504
SOLUTIONS OF A SMITHS PRIZE PAPER FOR 1871.
[537
But in virtue of the relation a 2 + /3 2 + y 2 = 1, vve have
da _ d/3 dy .
da „d/3 dy
da _ d/3 dy
■S^£ + ’dr 0 '
Hence subtracting the corresponding equations we have three equations, which are at
once seen to be equivalent to the two equations
d/3 dy dy da da d/3 _ ^ ^
dz dy ' da; dz ' dy dx ' '
equations which must be satisfied identically, whatever are the values of (x, y, z). The
equations have been obtained as necessary conditions; they are, in fact, the sufficient
conditions for a double system; for the line being unaltered in passing from P to P',
it remains unaltered when we pass to the following point P", and so on; that is, for
the passage to any point Q whatever on the line.
Coe. If the equation adx + ¡3dy + y dz = 0 be integrable by a factor, it must be
integrable per se: in fact, the condition that it may be integrable by a factor is
d/3 dy
dz dy
+ /3
dy da ida d/3
dx dz)^~^\dy dx
= 0.
But we have
and the equation thus becomes
that is, k = 0, and therefore
%-P-h &c„
dz dy
k (a 2 + /3 2 + y 2 ) = 0,
d/3 dy _ dy da _ da d/3 _ ^
dz dy dx dz 5 dy dx
Hence, also, if the lines cut at right angles a surface, we must have adx + ¡3dy + ydz
a complete differential.
The foregoing theory is given in Sir W. R. Hamilton’s “ Memoir on Ray-Systems. ’
6. If X = 0, Y = 0, Z = 0, W = 0 are four given conics in the same plane and
having a common point: show that, in the system of conics aX + bY + cZ + dW = 0, there
are in general four (improper) conics the equations of which may be taken to be
x 3 = 0, y 2 = 0, xz =0, yz = 0.