34 Frederick C Ferry. 



a pair of branches, of which pair a branch lies in either sheet; 

 such double points are actual but since the two branches lie 

 in diiferent sheets we will include them under h, letting 

 li = h^ -j- h^ , where h^ refers to such apparent-actual double 

 points and h^ to the properly apparent double points ; h^ < h. 

 It is evident that when S has f ' = and itself intersects 

 2 at either pinch point, there will in general be a cusp on 

 the curve of intersection; but since this kind of cusp has 

 one branch in either sheet and does not arise from stationary 

 contact of S and S of the ordinary kind, we will not include 

 this singularity in ß but will include it in h^ ; for it results 

 whenever one of the points of h., lies at either pinch-point. 

 Should we wish to distinguish this cusp from the other 

 points in hg we will designate its number by h^ , where 



With the aid of the ordinary Plückerian equations we 

 now obtain at once connecting our singularities, — 



h=i[m(m— 1)— r], x = i [r(r — 1) — 3m — n] , 



n =: 3 (r — m) -f- ß , y = x — (n — m) , 



a = 2(n — m) + ß, g = i [n(n— 1) — r — 3a] ; 



whence as a complete set in terms of p , q , H, and ß, we get 



1) m = p + q 



2) h = ^p(p-l) + q(q-l) 



3) n=3(2pq — q^ — p) — 6H— 8ß 



4) a=2q(6p — 3q — 1)— 8p — 12H — 15ß 



5) r = q(2p — q + 1) — 2H— 3ß 



6) x=:^[4pq(pq — q'-f q— 2) + q(q' — 2q' + 5q — 4) — 



(2H + 3ß) (4pq — 2q- + 2q - 1) + (2H + 3ß)' + 



2(3H+'4ß)J 



