ME. W. SPOTTISWOODE ON MULTIPLE CONTACT OF SUEFACES. 
253 
For osculation at the points P 1; P 2 , P 3 , 
6,:6 2 : 0 3 = BB'+2"-'l . 
: CO + 3"- '2 . l^P^ 
: DD' + l 71 -^ . 2 n-1 3P 2 23 , 
K'3"-'O0 2 — 2 n -'3 . 3"- 1 2P 023 3 = 0, 
M'l n_1 04— O , 3 n-1 0^ 1 — 3 n-1 l . p-'3P ol3 0=O, 
■R^-'O^ — Q 1“ -1 O0 2 — l n_1 2 . 2"- l lP ol2 0:=O. 
Similarly for osculation at the points P, P 2 , P 3 , 
0: 0 2 : 0 3 =FF'+2 n ~ 1 O . 2r-'OP* m 
: . 0 m-1 2P^3 
: KK'+0 m-1 3 . 2“ -1 3 Po 23 , 
D'3 n_1 l^ 2 — C 2 ,l-1 19 3 — 2 n_1 3 . S-^P^^O, j 
L'0 n_1 l^ 3 — O 3 n-1 10 -3"- 1 !) . 0 n - 1 3P OI3 tf 1 =0, 
R2 n-1 10 -P'O^-'l^-O"-^ . 2 71-1 OP ol2 0,=O. 
Similarly for osculation at the points P, P n P 3 , 
0:0,: ^LtZ+l—O . S^’OP^ 
: MM'+3" -1 l . 0™->lPL 
: OO'+0 n_1 3 . l-^P^ 
D3 n_1 20,— B' l n-1 20 3 — 1 B-1 3 . 3” -1 lP ]23 0 2 =O, 
F , O«- 1 20 3 -K3 m - 1 20 -3->0 . 0- 1 3P oa A=0 J 
Q , l*- 1 20 -P O“ _1 20,— 0“ -1 l . l n " 1 0P ol A=0. J 
And for osculation at the points P, P 1? P 2 , 
0:0,: 0 2 =PP'+1”- 1 O . 2 m-1 0Po 12 1 
: QQ'-f 2 n-I l . 0“-lP 0 2 12 
: RR'+O"-^ . 
C'2 n_1 30 1 — B 1” -1 30 2 — l m_1 2 . 2' 1-1 1P 123 0 3 =O 
( 12 ) 
(13) 
(14) 
(15) 
M'l” -1 30 -L O” -1 30,— 0 n_1 l . l” - 1 OP ol 3 03=O. 
Whence, finally, the two conditions for osculation at the points P„ P 2 , P 3 , together 
with simple contact at the four points P, P 1? P 2 , P 3 , will be 
{K'(CC' +3"- , 2 . l 7l-1 2P? 2 ,)3 n-1 0 — H(DD' + 1“ _ *3 . 2"- 1 3P 2 23 )2 n - 1 0} : 2 li -3 . S n ~'2P 023 
= {M(DD' + 1"~ 1 3 . 2 n - 1 3P 2 23 )l TC - 1 0-O , (BB'+2”- 1 l . 3 m - 1 lP 2 23 )3 n - 1 0} : 3"- 1 ! . l-'SPm > (16) 
= {B'(BB'+2"- 1 l . 3”- 1 lPf 23 )2 n - 1 0-Q(CC'+3’ l - 1 2 . l’- 1 2Pf 23 )l ? *- 1 0} : l n ~ l 2 . 2”- 1 lP 012 ,j 
