[645 
645] 
a smith’s prize paper, 1877. 
45 
imum, parameter 
rrounded by the 
n parameter, we 
val belonging to 
e other, the two 
re have a closed 
tie case supposed 
single point of 
imum parameter, 
it point. If we 
that this point 
light, each loop 
, and the figure 
rves. 
ifficulty that the 
nee appears from 
the difficulty is 
rential equations 
ng an attraction, 
lirectly with the 
ar plane through 
the problem is 
force varying as 
y similar to the 
circle into rings, 
attraction as if 
ir the radius of 
traction is = rrr 
3 surfaces to be 
of the equation 
= 0, where L is 
on of the order 
tired number of 
of L and of M, 
we obtain here 
e of r, we have 
function of the 
the first three cases the equation of the required conic in point-coordinates; and then, 
by changing these into line-coordinates, we have the equations for the remaining three 
cases. 
p = o: 5 points. The equation of the conic is 
(a, b, c, f, g, K$x, y, z) 2 = 0, 
and we have 5 linear equations to determine the ratios of the coefficients ; the number 
is therefore = 1. 
p = 4: 4 points and 1 line. Taking U= 0 and F=0, the equations of any two 
conics each passing through the four points, the equation of the required conic will 
be U + \V= 0, and the condition of touching a given line gives a quadric equation 
for A; the number is therefore =2. 
p = 3: 3 points and 2 lines. In the same manner, by taking U = 0, F= 0, W — 0, 
for the equations of any three conics through the three points; or if the equations 
of the lines containing the three points in pairs are x = 0, y = 0, z = 0, then the 
equations of the three conics are yz — 0, zx- 0, xy = 0, and the equation of any conic 
through these points is fyz + gzx + hxy = 0; the conditions of touching two given lines 
%x + yy + £z = 0 and %'x + yy + t,'z = 0, are 
V/ VI+ \/v + V^W£ = o> V/VI’ + V<7 vV + \/h vr = ^ 5 
we have thus the ratios \Jf : \/g : \Jh linearly determined in terms of VI) \/v> 
there is no loss of generality in taking VI) V!’ each with a determinate sign, the 
signs of V 1 ?) &c. being then arbitrary, we have 2 4 , =16 values of *Jf : \/g : y/h, and 
therefore one-fourth of this = 4, for the number of values of f : g : h; that is, the 
number is =4. 
11. This is a known theory; taking x u y u z x for the linear functions of x, y, z, 
which are such that 
(a, b, c, /, g, h\x lt y x , z Y f = (a, b, c, f g, K$x, y, z)\ 
then assuming x 1} y lt z x = 2% — x, 2y—y, 2£—z respectively, we have 
(a,...$2%-x, 2y-y, 2£-zj = {a,...\x, y, z)\ 
or, omitting terms which destroy each other, and throwing out the factor 4, this is 
(a, ...5|, y, £)*= (a, V, 15«) y, 
an equation which is satisfied identically by assuming 
cii; + hy +g£=ax + hy+gz . - vy + fit;, 
kg + by +/| = hx + by +fz + vi; . — A£, 
g^+fy +c£=gx+fy + cz-gt; + \y . , 
re obtained most 
;) by finding for