or, what is the same thing, it is 
{k 2 w 2 — 2kwt + <r 2 ) 2 [k 2 w 2 0 + kw (— 2t0 + v 2 — yfr 2 ) + <r 2 6 + 2avyfr + 2t^Jr 2 } 
— 36 {k 2 w 2 — 2kwt + a 2 ) {kwv + a\fr) (4kwd — -vp) 
— 32 (kwv + ayfr) 3 
— 108kw (4kw6 — y\r 2 ) 2 = 0. 
67. This is 
{kw) 6 . 6 
+ (kwУ , — yjr 2 — 6i0 + v 2 
+ (kw) 4 . cr 2 .3# + ayfr . 2v + yfr 2 .6t + 121 2 6 — 4tv 2 — 1440v 
4- {kw) 3 . — 2(r 2 yfr 2 + cr 2 (2u 2 — lOt0) + en|r (— 8tv — 1440) + ip (— 12i 2 + 36v) 
- 81 3 0 + 4t 2 v 2 + 288tv0 - 32v 3 - 17280 2 
+ {kw) 2 . a 4 .30 -f a- 3 \[r . 4v + cr 2 ip. 121 + cnp.37 
+ cr 2 (12£ 2 0 — 4tv 2 — 1440u) + o-^ (+ 8t 2 v + 288i0 — 96ir) + ip (81 3 — 72tv 4- 8640) 
+ {kw) . — cr 4 ip + cr 4 (— 6t0 4- v 2 ) + a 3 -yfr {— 8tv — 1440) 
+ o- 2 ip {- 81 - 90u) + <rÿ 3 . - 72 4 ip. - 108 
+ {kw) 6 . <t 3 {0, 2v, 21, 4^cr, ip) 3 =0, 
which, reducing the last term, is 
{kw) 6 Imnxyz 
. 
— 4a 3 Xyv {y — z){z — x) {x - y) {ny — mz) {lz — nx) {mx — ly) = 0. 
68. I verify the last term in the particular case z — 0 as follows : the coefficient 
of a 3 is 
(0, 2n{l + m)xy, 2 {m + n) x + 2 (n + l) y, 4*§Хх + цу, nvxy) 3 , 
which is 
= 2n 2 vx 2 y 2 {(£ 4- m) {Xx 4 /¿y) 2 4- [(m 4- ft) x + (ft 4- l) y\ {vXx 4- ¡ivy) + 2nv 2 xy} 
= 2n 2 vx 2 y 2 {[(/ + m) X 4- {m + ft) v\ Xx 2 
+ [2 {I + m) Х/л + {m 4- ft) /iv + {n + l) vX 4- 2nv] xy 
4- [{I + m) fi + {n + l) v] fiy 2 }, 
which, substituting for X, /л, v their values m - n, n — l, l- m, is 
= 2n 2 vx 2 y 2 . - 2Xy {x — y) {mx — ly) ; 
or for z = 0 the coefficient of cr 3 is 
= — 4Xfxv n 2 x 2 y 2 {x - y) {mx - ly), 
agreeing Avith the general value 
- 4Xnv {y -z){z- x) {x - y) {ny - mz) {lz - nx) {lx - my). 
50—2