DIFFERENTIAL INVARIANTS OF SPACF. 
283 
character of the calculations needed for the construction of the differential invariants 
of the third order may be gathered from the fact, that they require the complete 
solution of a Jacobian system of fifty-seven simultaneous partial differential equations 
of the first order and the first degree. 
For such analysis as will here be given in detail, we need the following particular 
cases of the preceding general results :— 
i7 ~ -f- 
(It 
~~ ^!/i ~ -F 2 /'(giQ, 
“ (/f ~ +/Puio + +/Coon 
da 
100 
dt 
da, 
010 
dt 
dttQQl 
dt 
dh 
100 
dt 
dh 
_ 010 
dt 
— = f/^ioo + «^001 + ./i?ioo ~b ^'^001 + c^ioo + I/Coon 
— + <^<^010 + + ^^Poio +/’Cioo + £/Coio; 
= Sttioo^ioo + 2(:t|.(,o + ('-fiiu + ^^boo) ’yioo + + (^hioi + ^S'loo) ^loo + 
2^010^100 "h ^^loufoio d~ ^*^110 "b 2/?oio'’?ioo “b ^bio'‘?oio "b ^^^^iio 
"b 2^010^100 “b ^ffoiCoio “b 
-'^^001^100 + ctioo^ooi + -«^ioi + '^KoiVim + «oio^uoi + ^^'Viui 
+ ^i/ooi^ioo + 001^001 + 2pCioi; 
^fioo^ioo “b 2/t^ou^oio “b ■^^^fiio “b ^oio^iuo “b 2^>ioo^oio "b 2^'’?iio 
+ ^001^100 "b 2 / 100^010 + 2/C 110’ 
(^100 "b 2 /i-oio) ^010 d" 2 a;^ 020 "b 3 ^oio'’?oio “b '^^Vozo "b (^001 + 2./010) ^nio + 2 /&„, 
db 
001 
dt 
dc 
I on 
dt 
dc, 
dt 
mo — 
__ 
dt 
2^boifoio "b ^fioo^noi "b 2 /a^oii d" “b ^oio'’?uni “b 2617,,] | 
“b 2/001^010 “b ^ooiCuoi "b 2 y’ 4 )ii ’ 
<^'100^100 “b 2pi^oo^ooi d- 2p|^ioi -b ^'oio'’?ioo d- V iou''?ooi + 2yp 
101 
-b ^001^100 -b 2<^ioo^ooi ”b 2 c{;^q2 , 
*^100^010 d" 2poio^ooi "b 2p^oii d“ <^oio'’?oio “b 2yoioPooi “b 
“b <^001^010 “b 2co;^o^ooi "b 2cyQji, 
(^100 d“ 2pooi) ^001 d" 2ploo2 “b (cqio “b 2y’ooi) ’?ooi ~b ^fvoo2 “b ^<^001^001 "b '^^Coo-z ’ 
2 o 2 
