ME. W. SPOTTISW CODE ON MULTIPLE CONTACT OF SUEFACES. 
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employ the equations (12) and (16) ; and for this purpose it is necessary in the first 
place to develop the formula □ f 2 U ; viz. we have 
□ ? 2 U=(02) 2 0 m - 2 P-2 .01.02. 0”- 2 12+(01) 2 0 w - 2 2 2 
-02 . 0 n -'2 . l 2 +(02 . O’ 1 - 1 ! + 01 . 0’ 1_1 2)12 — 01 . O’ 1 - 1 ! . 2 2 
But having reference to the conditions □ 20 U=0, □ 21 U=0, . . we may put (0 being 
an indeterminate quantity) 
01=^0"- 1 1, O2=0O n-1 2, (23) 
whence, by substituting these values in the developed form of the equation □ 2 2 U=0, 
and dividing throughout by 0, we obtain 
0{( 0»- 1 2) 2 0 re - 2 l 2 -2 . O’ 1 " 1 ! . 0- x 2 . 0’ l - 2 12+(0»- 1 l) 2 0”- 2 2 2 } 
=(0 m - 1 2) 2 l 2 — 2 . O’ 1 - 1 ! . O-^ . 12+(0 n " 1 l) 2 2 a } 
But since P 15 P 2 are supposed to lie on the quadric V, we have 1 2 =0, 2 2 =0 ; so that 
the equation last above written is reduced to the following, viz. 
2 . O— 1 ! . O*-^ . 12 = -4{(0 b - 1 2) 2 0’ ! - 2 1 2 - ..}=0 
or, as this may be written for greater symmetry, 
0 B+1 1, O’ 1 " 1 2 
O’ 1 - 1 , o >l - 2 l 2 , 0 n - 2 12 
O*-^, 0 n - 2 21, O’ 1 - 2 2 2 
O’ 1 - 2 0 2 , 0"- 2 01 , 
0”- 2 10 , O’ 1 - 2 l 2 . 
0”- 2 02 
0 »- 2 i2 
0”- 2 20 , 0”- 2 21 , 0”- 2 2 2 
=Q [O', 1, 2] suppose; 
and one expression for three-branched contact at the point P will be 
2.O’ l - 1 l.O ra - , 2.12=0[O', 1, 2]. 
A similar transformation may be applied to □ 2 3 U, . . ; and also to □ 12 □ 13 U, . . ; but 
the latter forms, which are rather more complicated, are not necessary for the present 
purpose. This being so, if the surfaces touch at the four points P, P,, P 2 , P 3 , we may 
take for the conditions of osculation -at the point P the three equations □ 23 U = 0, 
□ 21 =0, □ j 2 U = 0, which being transformed in the manner above explained give the 
following results, viz. : — 
2 . 0”- 1 2 . 0”- 1 3 . 23=0[O', 2, 3] ^ 
2 . O” -1 3 . 0”- 1 1 . 31=0[O', 3, 3] l; ..... . (25) 
2 . (P -1 1 . (P -1 2 . 12=0 [O', 1, 2] J 
whence, eliminating the indeterminate quantity 0, we obtain 
23 : 31 : 12 = 0*- 1 1[0', 2, 3] : 3, 1] : 0"- 1 3[0', 1, 2]. 
