DIFFERENTIAL INVARIANTS OF SPACE. 
29!) 
Denoting this l^y D, we ought to l)e al:)le to express D alge 1 )raicall_y in terms of the 
seven preceding forms; as a matter of fact, is a polynomial comlnnation of the 
forms. 
18 . The preceding determination of the differential invariants of the specified 
order has iDeen based upon a knowledge of the complete system of concomitants of 
two ternary quadratics. When we pass to higher orders, the last stage in the 
determination of the differential Invariants could he com})leted without fuiTher 
calculation, if we knew the complete system of concomitants of certain simultaneous 
ternary qualities some of which are of order higher than two. But, in general, such 
knowledge is not at present possessed; in its absence, some other method of attaining 
the end is necessary. Such a method can he devised in connection with the difterential 
equations; as applied to the two quadratics, it is as follows. 
As has heen pointed out, the equations determine the invariants and the contravari- 
ants of two simultaneous ([uadratics, tlie contragradient varialiles being 
Let such an one he 
O' = Voni + • • • , 
where t, q, q, . . . are independent of <^jqq, (^qqi ; fii^d n is a whole number, whicii 
in the case of an invariant is zero, and which, when t is known, can always he 
determined by inspection. Then when this value of cr is substituted in the ordinaiy 
way in the first six erpiations, it appears that t satisfies the four equations 
(0 
q{0 
2/<f+ 6?! 
?a cli 
/■f+2hf + bj' + f f 
• eg ?a ch dg 
^ ct . dt at . n dt . ct 
?a 
U 
dh 
.dt 
cf 
ra 
cf 
ah 
eg 
?f 
+ , + c „ + g ^ + 2f 0-, + c ^ 
cl) cf fh rb ft 
= h 
ct 
0.7 
+ h + h f + b 
ac 
eg 
cC 
= 0 , 
= 0 , 
= 0, 
0. 
The third of the former six equations gives, inerely liy processes of difierentiation, 
the succession of coefficients for ascending powers of 9,^0 ; and the fifth of them gives 
the succession of coefficients for ascending powers of The third and tlie fifth 
combined give all the coefficients when t is known. When this determination is 
completed, the seventh and the eighth equations are satisfied identically ; and tl)e 
ninth is satisfied in association with tlie proper value of g. 
It thus is necessary to consider the above set of four equations. Wlien we associate 
the equation 
ct 
2 Q 2 
+ fi 
ct 
ah 
a^ 
og 
= u. 
