MR. W. SPOTTISWOODE ON THE CONTACT OF SURFACES. 
267 
This may be regarded as a condition to be fulfilled either by «, b, c, the direction 
cosines of the axis, or by x, y , z, t, the coordinates of the point. Taking the first view, 
au J rbv-\-cw = 0 is the condition that the axis shall lie in the tangent plane of U, and 
au J r bv J r cw = 0 the condition that it shall lie in the tangent plane of Y ; hence (31) 
expresses the condition that the axis shall lie in the intersection of these planes. 
On the other hand, regarding a , b, c as given, the equation (31) will represent a 
surface whose intersections with U and V will determine the points of three-pointic 
contact about a given axis. The degree of this surface is 3(mfi -n — 3); and the number 
of points will therefore be 3 mn(m+n — 3). 
Lastly, the equation (31) becomes independent of a, b, c if 
0 = 0, 0=0, (32) 
which will consequently express the conditions that a three-pointic contact may subsist 
in some plane about any axis. The degrees of these equations are 2(n— l)-(-3(m — 2), 
and 2 [in — l) + 3(w — 2) respectively. Points for which such contact will subsist for any 
axis do not in general exist when U and V do not touch; but the condition for their 
existence will be found by eliminating x, y , z, t from the equations U=0, V = 0, 0 = 0, 
0 = 0 . 
§ 3. Modes of Contact. 
Hitherto we have considered only the contact of the curves of section of the surfaces 
U, V made by definite planes passing through an axis. If, however, in the equations 
(18), which express the conditions for the contact of these curves, we equate to zero the 
coefficients of the various powers of the quantity to' ; w, which determines the azimuth, 
we shall obtain anew series of conditions. And the fulfilment of these conditions will 
ensure the subsistence of contact, of the degree under consideration, independently of 
the azimuth of the cutting plane; or, in other words, for all plane sections round the 
point P whose planes of section pass through the axis, such contact may be called 
circumaxal ; and, in particular, contact which holds good in this manner for a single axis 
might be termed uniaxal contact ; that which holds good similarly for two axes might 
be termed biaxcd contact ; and so on for a greater number of axes. But before entering 
into this question, it will be as well to establish a theorem relating to the number of 
sections necessary to ensure uniaxal contact. 
Eeturning to equations (18); the second, viz (w' □ — toD')V = 0, expresses the condi- 
tion for two-pointic contact. Suppose that this holds good for more than one value of 
ns : to, say, ns \ ; to, and to ' 2 : m.,. Then, writing down the equation for each of these values, 
we may eliminate the coefficients and obtain the resultant, 
But as to', : to, and ns'. 2 \ns. 2 are by hypothesis different, the above equation cannot be 
satisfied, and consequently the coefficients of to' and to in the equation under consideration 
must separately vanish. But the evanescence of these coefficients expresses the con- 
mdccclxxii. 2 o 
