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Proceedings of the Royal Society of Edinburgh. [Sess. 
and there were also six invariants 
I 4 3 I 9 5 I 3 5 h 5 E » h ’ 
involving only the coefficients of u{V). 
All of these can be included in the complete aggregate required : but 
they are not sufficient in number to constitute that aggregate. They 
must therefore be supplemented by others. 
34 . It is desirable to have a full set of pure invariants, independent 
of one another ; for they involve the coefficients of the original form and 
the coefficients of t6(P), these being combinations of derivatives of the 
ten original coefficients. Thus the pure invariants are the differential 
invariants of the original form. As there are fifteen equations and 
thirty-one coefficients, we shall have sixteen members in the full set of 
independent quantities ; the single relation between the line-coefficients 
can be introduced later. 
These differential invariants can be derived from the six differential 
invariants already obtained ; no further integrations are necessary, because 
they can be constructed by means of a known theorem in the general 
theory of concomitants of quantics. We have, on the one hand, the line- 
co variant of the original quadratic form, represented by 
A^(d, m, . . ., a jjP,, . . ., P,y, 
and the associated line-covariant, involving the Riemann-Christoffel 
symbols as its coefficient, in the form 
M, . . ., A$P,, . . ., Pjy 
both being quadratic in the line-variables. Hence 
u /xA = (P) + yud , M -f- , . . ., A -|- ^a ^ P^ , . . Pg)^ , 
where /x is any arbitrary parameter, is also a concomitant of the system, 
The invariants of any concomitant are invariants of the system. We have 
invariants 1^,12,13, I4 , I5 , Ig of ib alone, as well as the discriminant A of 
the original quadratic form S ; consequently, these quantities 
\-i{u-\- fJiA .) , l.^^u + /xA) , l^{u -f- /xA) , I^{u + /xA) , + jxh) , + /xA) , 
being the six invariants of u + /xA, are invariants. This result holds for 
all values of the parameter /m ; and therefore the coefficient of every power 
of jui in each of these six modified invariants is itself an invariant. 
To express these invariants, we adopt an operator D, to denote 
