of the Line of the Second Order. 29 
Throughout I have written R with the positive sign; but 
this is done to avoid the interference of the double sign 
derived from this source with others that occur in the investi- 
gation: but the change is easily made for the other case from 
the forms here given. 
When Q is absolutely greater than R, or (a —b cos a+ ce)? 
greater than (a — cos a+ c)? +(6?—4a¢) sin? «; or again, 
6°—4 ac negative, the curve (3.) is the ellipse referred to its 
centre, and /’ is known in this case to be essentially negative. 
Wherefore R must have the sign that will render H + R and 
K + R both positive ; that is, R is to be taken positive. 
When, on the contrary, 4? — 4.a¢ is positive, or the curve 
is the hyperbola, /’ may be either negative or positive, accord- 
ing as the hyperbola is the primary or the conjugate one. 
These two cases will require that R shall be respectively ne- 
gative or positive. 
Second. ‘Tur PaRaBo.a. 
The preliminary transformation of coordinates cannot in 
this case be made in the same manner as in the preceding 
one. I have therefore deduced the truth of the theorem 
without the aid of any transformation. However, as the ex- 
pressions, though greatly simplified in the parabola by the 
relation 6? — 4.ac = 0, are not so simple as when the origin 
is transposed to a point z, y, in the curve, I shall here give the 
investigation according to this latter method. The general 
investigation might, indeed, be conducted in an analogous 
manner; but for the ellipse and hyperbola this would have 
led to expressions much more complex than we have had occa- 
sion to use in the method already developed. 
The equations to be resolved become, under this transfor- 
mation, 
BeAr NO? BUNA Oh ag pelt ahd) “ais vinie Dab yay ho) ee) 
Bi COs a) 2m" PG Sie ce... s/h as) oy) ¢; fai 1, (2s) 
Cane — ang ata® ay) et.al ake UP Man toe eg.) 
J=— 2 (ki + h'cosa)—2m*pr'sin®a«  . (4) 
f=—2uv (h' +k cosa)—2m*gr'sinra . . (5.) 
O=v(W? + 2NK cosa + k?)—mr?sinr?« . (6.) 
where d! =2ay,+bu,+ d N=h—xz, 
@=2ca,+ by, +e ryt » (7) 
From (1, 2.) and (2, 3.) we get, as in the former case, 
“ee 
oer a jee ye Me AM a (8.) 
2h 
2 a eae e . . . . . . 
Or = Ok 1 (9.) 
since here Q = R. 
