153 
1921-22.] Concomitants of Quadratic Differential Forms. 
where F, G, H, K denote quite general independent functions, are as 
follows : — 
(i) The general substitutions are adequately secured by the retention 
of the complete aggregate of infinitesimal substitutions. 
(ii) Each of the fundamental infinitesimal substitutions, obeyed by a 
function, leads to a linear partial differential equation of the 
first order which the function must satisfy. 
(iii) When properly arranged, the full system of these linear partial 
differential equations is a complete Jacobian system.* 
(iv) The original group needs to be ‘‘ extended ” (in Lie’s sense of the 
word), so as to include derivatives of all quantities initially 
subject to variation. 
(v) A function can only belong to the body of invariantive functions 
if it satisfies all the equations in the complete Jacobian system. 
(vi) The number of algebraically independent members in the body of 
invariantive functions is the excess of the number of variable 
quantities t involved over the numbers of equations in the 
Jacobian system. 
(vii) A function of the variable quantities involved may satisfy a sub- 
group, or more than one sub-group, of the equations in the 
Jacobian system ; when it satisfies all the sub-groups except 
that which involves derivatives with respect to the original 
coefficients of the differential form, it can be a semi-invariant 
(such as the leading coefficient in an invariantive form). 
After this statement of the method of analysis to be adopted, it is 
manifest that consideration must, lirst of all else, be given to the infini- 
tesimal substitutions and to their effect upon all the variable quantities that 
can occur. 
Effect of the Infinitesimal Transformations. 
9. We have taken the most general transformations to be 
o 
iTg , = I , G, H, K {x^ , , x.^ ^ y 
and all the quantities concerned are to be subject to them. We denote, 
temporarily, the modulus of transformation by Q ; thus 
0 = J 
F, G, H, K 
"2 ’ 3 ’ 
* For the nature and properties of a complete Jacobian system, see my Theory of 
Differential Equations, vol. v, ch. iii. 
t For the purpose of estimating the number, a coefficient of the quadratic form and all 
its derivatives are reckoned as independent. 
