96 Scientific Proceedings, Royal Dublin Society. 



(See Salmon's " Conic Sections," Art. 267.) It follows from this 

 that being given five points we can obtain as many other points 

 on the conic as we wish, and so can construct for its centre and 

 axes. 



The method of doing this is exhibited in Plate YIII. 



Let the five points be A, B, 0, JD, E. Take any one of 

 them, E, and draw ^iT through it parallel to the line joining any 

 other two, AB. Join B with one of the two remaining points, C, 

 and let BG cut EK in a. Join DE, and let it intersect AB in j3. 

 Join aj3, and let it cut CD in y. Join Ay, cutting EK in F. 

 Then i^ is a sixth point on the conic, and EF and AB are a pair 

 of parallel chords. Hence the line GG\ joining their middle 

 points, is a diameter. 



Again, draw EL parallel to GO', to meet i?C in a. Join a'j3, 

 and let it cut CD in y'. Join Aj\ cutting EL in F'. Then F' is 

 another point on the curve, and EF^ is consequently a chord con- 

 jugate in direction to EF. Hence FF' is a diameter, and the 

 point 0, in which it intersects G0\ is the centre of the conic 

 passing through A, B, C, D, E. If, now, only the centre be required 

 we may omit the rest of the construction. What we have done 

 up to this is susceptible of great accuracy, as we have had only 

 to draw straight lines from point to point, and to draw a pair of 

 lines parallel to given directions. The rest of the process, however 

 — that, namely, which is required to determine the axes — is of 

 a more complicated nature, and consequently more liable to intro- 

 duce error, although I think in most cases the axes, so determined, 

 will be more reliable than those depending on the mere judgment 

 of the draughtsman. 



To determine the axes we draw EA (or FB) to meet GG' in X 

 and AF{o'r BE) to meet GG' in Y. Then, X and F being harmonic 

 conjugates with respect to the curve, if we take OM (= ON) a 

 mean proportional between OX and OY, the points ilfand A^ will 

 lie on the ellipse. 



Again, if through we draw i/jBT' parallel to EF, and draw 

 ME (or NF) to meet RIF in X', and draw MF' (or NE) to meet 

 im' in Y^; then, since X^ and Y' are harmonic conjugates, if we 

 take OJf a mean proportional between OX' and 0Y\ the point IF 

 will lie on the ellipse, and OM and OM' will be a pair of con- 

 jugate semidiameters. It only remains, then, to draw through M' 



