28 
ME. W. SPOTTISWOODE OX AX EXTEXDED FOEM OF 
or 
d d^ 
dx'^ • -52 ^2 + 2s S,U., ^^2, 
d^ d^ / d ^ \ 
'^2 ^^3 — SSi 2c 2 j 
^ ^ ^ ~ “0 
= m(zCi — ai)(w2 — 2a2). 
Again, taking the first two equations of the quadratic system above written, and ope- 
d d 
‘^dy’’ ^^dx 
rating on s<^y-> respectively, we have 
ss 
d^ 
d 
-,S,-7-z=SS, S, 
d^ 
^ dx^ ' ^dy ^ ^dx'^ dy 
-2 sSiCC 2 
2 s Si 
d 
d^ 
• dx^ </w+ + 2sSi ^2 
Hence 
d^ . 
dx dy '^^dx ^-5 ^2 yy 
d^ d'^ r d 
d^ 
dx dy 
d^ 
dy 
d^ 
'■dx'^' 
3SSiS2^^2^^ — 2/32)+2sSi^_^^^ (S 2 2 cc 2 ) 
— cci)(v 2 — 2 / 82 ) + { ^^(v, — /3,) + i;(!q — cci) } (?<2 — 2cc2) 
- z=u{u^ — a,){v^— 2 ^^)-\- u{v^ — — ‘^<^■ 2 ) — <){v^ 2 — - <^ 2 )- 
Hence the cubic system, 
5S1S2 ^3 “^(^i " 2 a 2}5 
d^ 
3 s Sj S2 y ^<2 ” u{Ui 2/82) 
“h if-ivi 2C62) 
-jrv{ui — ai){u^ — 2062), 
^^dx dy^~^^^^ /^i)(^2 20.2) 
— cc,)(v2— 2/82) 
+?<r;,— /8,)(r;2— 2/82), 
d^ 
ss,s,-^=v(v,—l 3 ,)(v,— 2 ( 3 ,), 
in which the ss, us, vs, as, ( 5 s may be furnished with double suffixes, as was done in 
the quadratic system ; and then the sum of the four equations would give the r alue of 
For the general case, let 
Nj„sSi . . 5 i_, , . _ =A, 
