of the Line of the Second Order. 2% 
ce Saat Gtemte. Sees ole ee ah Ge) 
O=2 (kK +h cosa) + mprisinta. . . . « (12) 
O= ue (i +k cosa) + mgr'sina . . . se (13.) 
f=? (hi? +20 cosa +k?)— mr? sin®a . « (14) 
The quantities 1', kK’, p, 9, r', m being determined from these 
equations, the proposition will be established. 
Multiply (10.) by cos « and subtract the result from the 
sum of (9,) and (11.); then 
a—b cosa+c=2 wu? sin? «—m? sin? « (pr—2 pg cosa+q’), or 
Q=(2—m?)uvsin®a . - « » (15) 
Again, from the same three equations, we have 
mpsinta=u?—a . - » « « (16) 
2m? pqsinta=2u?cosa—b . . .« (17.) 
m? q? sin? @= Ur Cu sw «2s ()8s) 
From these, since 
(2m? p q sin? «)?=4: (m? p? sin?) (m? q? sin? «), we get 
(2 u? cos « — 6)? = 4 (u® — a) (uw? — Cc), or 
4.14 sin? a — 4(a—bcosa« + c)uw = b? — 4a; or again, 
; Q+R 
2u? sin? 4 = + 2 — ~_ Lele. i 
2u? sin®?a = Q + R, and u its (19.) 
Resolve (15.) for 2m, and insert (19.) in the result; then 
Eaeyaeb i 
Cr ae NN - (20.) 
the constant ratio, or ** determining ratio” is hence found. 
For the determination of p and g, substitute the values of 
m2 and u2 from (19, 20.) in (16, 18.) ; then after very slight 
reduction we obtain 
P= Tein a 4Rsint a rier i ewe LY, 
fe _(Q—2esintatR) (Q+R)_(K+R) (QR) G9 
1 =m? sin? a 4 Rsinia ~  4Ksinta eh 
The signs of p and g deduced from these are double, and 
they are in each case to be so taken (like or unlike) as to 
fulfil (17.), viz. to be alike when 2 u* cos a — 4, or (a+¢ + R) 
cos « — bcos 2 wis positive, and unlike when this quantity is 
negative. In either case the two values express sameness of in- 
clination to either axis of coordinates, and hence parallelism 
of position. Wherefore in the ellipse and hyperbola there 
are two such lines as that of which the proposition affirms the 
existence. 
It would also be easy to show that these lines are perpen- 
