MR. W. SPOTTIS W OODE ON THE CONTACT OE SURFACES. 265 
If the surfaces touch at the point P, the equation (w'n — is satisfied iden- 
tically, and there will in general be two directions, determined by the equation 
(^□-^□') 2 Y=0, (27) 
along which there will be three-pointic contact, as will be further noticed in the 
next section. The condition for a four-pointic contact will then be obtained by elimi- 
nating •cj' : ra- from the equations (27) and (26). These considerations may readily be 
extended to higher degrees. 
It will perhaps be worth while before proceeding further to evaluate the expression 
(24) ; viz. the condition to be fulfilled in order that when two surfaces meet at a point 
P but do not touch, the curves of section made by some plane passing through a given 
axis through P shall have three-pointic contact. With a view to this, let us take the 
following form for □ , viz. 
□ — esc), -j- /3c).,-}- yc> :1 , □ ' = -f- [3'h . 2 -p yA (-"8) 
Then the expression to be developed will be 
(AV+. .) 2 (^+/3 2 ^+. .+ 2/3 y (&A+U)+. .)V 
-WV + . .)(Av+..)(2«^+2pi+. W +/3'y)(*A+W+. -)V 
+(«*iV +..)W+^+- .+2/3Vm+^A) + . .)V. 
In this it will be found that the coefficients of a' 2 a 2 , /3 ,2 /3 2 , . . vanish, and that the coeffi- 
cients of a 2 / 3' 2 , — 2aa'/3/3', a' 2 / 3 2 are all the same, viz. 
(^V)AY - 1 ; V& 3 V($ A + )V + (^V) 2 o;V. 
Hence the whole expression may be reduced to the form 
.+2^P-W)P-^,) + . .}Y, . . (29) 
it being understood that the operations cb, c> 3 do not affect the quantities o,V, cbV, § 3 V 
so far as they appear explicitly in the above expression. In order to calculate the 
coefficients of the powers and products of a, b, c, we have 
a.v 
— VW 
—wv , 
aY =wu 
—uw , 
o 3 V =uv 
— vu , 
C$,M 
=vv' 
—ww 1 . 
h,u =wu 1 
—uv' , 
b 3 U =uw' 
— VUi , 
dp? 
=vu' 
—wv x , 
'6 2 v =ww' 
—uu ' , 
0 3 V — Ul\ 
—vw\ 
\w 
=vw t 
— WU ' , 
b 2 W =ivv' 
— MW,, 
b 3 iu — Uul 
—vv ' , 
— V'iVy — 2vWU’-{-W 2 V 1 -\-w{vit! —WVj)— v(vw l — WU 1 ), J 
= UlVu’ -\-Vlw' — W 2 w' — 2lVW l J r u(vW 1 — wu') — w(vv ' —ww 1 ), I 
_____ _ j 
— UWU 1 - {-vwv 1 — w~w ' — uvw l -{-w{ww' — uu’ ) — v(wv' —UWi), j 
c$ 2 V=w 2 m,— 2ivuv'-{-u 2 w 1 -f u(wv' —mv l )—w(wu 1 —uv' ), ) 
each of which consists of two parts ; viz. the first involves the second differential coeffi- 
