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.