270 
ME. W. SPOTTISWOODE ON THE CONTACT OF SUEFACES. 
triple point at P, and along each of the branches the contact will be four-pointic. This 
may be extended generally; but there will be occasion to return to the question hereafter*. 
It having been proved that if jj-pointic contact subsist for_p plane sections about an 
axis it will subsist for all plane sections about that axis, the question naturally suggests 
itself whether there be not a corresponding theory as to the number of axes about which 
there must be circumaxal contact in order that it may subsist for all axes. In uniaxal 
contact it is supposed that from the point P at which the surfaces meet a series of curves 
are drawn (on both surfaces) lying in planes passing through the axis, and that contact 
of the degree under consideration subsists between every curve on U and the corre- 
sponding curve on Y. If the circumaxal contact be multiaxal, we are supposed to take 
other axes through P, and draw other series of curves in planes passing through these 
axes respectively ; and the question is, whether it be necessary that the contact shall 
subsist for a definite number of these series of curves, in order that it may subsist for all 
such series. In the latter case we shall call the contact superficial ; commencing with 
two-pointic contact, and taking the form AS, -j- B£ 2 -{- C§ 3 for □, we obtain, on equating 
to zero the coefficients of ■&', vs respectively, 
«S,V -b/3S,V -f yS 3 V =0,1 
L (42) 
AV+/« 3 V+"AV=0;J V ' 
and applying to these the transformation (14), we deduce the following forms, 
(bl 3 -cZ 2 )V= 0, (cS,-aS 2 )V= 0, («S i -iS,)V= 0, 
or 
lF:l.y-.\V=a:b:c, (43) 
If this contact holds good for a second axis (say «„ Z>,, c ,), we shall have also 
&iV: S*V : lfi r =a l ; b , : c x (44) 
But since the two axes by hypothesis do not coincide, (43) and (44) cannot both be 
satisfied except on the conditions 
&,V=0, S 2 V=0, S :i V=0 (45) 
These conditions are in reality only two in number, in consequence of the identical rela- 
tion + «;S 3 =0. This shows that if two-pointic contact be biaxal it will be super- 
ficial. But inasmuch as the directions of these axes are arbitrary, we may take for one 
of them the tangent to the curve of intersection of U and Y through P ; hence, setting 
this axis aside, and reckoning only arbitrary axes, we may state that if two-pointic contact 
be uniaxal it will be superficial. This, of course, is merely the ordinary property of 
common contact. 
At the risk of being tedious on so simple a question, I venture to point out that a 
result substantially the same may be deduced by a geometrical consideration from the 
equation (A<$, -f-Bd 2 + Cd 3 )V=0, without the intervention of the suppositions (45). Take 
* Since writing the above I have found that a similar theorem was enunciated by M. Moutaed. Poxcelet, 
^Application d’ Analyse a, la Geometric,’ 1864, tom. ii. p. 363. 
