TRANSACTIONS OF SECTION A. 479 



loounding curve intersects the axes, and in other transition-cases, which will he 

 explained in § 8. 



3. Order-quartics, whether singular or non-singular, have been classified by Dr. 

 Zeuthen, of Copenhagen (' Matheaiatische Annalen,' 1874), according to their 

 depressions, characterised by a bi-tangent and two points of inflexion ; such pits 

 are termed by that eminent geometer, /b/m (although ybye« might be thought more 

 expressive). So class-quartics may have four or fewer stirrup-like excrescences. 



Def. — A stapes or stapete is characterised by two cusps and a crunode. By a 

 stapete-point is meant such an excrescence in its nascent state, just as a folium-point 

 (foveate) or a point of undulation is an incipient depression. The stapetes and 

 folia are reciprocal, and either do or do not constitute singularities, just as the curve 

 is regai'ded by its class or order. 



Ex. a* + 4/33y = ; 27 (apY + 64 {bqY cr = o. 



These equations denote the same quartic, with one stapete-point and one folium- 

 point. In passing from one form to the other, 8 dimensions are lost : for (a, ^) is a 

 triple stapete-point with three singularities, and (a, y) is a point of undulation with 

 none. Contrariwise q is the same foliiun point with a triple tangent, and 7- a 

 point with no singularity. The same contradiction and parallelism occiu' in cusped 

 cubics, which are always inflexional. So that these conclusions may be gene- 

 ralised. Since 



a" + M^"-!-)/ = o; (n - l)»-i (apY = (- nhq)"-'^ cr, 



denote the same curve, n(n — 2) dimensions are lost by the mergence of cusps and 

 nodes, or of stationary and bi-tangents at the points B, C. 



4. The positions of the quadi'uple focus P, and of the foci Q, E. of the satellite- 

 conic, will be distinguished in five families of groups. 



I. P, Q, R collinear : Q, R coincident in the centre of a satellite circle. 



n. P, Q, R collinear : Q, R the foci of a satellite conic. 



III. P, Q, R not collinear, but Q, R at an infinite distance. 



IV. P, Q, R not collinear, but Q or R at a finite distance. 

 V. P, Q, R unrestricted. 



The special forms should be noted, when P is at an infinite distance. 



5. The process adopted will be exemplified in family I., thus represented by the 

 Boothian equation. 



P is the origin ; PQ = a, the distance of the double focus of the satellite conic. 

 By partial difierentiation, 



o = $a- «!)' + 2X|(f + ^^) - «(1 - a$) (f + r,^) 

 o = ,;(l-«|)^ + 2X,(^ + ,^). 



The factor (»; = o) alone yields bitangential values. 



It r, = 0,2k = ^ (1 - ai) 



2X|' + (1 - a$) (I - 2fl^) = o. 



If it be thought necessai^y, the equation to this unicursal bounding curve will be 

 foimd explicitly to be a quintic 



^-2. _ I8a3 + sexV - ( - - 14a« + 4X V (a'^ - 8X) 



But hereafter the explicit equation to the bounding curves will be rarely deter- 

 mined. 



6. At a singular point on the quintic y = o =— ; there are two cusps, one 



at infinity, when ^ = o, at the extremity of the (X) axis, and another, when 

 3«| = 2, 8X = a\ 27a\ = 2. 



