338 
ON THE CENTRO-SURFACE OF AN ELLIPSOID. 
[520 
whence in the equations for rj, r) 1 , substituting the values ^=^ = — a 2 , we have 
that is, 
or 
1 '• V : Vi = ft4 P ~ 2a 3 Q + 3R, 
: — a 2 R + a 6 (P — 2a 2 ), 
: — a 2 R + a 6 (P — 2a 2 ), 
1 : V : Vi = — (Pfiy : a 4 /3y : a 4 /3y, 
V = Vi = -a 2 . 
40. II. a 2 + £ = 0 without a 2 + £ x = 0, consequently a- -4- % = 0 ; writing £ = — a 2 , in 
the relation between (f, f x ), this is 
6P + 3Q (& - a 2 ) + P (&■ - 4a 2 £ + a 4 ) - 3a 2 £ (£ + a 2 ) = 0, 
viz. this is 
(b 2 + c 2 - 2a 2 ) + £ (- a 4 - a 2 b 3 - a 2 c 2 + 36 2 c 2 ) + a 2 (a 4 - 2a?b 2 - 2a 2 c 2 + 36 2 c 2 ) = 0, 
where the left-hand side should divide by £ x + a 2 ; the equation in fact is 
(£i + a 2 ) (b 2 + c 2 — 2a 2 ) + a 4 — 2a 2 6 2 — 2a 2 c 2 + 3b-c 2 } = 0; 
or, what is the same thing, 
(& + « 2 ) {(& + a 2 ) (£ - y) - 3 /3y} = °> 
whence 
ffi = — a 2 + 
3/3y 
£-7' 
41. Considering these values of £, as given, the verification of the value 
« = — a 2 , and determination of « = — a 2 4- , i s somewhat complex. 
(p - 7) 3 
Writing for a moment A = — ~—— , we have 
1 : V : Vi = P (a 4 + a 2 A) — Q (2a 2 + A) + 3P 
: — P (a 2 + 2A) - (a 2 + A) 3 (2a 2 - P + A 
: — R(a 2 — A) — a 6 (2a 2 —P + A). 
The first term is 
a 4 P — 2a 2 Q + 3R + A (a 2 P — Q), 
which is 
= — a 2 (3y + A (a 4 — b 2 c 2 ); 
and for the value of rj 1 , proceeding to the third term, this is 
— a 2 R — a 6 (2a 2 — P) -f A (P — a 6 ), 
which is 
= a 4 /3y — a 2 A (a 4 — b 2 c 2 ), 
so that without any further reduction r] l = — aK