520] ON THE CENTRO-SURFACE OF AN ELLIPSOID. 333 
31. Developing and reducing, and omitting this factor, the final result is 
6P + 3Qp + P (p 2 + 2q) + 3pq = 0, 
where as before P, Q, R denote a 2 + b 2 + c 2 , b 2 c 2 + c 2 a 2 + a 2 b 2 , a 2 b 2 c 2 , respectively; that is, 
6P+3Q(|+p) + P(p + 4fp + p 2 ) + 3(| + p)£p = o, 
or, as this may be written, 
QR + 3 Qg + Pp 
+ P (3Q + 4Pf + 3p) 
+ £ 2 (P + 3f ) = o, 
viz. the parameters p p have a symmetrical (2, 2) correspondence. 
32. From the equations {a 2 + f) 3 (a 2 + 77) = (a 2 + fj) 3 (® 2 + i?i)> &c., we have 
2 (& 2 -c 2 ) {(a 2 + f) 3 (a 2 + 17)-(a 2 + p) s (a 2 + 17O} = 0, 
2& 2 c 2 (Z> 2 - c 2 ) {(a 2 + f ) 3 (a 2 +17) — (a 2 + p) 3 (a 2 4- i?^} = 0 ; 
and observing that the term in { } is 
a 6 (3f + v - 3p — 17^ 
+ a 4 (3p 4- 3p7 -3p 2 -3p*) 
+ a 2 ( p + 3pi?- fr-Sgtih) 
+ ( pi? -p 3 i?i), 
these are readily reduced to 
(3| + 7? - 3p - i?i) P + (3p + 3p? - 3p 2 - 3pi?j) = 0, 
(3f + 7/ - 3p - 1?j) P + pi? - P 3 7?l = 0, 
or, what is the same thing, 
3(f-P)(P + l+P) + ^?(P + 3p-7? 1 (P + 3p) = o, 
3(£-p)P +1?(P+ P)-1? 1 (P+ P 3 ) = 0, 
and if we hence determine the ratios 3(f— p) : 7? : 7? 1} the first of the resulting terms 
divides by £ — p, and we have 
3 : 7? : i?i = -P(P + fP + P 2 ) + 3P- 3fp(f + p) 
: P(2p-|)-p 3 (P + ^ + P) 
: P (2£ - p) — P (P + l + P). 
Hence observing that by the relation between p p the first term is 
= 3 {P£p -fi Q (f + p) + 3P}, 
the equations become 
1 : 7? : i?i«PfP+Q(f+P) + 3P 
: P(2p-f)-fi*(P + f + fi) 
: P(2f-P)-P (P + f+P);