8 Mr. 'T. S. Davies on the Foci and Directrices 
dicular to one of the conjugate rectangular axes, did space 
allow us to dilate. 
We shall now proceed to the other three equations (12, 
13, 14.) which, together with m, p, g already found, involve 
A, Ry’. 
From (12, 13.) resolved for h!' and x’, we get 
: m? r! (q — p Cos a) 
h = So eae aS 
__ mr! (p— q cosa) 
i= 5 see, pele 
Also, multiply (12.) by #! and (13.) by 7', and add; then 
u2 (Wi? + 2h! cos @ + kh?) = — m? 7! (pk'+qh') sin?a . (25.) 
Insert (23, 24.) in (25.) 5 whence there results on reduction, 
u2 (Wi? + 2h cosa+k?)=m'r? sint?a . . (26.) 
Write for the left side of this equation its value on the right 
in (14.); then we immediately obtain by means of (20.), 
1 2 41 
aera MEE a CUR ASAE «, (2s) 
m? (m?—1)sin?a 2 R(R—Q) sin? a 
This gives two values of 7’, and since they are equal, the 
two lines to be found are symmetrically situated with respect 
to the centre. 
Again, substitute the values of 7’, p, g, u? in (23, 24.), and 
we obtain the values of #! and k’; viz. 
watal at VH+R.cosa— Vv K+R : (28.) 
¥ b—A0a¢ sin? « 
Hat a/ Pf eK Boe eee 
~ b?—Aac sin? 
These show that the points to be found are also symmetrically 
situated with respect to the centre, and with respect to the 
axes of coordinates. It also appears from these equations 
that 2’ and I’ are always of different signs. Also, since 
2Qac—bd pr, 20d=— be 
faaae ES" t eae 
we get by merely substituting the value of 3 /’ from (6.), the 
value of 7 from (5, 27.), and those of p, g from (21, 22.). The 
complete solution of the system of equations, and therefore 
the demonstration of the theorem, is effected. 
With respect to the conditions of reality of these expres- 
sions, little need be said here, as the subject is discussed in 
several treatises on lines of the second order with sufficient 
amplification, the functions here involved occurring in almost 
all inquiries relating to these curves. 
ca (23.) 
es 
