266 
MR. W. SPOTTIS W OODE ON THE CONTACT OF SURFACES. 
cients of V, the second the first differential coefficients of V. This being so, the first 
part of the coefficient of c 2 , viz. (^V^— ^y^)‘ 2 V, will be 
= {wu — uwf{ v 2 'Wi — 2 vwu' + w 2 v l ) 
—2 {wu — uw)(vw — wv){wuu' -\-vwv' — whv' — uvw x ) 
+ {vw — wv) 2 {whc 1 — 2 wuv' + u 2 w , ) 
—w 2 {{vw —wvfu l T {wu — uwfv x -{- (uv — vufw 1 
+2 (wu—uw)(uv—vu )u’ 
+ 2 {uv — vie ){vw — wv)v' 
+ 2 {vw —wv){wu—uw)w'} 
w', 
v', 
u , 
u, 
(m — l) 2 
M 1 } 
«/, 
M, 
w', 
H , 
m', 
0, 
V, 
w'. 
»1 , 
u ' , 
m', 
v 1 , 
u\ 
IV 1, 
w, 
w, 
v'. 
m', 
w>, 
u ' , 
W, 
u , 
V , 
w , 
• 
• 5 
1', 
m', 
u ' , 
^ 1 5 
u , 
V , 
IV, 
• 
• 5 
u , 
, 
w , 
^ , 
• 
and the second part of the same expression 
tv-P 
(m — l ) 2 
O suppose 
= {tVU — UW) 2 { — WWV t -{-{vW-^-Wv )^ — VWWt } 
— (wu-ww){vw—wv) { — {wu-\-uw)u! — {vw + wv)v' + 2 www' + {uv+vu)w x } 
-\-{vW — Wv) 2 { — UUWi -f- {WU -f* UW)v' — WWUi } 
= -ot{ {vw — wvju l + {wu — uwfv l -f- {uv — vu) 2 w 1 
+2 {wu — uw){uv— vu)u' 
+ 2 {uv — vu){vw—wv)v' 
+ 2 {vw — wv){wu — uw)w ' } 
wwt 9 - 
M„ 
w', 
v' , 
, 
u 
*> 1 , 
v ! , 
m'. 
V 
w' , 
IV 1, 
u' , 
w 
u ' , 
h x , 
k 
u , 
v , 
w , 
* , 
. 
ivwt 12 
Q suppose. 
Hence the whole expression (29) 
= {au-\- bv + cw)f^(au -\-bv-\- civ) {au 4 -bv+ civ) • 
Hence the condition for a three-pointic contact in some plane about a given axis will be 
O _ _ _ Q 
(au + bv+cw) —{au+bv-\- cw ) = 0 . 
( 31 ) 
