262 
ME. W. SPOTTISWOODE ON THE CONTACT OE SURFACES. 
( 8 ) 
(9) 
then it will be found that 
□ V=(«S 4 +^5+cS 6 +/'^i +^2+^3)^, 
cc □ V= (Bc$ 6 — C^ 5 -j-DS,) V = (ax' □ , — wu 1) V, 
y □ Y = (C2> 4 - A^g + mjv = (w> □ 2 - O- □ ’ 2 ) V, t 
z □ Y = ( AS 5 - BS 4 + DS 3 ) V = (® ' □ 3 - ®- □ ' 3 ) V, | 
mY=(A^ + B^ + Ca 3 )Y=(^a 4 - w D')V, j 
where 
□1=^6 — 7&5 + S&1, □1=^6 — 7 ^ 5 + 
□ 2 =y& 4 — aS 6 + ^ 2 , □2=7^4 — of&g-f- c)A>, 
□ 3 =ai s — j35 4 + ii 3 , □ 3 =a , S 5 — 0'& 4 + %%, 
□ 4 =a&i+0M-y& 3 > D^A+^+A' J 
Lastly, the operators & 2 , . . are subject to the following identical relations, viz. :- 
V^g W^ 5 ~j“^l -—65 
w\ — ul 6 =0, 
116 5 — v}> 4 -f- /i'o 3 =0, 
u\ +^ 2 + w& 3 =0, 
by means of which we may always eliminate one of the three operators entering into 
each of the expressions (8). In fact, the following values would express the result of 
such an elimination : — 
k{Bd 6 -C^ + m i )=(B/c-Bv)\-(C/c-'Dw%,^ 
k(C\-A\+m 2 )={Ck-Dw)\-(Ak-I)u)\, (11) 
Jc( A £ 5 - B \ + DS 3 ) = (AJc - Dm % - (B Jc - D v % ; j 
so that in the case where □ V = 0, we should obtain 
( 10 ) 
^V:^V:S 3 V:i 4 V:S 3 V:J 8 V= 
(12) 
A, B, C, D, 
U, V , w, Jc. 
There is one other mode of transformation which, on account of its utility, may properly 
find a place here. If U 0 , U 1? be the same linear functions of «, 0, y, &, and a!, 0', y', l', 
respectively, say 
U 0 = («, (3, y, S),| 
U, = K f 3 ', y', V), j 
(13) 
then it will be found that 
(U 0 , U,)K -«) = ( . 
c, —5, 
fh 
(U 0 , U,)(0', -0)=(-c, 
a, 
(U 0 , U,)(y', -y)=( b. 
- a , 
70, 
(u 0 ,u,)(y, -i)=( /, 
)• 
( 14 ) 
