286 ON THE DEDUCTION OP THE GENERAL EQUATION. 



Olenidce — Buckler of moderate size, but comparatively short. Gla- 

 bella small, narrowing anteriorly. Facial suture terminating at the 

 lower margin of the buckler. Body-rings 12-17. Pleura; spined. 

 Pygidium small. 



Phacopsidee — Buckler and pygidium generally large ; the latter 

 with well-developed axis, ofteu terminating in a spine. Glabella 

 lobed or pustulated, widening anteriorly. Facial suture terminating 

 at the sides of the buckler, about on a level with the eyes : these 

 latter very visibly reticulated. Body-rings 11-12. Pleura- rounded 

 or spined. 



Cerauridce — Buckler large, horned. Pygidium with short axis, 

 and with horns or spines. Glabella widening above, furrowed. Facial 

 suture and body- rings as in Phacopsidse. 



LickasidcB — Buckler broad, but short and somewhat pointed. Gla- 

 bella prominently oval, with several accessory lobes. Facial suture 

 terminating at the lower margin of the buckler. Pygidium with 

 short axis, and denticulated or spined limb. Body-rings 11. 



Acidaspidce — Glabella in separate lobes, strongly pronounced. 

 Buckler broad, and somewhat short. Pygidium small, or of mode- 

 rate size with short axis, and spined or denticulated limb. Body- 

 rings 8-10. Pleurce spined. Entire shell more or less ornamented. 



Paradoxidcz — Buckler large, horned. Glabella well-developed, 

 widening above. Pygidium very small. Body-rings 11-20. 



Appendix. — Agnosti. Small inconspicuous forms, exhibiting in 

 general a couple of nearly similar shields (buckler and pygidium) 

 separated by two or three thoracic segments. When more fully stu- 

 died, the agnosti will be found, probably, to comprise a distinct 

 group, embracing several families. 



ON THE EEDUCTION OF THE GENEEAL EQUATION 



OF THE SECOND DEGBEE IN PLANE CO-OEDINATE 



GEOMETEY. 



BY J. B. CHEBBIilAN, M. A. CANTAB. 



laOPESSOB OB HATUEAL PHILOSOPHY, UKIVEESITT COLLEGE, TOEOSTO- 



Read before the Canadian Institute, January Id, 1856. 



The general equation of the second degree in plane rectangular 

 co-ordinates, under the form 



ax 2 + 2 cxy + by 2 + 2 dx + ley +f= 0, where a is essen- 

 tially positive, and where the quantity 



\ (a + b) * —4 (ab— c* ) l* 



