MR. W. SPOTTISW OODE ON T1TE CONTACT OF SURFACES. 
275 
This theorem for the case of three-pointic contact was given by Dupin, ‘ Developpe 
ments de Geometric,’ p. 12. 
§ 4. On Surfaces having Superficial Contact with given Surfaces. 
It is well known that at any point P of a surface U we may in general determine a 
plane V touching, or, in terms of this memoir, having two-pointic superficial contact 
with U. This suggests the question whether surfaces V of other degrees may not be 
determined having superficial contact of higher degrees with U at a point P. 
The number of conditions for a 1, 2, . .^-pointic superficial contact has been shown 
above to be 
1 , 3 , 6 , 
p(p + i) 
' 1 . 2 ' 
Now the number of independent constants in the equation of a surface V of the degree 
1,2,.. m, is 
o q 1 q (?» + 1 ) (m + 2) (m + 3) - _ 
i i) > 
so that, employing the equations which express the conditions of contact for deter- 
mining the constants of V, we shall meet with the following cases. First, if the number 
of conditions be equal to the number of constants, there will be a determinate surface 
V having a superficial contact of the degree under consideration (say p) with U at the 
point P. Secondly, if the number of conditions exceed that of the constants by unity, 
the constants may be eliminated, and the result will be an equation between the coor- 
dinates ; in other words, an equation to a surface which will cut U in a curve at every 
point of which a surface Y may be drawn having ^-poin tic superficial contact with U. 
Thirdly, if the number of conditions exceed that of the constants by 2, we may eliminate 
the constants in two ways, and obtain two resulting equations, which will represent two 
surfaces mutually cutting U in a finite number of points, at each of which a surface V 
may be drawn having ^j-poin tic superficial contact with U. Lastly, if the number of 
conditions exceed that of the constants by more than two, we shall obtain a number of 
resulting equations equal to that excess. From these, together with the equations U = 0, 
V=0, the variables maybe eliminated; so that the number of resultants less 2 will 
represent the number of conditions which must subsist among the constants of U, in 
order that it may be possible to draw such a surface V. This being the case, there will 
be a determinate surface V of the degree m having j>pointic superficial contact with U, 
(1) at any point on U, or (2) along a certain curve on U, or (3) only at a finite number 
of points on U, according as the expression 
P(P + 1 ) (m+lKm + 2)(m + 3) , n ^ 0 . 
1.2 1.2.3 + 1 — L 
or, clearing denominators, according as 
3p(p + l)— m 3 — 6m 2 -— llm=0, 6, 12 
2 
MDCCCLXXII. 
P 
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