400 ?rofp:ssor a. r. forsytpi ox the differextial ixyartaxts 
same general character as those in §§ 14-22 ; their aim is to obtain the one solution, 
other than V and V-, of tlie tAventy-eiglit partial differential equations satisfied bv 
differential invariants of deformation, which are of order not higher than three. The 
mode of dealing with such a system of equations has been amply illustrated in Part I. 
of the memoir; accordingly, only the results of the analysis will he given, 
e denote by F, F, F , A, A , Aquantities defined in ^ G ; and aa'c Avrite 
u = p:,, _ + Goo + r'T:,o - (2r' + A") E,, + fe„., 
+ 2A'Th„ - 2 A'Goo + 
r = Koo - 2F^, + G,i + T"E,, - 2F'Eoo + 2FF,.o, 
+ A^'Goo - (r + 2A') Gn + AG,,,, 
^ ” 2Fji 4- Go,,, 
these being simultaneous solutions of the eighteen equations, Avhich correspond to the 
vanishing of the derivatives of ^ and rj of order 4 and of order 3 in the various 
arguments (§ 13). Further, Ave Aviite 
P — E''(— 4 EoiG^oCtoi + SF^o^HoGoi — 4 :Gj|/) 
+ EF(- 4EioGioGoi + 8 E,„FoA,i “ 16F,oF„iGjo - leFoyGoi + lOF^oGop 
+ -ll\,iGj,p + ^KoiFoiG]o) 
+ EG(- 2E„EoiG,o 4- f->K,oFioG,o 4- 4E„Fo41o, - 4E,oG,o^ - 4Eo,FjoFoi 
- 2E„44,Go, + SFoiF,,= - 4 F,/Go, - ^Kop44o) 
4- t'l — 2 EjoEoiG(ii. 4- 4:E^oF(ji^Ho + lOE^ijFioGoi — 4E,oG]o~ — ]2 Fio'T4o 
4- 24F,/F„ - GEo,F,Ao - 12EoiF,oFo, ~ 2E,p\-4o) 
4“ f G^SEjoEojFoi “ 4 Ejo'Goi + TFjoiEo^Gjo 32E2oFji,Foi 4" l^EojFjoG^o -F SEopFj,,) 
+ G'(8E,FFoi - 4Eo;rG,o - 4EorE,o), 
10*- uy’^lii 
./ = K~(8F,„G„i' - 4E,„G,„’ - 4G,„G,„*^) 
4* EP ( 4E,,,Gui~ ~ il^PjoPoiGin 4^ 8F]oG|i|G„| 4 - 4P-imGj|,G,„ -F 1 gF4)iFoiG 
4- EG (Gp^oJhiiGoi — 2F_.|oGii,G(:n + 4P4jP'’^,44^ + f*l'’iul''’ui' “ ^FjoFoiC 
- 4EorG,, - 2 Eo44^g.o - 4 E 01 F 0 P - 2Eo,Gur) 
4- F-(10EoiFo4loi - 2 EjoG,,Goi + 4EoiFioGoi + 24Fof-F^o “ F2P\oFoiG 
2PvoiG|o~ *1 Ko4oiC»^io 4EofGoi — 12Eo4''oi') 
+ FG (— 4E^oEoiGoi — IGEjoFoP 4" SEjoFo^Gjo — IGEoiFjoFoi 4- SEo^FjoG 
+ IGEoF^Foi + 4EopG,o) 
+ G- (8EjoEo,Fo, - 4EjoE,-,44oi - 4P;-,F). 
1)1 
'10 
10 
