50 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



which puts in evidence the tangent >; — fi'z. It is easy to see that the equation 

 may be written in any one of the four forms 



a//(£ - 72) + \A»(? -fr) + J- |-f, (m - /) (i» - a'z) 



P 



Vnrl-az) + </l(%- jfe) + J- J^C™ " (« - ffO = 







*/*(, -«' 2 )+ A/m(»- S': y + N /" tr 5 " (» l - (5 * ■*) = ° - 



|8-a 



^»(,-«fr)+ a//(„-/3V) + J-j^ (m -0(1- /3:) = " • 



viz., in forms containing any three of the four radicals s/£ — az, s/£ — 

 s/rj — az, Jn—fiz. The conic is thus expressed as a trizomal curve, the 

 zomals being each a line, viz., they are any three out of the four focal tangents; 

 the order of the curve, as deduced from the general expression 2 V_ V, is = 2 ; so 

 that there is here no depression of order. 



115. But the ordinary form of the focal equation is a more interesting one: 

 viz., A, B being as usual the squared distances of the current point from the two 

 given foci respectively, say 



A = (| — a:) i) — a'z) , 



B =( £_3:)„_/3'z), 



then 2a being an arbitrary parameter, the equation is 



2az + a/A + a/B = , 



viz., the equation is here that of a trizomal curve, the zomals being curves of the 

 second order, that is, the zomals are (r = 0) the line infinity twice, and the line-pairs 

 AI, AJaudBI, BJ respectively : the general expression 2" ~~r gives therefore the 

 order = 4 ; but in the present case there are two branches, viz., the branches 



2az + a/A - a/B = , 2az - a/A + a/B = . 



each ideally containing {z = 0) the line infinity ; the curve contains therefore 

 (z 2 = 0) the line infinity twice, and omitting this factor the order is = 2, as it 

 should be. 



116. To express the equation by means of the other two foci A» B u writing 

 the equation under the form 



A + B + 2a/AB - 4a 2 2 2 = , 



and then if A 1? B x are the squared distances of the current point from A x , B x 

 respectively, we have {ante, No. 65). 



AB = \B lt 

 A + B - A a - Bj = kz> , 



