[654 
86 
ON CERTAIN OCTIC SURFACES. 
it is easy to find 
that is, we have 
where 
t № 
x A h 2 (k — l) 8 ’ 
$ ^F 1) ’ U = 11 K/* + 9) ( ke V ~ ^ + h (k - If 0V], 
(bk 2 6 2 — a) (6y — #) 2 + /i (k — l) 3 0 2 w 2 = 0. 
If for greater convenience we write #=-</>, then this formula becomes 
-, u = n [<AV+02/ ! ) (h -1 > a +* - i) s m, 
where 
(bk 2 x 2 (f> 2 — ay 2 ) (</> — l) 2 + h (k — l) 2 w 2 <f> 2 = 0, 
or, what is the same thing, 
h(k— l) 2 w 2 
bk 2 x 2 
(f> 2 = 0. 
Suppose that the terms in U which contain z 2 are = Sz 2 ; then we have 
t h(k jJ y ® = 201-11' (fx^ + gf) m - 1)>, 
or, what is the same thing, 
/)4/*4 /y>4 
0 = A(F~i7 ^ ^ 2ir (Z^ 2 + #Z) (&</> - l) 2 , 
where IT' refers to the remaining three roots <£», <£ :; , </> 4 ; this may also be written 
<H> = 
b 4 k A x A 
h(k — l) 6 y* 
n (fx 2 <fr + gy 2 ) (kef) - l) 2 .2 
P 
(fx 2 (f) 2 + gy 2 ) (k<f) - l) 2 ‘ 
Hence, observing that we have identically 
Z-6lv) - 1> S + P = tt~ *.) (■#■ - « «- - «(■#> - 
and writing <t> = ± iy = {i = V(— 1) as usual), we find 
æ \(J) to 
n {<f)x V(/) ± iy V<Z)} = h - k hla —~ [c [x V(/) ± iy >J(g)Yfgw*\ y\ 
U(kef)-1) 
whence, writing for shortness 
bk 2 
(k — l) 2 
~lx 2 ^ ~ ^ + hw ^ 
A = [o{x V(Z) + iy V(£) 2 } -fgw 2 ] [c [x V(/) - iy s/(gf} -fgw 2 l 
= (?f 2 o& + c 2 <7 2 ?/ 4 + / 2 gHv 4 + 2cfg 2 y 2 w 2 — 2cf 2 gx 2 w 2 + 2 c 2 fgx 2 y 2 ,