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XII. On the Contact of Surf aces. By William Spottiswoode, M.A., Treas. R.S. 
Received January 18, — Read February 22, 1872. 
In a paper published in the Philosophical Transactions (1870, p. 289) I have considered 
the contact, at a point P, of two curves which are coplanar sections of two surfaces 
(U, Y), and have examined somewhat in detail the case where one of the curves, viz. 
the section of V, is a conic. In the method there employed, the condition that the 
point P should he sextactic, involved the azimuth of the plane of section measured 
about an axis passing through P ; and consequently, regarded as an equation in the 
azimuth, it showed that the point would be sextactic for certain definite sections. It 
does not, however, follow, if conics having six-pointic contact with the surface U be 
drawn in the planes so determined, that a single quadric surface can be made to pass 
through them all. In fact it will be shown in the sequel that when this is possible, the 
quadric in general degenerates into a pair of planes. The investigation therefore of the 
memoir above quoted was not directly concerned with the contact of surfaces, although 
it may be regarded as dealing with a problem intermediate to the contact of plane 
curves and that of surfaces. 
In the present investigation I have considered a point P common to the two surfaces 
U and V, an axis drawn arbitrarily through P, and a plane of section passing through 
the axis and capable of revolution about it. Proceeding as in the former memoir, and 
forming the equations for contact of various degrees, and finally rendering them inde- 
pendent of the azimuth, we obtain the conditions for contact for all positions of the 
cutting plane about the axis. Such contact is called circumaxal ; and in particular it 
is called uniaxal, biaxal, &c. according as it subsists for one, two, &c. axes. If it holds 
good for all axes through the point, it is called superficial contact. 
It would at first sight seem that there should be a similar theory as to the number of 
axes about which contact must subsist in order that it may subsist about all axes, or be 
superficial. It is, indeed, found that if two-pointic contact be biaxal, or if three-pointic 
be triaxal, &c., the contact will be superficial. But this would prove too much, as it 
would give four conditions instead of two for two-pointic, six conditions instead of three 
for three-pointic, &c. superficial contact ; and, in fact, it turns out that there are always 
in two-pointic contact one, in three-pointic two, &c. axes (viz. the tangents to the 
branches of the curve of contact through the point) about which the contact is circum- 
axal per se, so that the theory in one sense disappears. But as it at first had a semblance 
of existence, it may still be worth while to have laid its ghost. 
At the conclusion of § 8 it is shown that the method of plane sections may, in the 
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