PROF. W. K. CLIFFORD ON MR. SPOTTISWODE’S CONTACT PROBLEMS. 
713 
The curve w 9 has one branch at A in each of the chief directions and one other 
branch. The curve t 6 has one branch in each of the chief directions and two other 
branches. 
The equations (5), (6), (7) have now become 
»!(»)=«* 
n\( W)=v 7 , 
n,t & =w 9 . 
The first two of these give us the curve already considered, 
(B 2 ).+=(B >).u 7 , 
Avhich has at A two branches in the chief directions and one other. The first and 
third give 
n , . (B 2 ) . w 9 t 6 u 6 , 
•a curve of order 12, having two branches in each of the chief directions and two other 
branches. Of the 108 intersections of these curves, then, 24+8 + 4 + 2 = 38 coincide 
with A, leaving 70 for the number of septactic conics. 
Part II.— THE CONTACT OF QUADRIC SURFACES WITH SURFACES OF ORDER 
I. Conditions of contact. 
Let A, B, C, D be four points forming a tetrahedron, whose tangential equations are 
0 =%aX,%bX, %cX, %dX 
respectively, their coordinates being a ir b t , <+ d t (i= 1, 2, 3, 4). Then the point 
P=A+0B+pC+0pD, 
whose coordinates are 
'p i ^a i + Qb i -\-(pc i -\-6<pd i (i= 1, 2, 3, 4), 
will, if we suppose 0 and <p to take all possible values, trace out a quadric surface whose 
tangential equation is 
0=AD-BfeQ 2 . 
To the pair of values 
0=0 , <p=0 , corresponds the point A, 
*5^ 
II 
8 
-e 
II 
o 
» B, 
3 = 0 , (£) = ao , „ 
„ c, 
II 
8 
-e 
II 
8 
„ D. 
The equation shows that AB, AC, BD, CD are generating lines. 
