682 PROFESSOR TAIT ON 
B’, y’. In this way we see that Mrnp1ne lines pass through each point of each 
of the two curves; and by a similar process that every line joining two points, 
one on the one curve the other on the other, is a M1inp1NG line. 
Another process is more instructive. Note that, by the equations of con- 
dition above, we have 
S’pp.= (Fara pi) (Pee — yi 
2 a 
Then our equations become 
SpmpSpep pita eta 
dq ee 
and 
(p” + ¢3) Sprapi + (pi t+ €1)Spap=0. 
If we eliminate p’ or pj from these equations, the resultant obviously be- 
comes divisible by Spawp or Spiwp1, and we at once obtain the equation of one 
of the focal conics. 
10. In passing it may be well to notice that equation (13) may be written in 
the simpler form 
S.p8paw8 + Spap=SCa’8 . 
Also it is easy to see that if we put 
0=pSBp—(a + p’)8 
SB0=0 , 
we have (11) in the form 
and by the help of this (13) becomes 
& =Spap . 
This gives another elegant mode of attacking the problem. 
11. Another valuable transformation of (5”) is obtained by considering the 
linear and vector function, y suppose, by which 6, y, 5 are derived from the 
system $8’, Uy’, Ud’. For then we have obviously 
p= ay B" + xo'x8" , : ; . . (5°) 
This represents any central axis, and the corresponding form of the MtnpInG 
condition is 
8.7xoa 740 =S8ya4y : ; : (7-5 
