TRANSACTIONS OF SECTION A. 539 



4. On Involutional (1 1) Correspondence. By Professor Genese, M.A. 



It is here proposed to say that there is an involutional (1, 1) correspondence he- 

 tween two pouits whose co-ordinates (say, areal) are .r : y : s, a : ^ : -y when if 



.f : y : s :: /i (a,0, y) :/, (a, ^,y) :/3 (a,^,y), 

 then 



a : /3 : -y : : /i (,r, y, z) if^ (.r, y, s) l/g (.r, y, s).^ 

 /„ jTj, /g denoting three distinct homogeneous functions. 



■ The following method gives five solutions of the problem suggested. Consider 

 the equations 



F, (a, |3, y) F, (a, ^, y) ' ^ 



Fg (^y^) _ F4 (-^Y^L^s) (2') 



F3(a,/3,y) - F,(a,i3,y) 



It is clear that .r : y : s : : a : /3 : y is one solution. If then Fj, F^, F3, F^ be so 

 chosen that (1) and (2) admit of only one other common solution containing a, ^, y 

 that solution will by symmetry determine an involutional (1, 1) correspondence. 



Let 0", C° denote the loci represented by 



F,{x,y,z)=yF,{.v,y,z) (3) 



Fs {'V, y,z)=^ F, (.r, y, z) . . . . (4) 



of the mth and nth degrees respectively. 



Then, y, ^ being chosen to make C" 0° intersect at a, /3, y, we have the curves 



(1), (2). 



Now C", C° intersect in mn points ; for the solution of our problem p = mw — 2 

 of these must be fixed points. Also C", 0" pass through m* and w^ fixed points 



respectively : 0™ is determined by "^ "1^ ' points, &c. 

 Five solutions are obtained : — 



w = 1, m = 2, p = ] . . . Aj 



n = \, 1)1 — 2), iy = \\ ... Aj 



n = 2, 7n = S, p = 4:} ... A3 



n = 2, w-2, j9 = 2 i . . . Bi 



n = 3, m = 3, p = 7 \ . . . Bj 



Interpretations 



Aj. Fj = 0, F2 = any two conies F3 = 0, F^ = any two straight lines (not 

 intersecting in a point common to F, = 0, F.j = 0) 



Ao. F3 = 0, F^^O any two straight lines Fi=0, F2 = two cubics passing 

 through the intersection of Fj = 0, F^ = 



A3. F3 = 0, F^ = any two conies Fj = 0,"F„ = 0, two cubics passing through 

 the points common to Fg = 0, F^ = 0. 



Bj. Fj = 0, Fj = 0, F3 = 0, F^ = represent conies ; but two of the points of in- 

 tersection of F] = 0, F., = must coincide with two on F3 = 0, F^ = 0. 



Bj. Fi = 0, Fj = 0, F3 = 0, F^ = represent cubics having seven points common. 



Note 1. 

 If, having obtained a correspondence, we put therein .r : y : s : : a : /3 : y, we 

 obtain the locus of points at which the lines (1), (2) touch. 



Note 2. 

 The analysis of Aj suggested the following algebraical exercise: — 

 ^^ £ = IL = I. =. /n+i(a>/ 3, y)+/„(a, ^, y) 

 1 /3 y /n+2(a,/3,y)-/« + l(a>^>y) 



where /„,/„ + !,/„ + „ are any homogeneous fxmctions of w, w + 1, « + 2 dimensions 

 respectively ; then will 



± = ^ = y_ = fn+x (-y> y> g) +./"» (-^'^ y> g) 



X y s fn+i tr, y, z) -/„ + , (.r, y, s) 



