260 
MR, W. SPOTTISWOODE ON THE CONTACT OP SURFACES. 
cases possessing most interest and importance, be replaced by the more general method 
of curved sections. 
In the concluding section a few general considerations are given relating to the 
determination of surfaces having superficial contact of various degrees with given surfaces ; 
and at the same time it is indicated how very much the general theory is affected 
by the particular circumstances of each case. The question of a quadric having four- 
pointic superficial contact with a given surface is considered more in detail ; and it is 
shown how in general such a quadric degenerates into the tangent plane taken twice. 
To this there is apparently an exceptional case, the condition for which is given and 
reduced to a comparatively simple form ; but I must admit to having so left it, in the 
hope of giving a fuller discussion of it on a future occasion. 
The subject of three-pointic superficial contact was considered by Dupin, ‘Developpe- 
ments de Geometric,’ p. 12 ; and, as I have learnt since the memoir was written, 
a general theorem connecting superficial contact and contact along various branches of 
the curve of intersection of two surfaces (substantially the same as that given in the 
text) was enunciated by M. Moutakd*. 
§ 1. Preliminary Formula ? and Transformations. 
Let U = 0, V = 0 be the equations of the two surfaces whose contact is the subject of 
investigation. Let their degrees be m and n respectively ; and let, as usual, 
<LU— u, <^U=y, cLU=w, cbU=#, 
’dl'U=u 1 , B|U=w 1 , B“U=w 1 , b; 2 U=^ n 
a,a.u=«', 3Au=t/ , 
d^u=m', 
Also let u, v, . . zq, . . u 1 , .. I 1 , . . represent the corresponding differential coefficients 
of V. 
Further, a , (5, 7 , c), cd, fi', 7 ', d being arbitrary quantities, let 
ax +/% + 7 " =•& , 
a'x -f- fl'y -j - fz -}~ 0 1 — 
a'zsr — avj'— A, 
— ^L/=B, 
y' ts — yur'~ C, 
h'ztx — =D. 
Also, forming the determinants in the usual order, let 
a, b, c,f, g, h— 
« , P, 7> & > 
a' ? ft', 
* Poncelet, ‘ Applications d’ Analyse a la Geometric,’ 1864, tom. ii. p. 363. 
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