440 
ME. A. CAYLEY’S IMEMOIE ON CUEYES OE THE THIED OEDEE. 
Take (^, y, z) as the coordinates of the satellite point, then we have 
x:y: 
where the parameters I are connected by the equation 
We have 
and it is easy to see that the function on the right-hand side must divide by 7f — 'C- 
hence x^-\-ff-{-z^ will also divide by 7f — and consequently by D(r — >/^)- 
We have 
and 
■ -3f(S»+»-) 
.+? 
+6?5¥r{-J‘-2V-!)'’+3f(i?+j)»)-3f} 
+i2Pirii { -,r¥+r)+3r>i¥-?*} 
+ 8 ^’ {_,r+ 3 i)*£*S‘-(>i’+r)?} 
¥.h(s”-p)=(;)'-2««+2:‘)(>iT+6;p>iT+i2;‘r>i2:+8?*?*). 
Adding these values and completing the reduction, we find 
aP+f+z'^ (i,»-r)(r-f)(f -!I*)= -P->;'-2:»+2,T+21i'r +25V 
+ 181fiiT 
+i2i»(f+,»+r)5,i: 
+8/”(«’+2*?*+Pii’); 
and we have also 
a!lz^(rf-V)(V-'e)(?-rf) = P,T 
+2i(p+>,=+e')f,t 
+ 4P(>,T+C’P+f5’) 
+8it¥r, 
and thence 
{A(a'»+y+c')+Rrys}-(,>-i;»)(r-P)(P-P) 
= _A(p+,>+i:*)' 
+ (121>A+ 41BXP+s*+r)592 
+(l8?A+(l+81»)B)|Vlr 
+ ((41>+81»)A+4PB)(,¥+i:’P+IV). 
