280 
ME. W. SPOTTISWOODE ON THE CONTACT OE STJEEACES. 
equations in all. These, however, are equivalent to only nine, as may be thus shown. 
Taking the first, and equating to zero the coefficients of the several powers of ■&' : w, and 
eliminating a, h, g, 1, w 7 e obtain the condition. 
%-Y.j j j 
ip 
xx ln , 
in yP 1 
2^^X.h2 
xV 
tl/X 2j 2 , 
wT u2 yV 
«X 1M 
— rP 
' A/J ~ 122? 
uY m -yP, 
'Z/'Sl 222 
■ aP 222 , 
vY 222 -yV, 
^^Z m 
12 
MZ122 
wZ 222 
-zP 
-zV 
-zP 
-zV 
111 ? 
112 ? 
122 ? 
222 ? 
«*T in — £P m , 
wT 112 — tfP I12 , 
? ,T — tV 
LVJL 122 CJL 1225 
WT 222 tV 222, 
(62) 
and it is not difficult to see that omitting the factor u 3 , and employing the relations 
«qX + w , Y-f-'y'Z + ^ / T=P &c., this expression may be reduced to the following form: — 
u , 
x ? 
y ? 
* 5 
£ ? 
= W 
X 
xx in? 
Y 
X 1115 
Zj ii? 
T 
X 1115 
P 
X 1115 
Xin, 
Y 
X 1115 
Z„1, 
T,n, 
sr 
XX 1I2? 
Y 
X 1125 
z m , 
T 
X 1125 
P 
X 1125 
X a2 , 
Y 
X 1125 
Z| 12? 
T 
x 1125 
X 
xx 122? 
Y 
x 1225 
Z122, 
T 
X 1225 
P 
X 1225 
XT 
^^1225 
Y 
X 1225 
Zj 22 , 
T 
X 1225 
X-222? 
Y 
x 2225 
Z 22 2? 
T 
X 2225 
P 
X 2225 
x 222 , 
Y 
X 2225 
Z 222 , 
T 
x 2225 
which vanishes identically, since uX 
in -\~v\ m + wZjn+^Tm — 
0 &c. 
The four equations derived from each of ( 61 ) are consequently reduced to three, and 
the whole number of equations connecting the ten coefficients of U to nine, the proper 
number for the determination of their nine ratios. 
This being the case, we may take as the equations for determining the ratios a : h : g : 1 
the following, viz. 
(wX m — a’P m )a+ («Y m — yP m )h + (wZ m — zP m )g+ («T m — *P m )l= 0 , 
(uX U 2 —xV n 2 )a+(uY u 2 -yV u 2 )h-{-(uZ n 2 — zP U 2 )g+(uT n 2 — tV n 2 )l= 0 , 
(uX U 2 — #P J22 )a + (wY 122 — _yP 122 )h + (wZ 122 — zP 122 )g + (wT 122 — £P 122 ) 1 = 0 ; 
and the quantity to which 1 is proportional will then be 
wX IU aP m , uY ni —yV in , uZ lu zP uu 
«X 112 a Pj i 2 , tl’Sl. ,12 y\ ]12 , uA ll2 zP 112 , 
WX 122 tlP 12 2, IlY. J 22 ?jV 122 , ul l22 sP 122 5 
which, omitting the factor w 2 , is equal to 
u , x , y 5 z ? 
= 
— (n— 2 )n + l't ,x , y , z , 
= — (n— 2 )n 
X Y 7 
- /x in? x in? x -'iii? 
P X X 7 
x 111? xx lll? x 111? ^111? 
^T m , X m , Y m , Z m , 
X v 7 
xx 112? x 1125 ^112? 
V X Y Z 
-*-112? xx 112? -"-112? ^1125 
7 'T Y V 7 
^ X 1 125 Xx 1125 x 1125 ^l 125 
X Y 7 
xx 122? x 122? xi 122? 
P X Y 7 
x 122? - tV 122? x 122? ZJ 122? 
^>22, X 122 , Y 12# , Z 1225 
Calling the coefficient of — ( n — 2 )n in the last expression (X, Y, Z), and forming similar 
