IX. On Multiple Contact of Surfaces. 
By William Spottiswoode, M.A., Treas. R.S. 
Eeceivcd May 24, — Eead June 17, 1875. 
In a paper “ On the Contact of Quadrics with other Surfaces,” published in the Pro- 
ceedings of the London Mathematical Society (May 14, 1874, p. 70), I have shown that 
it is not in general possible to draw a quadric surface V so as to touch a given surface 
U in more than two points, but that a condition must be fulfilled for every additional 
point. The equations expressing these conditions, being interpreted in one way, show 
that two points being taken arbitrarily the third point of contact, if such there be, 
must lie on a curve the equation whereof is there given. The same formulae, interpreted 
in another way, serve to determine the conditions which the coefficients of the surface 
U must fulfil in order that the contact may be possible for three or more points taken 
arbitrarily upon it ; and, in particular, the degrees of these conditions give the number 
of surfaces of different kinds which satisfy the problem. 
In another paper, “Sur les Surfaces osculatrices ” (Comptes Rendus, 6 Juillet, 1874, 
p. 24), the corresponding conditions for the osculation of a quadric with a given surface 
are discussed. 
In the present paper I have regarded the question in a more general way ; and having 
shown how the formulae for higher degrees of contact are obtained, I have developed 
more in detail some special cases of interest. 
For the convenience of the reader, I have in § 1 briefly recapitulated the principal 
parts of the two papers above quoted. In § 2 I have given, at all events, a first sketch 
of a general theory of multiple contact with quadrics ; in § 3 the particular cases of 
three-, four-, five-, and six-pointic contact are discussed ; and in § 4 some conditions for 
the existence of points of four-, five-, six-branched single ( i . e. not multiple) contact are 
established. 
Thus far the investigation concerns the contact of quadrics only with other surfaces. 
The concluding part of the paper is concerned with the corresponding problem for cubics, 
in which case conditions of possibility do not arise for either simple or two-branched 
contact, but are first met with for three-branched contact. The conditions in question, 
with some of their consequences, are here given ; but it is perhaps hardly worth while 
to prosecute the subject much further in this direction. 
It will be observed in the course of the paper that some of the numerical results 
must be taken as subject to limitations to be expected from further research ; but the 
intricate nature of the investigation will, I hope, be considered as affording some justifi- 
cation for submitting it thus rough-hewn to the notice of the Society. 
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