SECTION X. 
ON THE CURVATURE OF SURFACES. 
211. To find the requisite conditions for a contact of 
the first, second, &c, order, between two surfaces. 
If two surfaces, referred to the same origin and axes, 
pass through the same point, the co-ordinates of which are 
¿v, y, z; and if we change oo into oc + h, and y into y + /r, 
the equation to the first surface will give for the value of 
the new ordinate, 
z + ph + qlc + 1 (rhr + 2 shk + tk?) + &c. 
and the equation to the second surface 
z + Ph + Qk + 1 (Rh 2 + 2Shk + Tit) + &c.; 
the distance of the surfaces, measured in the direction of their 
ordinates, will therefore be expressed by 
(P-p)h + (Q-q)Ic + ^{(R-r)h 2 +2{S-s)hk + (T-t)k 2 }+&iC. 
If we suppose the equation to the second surface to 
contain a certain number of arbitrary constants, we may 
determine them so as to make the first terms of this dis 
tance vanish; and it will follow that any other surface, for 
which these terms do not disappear, cannot be situated 
between the two former with reference to the points which 
are contiguous to their common point; at least so long as 
we take h and 1c so small, that the sum of the terms of 
the first order may be more considerable than that of all 
the terms of succeeding orders. When we have P — p = 0, 
Q - q = o, the surfaces will have a contact of the first order ; 
if besides these, we have R — r = 0, S - s = 0, T — i = 0, 
the contact will be of the second order, and so on.