276 
ME. W. SPOTTISWOODE ON THE CONTACT OF SUEFACES. 
Now it is obvious from the signs of the terms on the left-hand side of this equation that 
the result cannot be positive ifjp be less than 2m; beginning therefore with p= 2m, we 
obtain 
— m 3 +6m 2 — ■ -5m=0, 6, 12; 
or, resolving into factors, we have the three cases 
m(m— l)(m— 5) = 0, 
(m— 2)(m 2 — 4m+3)=0, 
(m— 3) (m — 4) (m + 1 ) = 0. 
Next, let ])—2m-\-l ; then, substituting this value in the equation (57), we obtain 
— m 3 -l-6m 2 +7m+6=0, 6, 12; 
or resolving into factors so far as possible, we have the three cases 
to 3 — 6m 2 —7m— 6=0, 
(m— 7)(m-f-l)=0, 
m 3 — 6m 2 —7 m 4-6 = 0, 
the first and last of which give no solutions in positive whole numbers. 
This appears to exhaust all the solutions of (57) in positive whole numbers, lleca 
pitulating the foregoing results, we may form the following Table : — 
Degree of 
contact. 
Number of 
conditions. 
Degree 
of Y. 
Number of 
constants. 
Difference 
conds.-consts. 
Superficial contact 
possible. 
2 
O 
O 
1 
3 
0 
At every point on U. 
4 
10 
2 
9 
1 
Along a curve on U. 
6 
21 
o 
O 
19 
2 
1 At a finite number of 
8 
36 
4 
34 
2 
j points on U. 
10 
55 
5 
55 
0 
At every point on U. 
15 
120 
7 
119 
1 
Along a curve on U. 
Such is the general theory. But it is probable that it undergoes modifications in 
each particular case ; it certainly does so in the only case fully examined here, viz. that 
of a quadric having four-pointic contact with U. 
In fact, inasmuch as the equations, whereby the constants are ultimately determined, 
are linear the solution is in every case unique. But four-pointic contact will subsist if we 
consider the quadric to consist of the tangent plane taken twice ; and as the solution is 
unique, no other quadric can in general be drawn having four-pointic contact. Further, 
since a tangent plane can in general be drawn at every point on U, the quadric of four- 
pointic contact (viz. the tangent plane taken twice) exists generally ; and the condition 
(viz. difference, conditions — constants = 1) restricting it to a curve on U must be satisfied 
