26 Professor Hamitton’s Third Supplement 
(S*), and the remaining twenty-seven are determined, (when the variables and 
-coefficients of VY” are known,) by the six equations (G’), (B*), the three lefthand 
equations (G*), the six first (7°), the two first (A*), and the ten (L*) (1Z*) (P*); in 
resolving which equations it is useful to observe, that by (7°) and (G*), 
wo? 8a 
Ligure) owe (aa sig (T") 
= ovdy 
And the twenty-seven expressions thus found for the coefficients of 7 of the two 
first orders, on the supposition of a previous elimination of one of the seven related 
variables, may be generalised, by (@Q*) and (#*), into the twenty-seven relations 
already mentioned as existing between the thirty-five coefficients on any other suppo- 
sition ; which supposition, if it be sufficient to determine the form of JV’, will give 
the eight remaining conditions analogous to the conditions (S*), that are necessary 
to determine the coefficients sought. 
If, for example, we determine JV” by supposing it made homogeneous of the first 
dimension with respect to a, 7, v, we shall have the eight following conditions, 
ow ow ow 
ae ? 3 
Pao bite this oS W, (U*) 
and 
PH BIA 6S Blane 
© 352 Ru abr She 4 
BB aly BM gin OW ails 
7 dadr 3 or? y Strout” 
BW ge, Wag ey M, ang 
Solus, ein 
ow Ow ow ow 
TGS | Se ke ee cv") 
Sw yw SW _3w 
Seay Sy ge ae 
ev ew EW 8w 
* Bode! * 7 Boe ’ BBe" BE"? 
Jou | ew | ew _ aw 
dady ordx dvdx dx’ 
to be combined with the twenty-seven which are independent of the form of JV, and 
are deduced by the general method already mentioned. But this supposition of 
homogeneity appears to deserve a separate investigation, on account of the symmetry 
of the processes and results to which it leads. 
