992 
PROFESSOR A. CAYLEY ON THE SINGLE 
4 (a 2 — S 3 ) 2 W'X = 
aS [(p+r)(p +r') + (q+s)(q'-\-s')~] — a 2 (p-\-r) (q'+s) — h~{q+s)(p'+r'), 
4 „ X'W= 
>) £ > 5 >5 ~\ ^ : 5 5 5 5 
4(/3 2 —y 2 ) 2 Z'Y = 
/ 3 y[ (j-> - r) ( p -r) + (q - 5 ) (</' - s')] - /3 2 (p - r) (q - s')-y~(q-s) {p -r), 
4 „ Y'Z = 
,,[ „ » ]-y~ 5 , —F 
whence 
4 (a 2 — S 2 ) (' W'X — X'W ) — — [_(p + r) (q' + s') — (p + r) (q + s) ], 
*(?- rKZ'Y -Y' Z)=—[<i>—H(-z'-L-(P-'•')(?-»)]. 
and substituting in the expressions for Q and S 
4(a—S 2 )(/3- —y°)Q = 
- (f3~ -y%(p + r) {q'+s') — (p +r) (q +«)] + («° - &)[p ~ r)(q - s') -(p- r)(q - s)], 
4 „ E= 
55 [ 55 ’ 55 J 55 C 55 >5 ]• 
133. Hence collecting and reducing 
4(«=-8?)08»-/)P = 
— (a 2 — /3-+ y~ — S 2 ) (pp' — qq + r / - 
4 „ R= 
(a 2 +/3 2 -y 2 -S 2 )( 
4 „ Q = 
(a 2 —/3 2 -f-y 2 —S 2 )( 
4 „ S= • 
— (a 2 + /3 2 —y 2 —S 2 )( 
■««') + (a 2 + /3 2 — y 2 — S ~)(pr' +pr — qs — qs), 
)-(a 3 -£ 2 +y 2 -S 2 )( 
) — (ot~+/S 2 —y 2 — S 2 ) ( 
)-{-(a - —/Y + y - —- S-)( 
), 
); 
we have p 0 ( =cp) = a 2 — /S 2 -f y 2 — 8 2 , r 0 (=c 8 2 ) = a 2 +/3 2 —y 2 — S 2 , and thence 
V — jV = 4 (a 2 — S 2 ) (/3 2 - y 2 ); 
the equations hence become 
