274 
ME. W. SPOTTISW OODE ON THE CONTACT OF SURFACES. 
degree, the equations above written (52) cannot be satisfied except on the conditions 
&*V=0, S 2 V=0, . . ^ 3 V=0, . ., which are in fact the conditions for superficial contact. 
There is another more general way in which the subject may be regarded. In fact if 
a, 0, y (or, if we prefer so to state it, if A, B, C) no longer have the significations 
originally given to them, but represent the differential coefficients of an auxiliary 
surface W ; say, if 
a,W=u, d,W=v, a,W=w, d t W=k, (54) 
then the equations 
V=0, (uS 1 +vi 3 +wi 8 )Y=0, (u^+v^+w^) 2 Y=0, (55) 
will no longer express the conditions for two-, three-, . . pointic contact of the curves of 
section made by the plane (a, (3, y) or the plane (A, B, C), hut the contact of the 
curves of section made by the surface W. And as the surface W is perfectly arbitrary, 
the formulae will apply to any curve drawn at pleasure from the point P on the surface U. 
It is to be borne in mind that in expanding the expression for three-pointic contact we 
shall obtain 
(u&j+ . .) 2 =(uS,+ . .)u&jV + («&,+ . .)vd 2 V+ . . 
+ (u-iJ+Y 2 il+ . . 2 uv&j& 2 + . .)\ ; 
but in the only case which possesses much interest, viz. when the two-pointic contact at 
the point P is superficial, we have S,Y = 0, ^ 2 V=0, ^ 3 Y=0 ; and consequently 
(u&i+ • •) A =(u 2 ^-f-v“^ 2 -)- . . 2uv£ 1 & 2 -f- . .)Y, (56) 
which is of the same form as the expression derived in the case of plane sections. And 
as the operators c$,, e> 2 , . . are unchanged, and are subject to the same identical relations 
as before, the conditions of contact now considered will be susceptible of the same 
transformations (the transformations (13-16) excepted) as those considered before. 
From these, therefore, we may draw the following conclusion : — 
Consider two surfaces, U, V, having superficial two-, three-, . . pointic contact at a 
point P ; from P draw any number of curves arbitrarily on U ; two, three, . . consecutive 
points of these curves will, in consequence of the superficial contact, lie also on V. This 
being so, if for any three, four, . . of the curves an additional consecutive point lies on V, 
then the same will be the case for all the curves, and there will be superficial three-, 
four-, . . pointic contact between U and V at the point P. 
This may be also stated in the following form : — 
If two surfaces, U, Y, have two-, three-, . . pointic superficial contact at a point P, and 
if through P we draw any number of surfaces arbitrarily, the curves of section on U 
and V which correspond to one another will, in consequence of the superficial contact, 
have two-, three-, . . pointic contact. This being the case, if any three-, four-, . . corre- 
sponding curves have three-, four-, . . pointic contact, then all will have three-, four-, . . 
pointic contact ; and there will be three-, four-, . . pointic superficial contact between U 
and V at the point P. 
