V

de

2

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2

sin a

tanr

where

4u k=

(1+)2–2?' i. e,

tan R
4

cos R sin r
k=
tan R
sina

- 1
tanr

cos Rsinr Suppose that for y=0, x=B, it is easy to see that

sin? (R-a) — sinr

cos? R sinar Let the corresponding value of Obe O'=u, i. e, suppose that for =0, we have O=«, then

cos (R+a) 1

sin? (R-a-sinor cos (R-a) cos? R sinar

(

+

[ocr errors]

tano 8=

COST

[ocr errors]

tan?a=

1

COST

II

=

cos r- cos (R+a) cos?r— cos? (R– a)
cos r- cos (R-a) cos? R sinar
cosr – cos (R+a)) (cosr+cos

(R—a))
cos2 R sinar
(cos r + sin R sin a)? - cos? R cos? a

cos2 R sinar
(cos r sin R+ sin a)? — cos? R sin?,

cos? R sin?? i. e.

tan R

sin
tan?
a = (tan
+

-1,
tanr

cos R sinr whence

tan R
(tan +

tanr
And a having this value, the finite relation betwcen 0, b' is

FᎾ =FᎾ +F. By comparing with the corresponding formula in plano, we arrive at Richelot's conclusion, that the formulæ for the sphere may be deduced from those in plano by writing in the place of

tan R the functions

tan r' cos R sin r, respectively.

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sec a =

sin a
R sinr

COS

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' 2 Stone Buildings, October 1, 1856.

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