142 LOTKA. 



to circumstances, there may be one or more applying to the system 

 under consideration. They commonly involve the initial values Ai, 

 A2,. . .An of Z], Xi,. . .Xn as parameters, so that they are of the 

 general form 



$ (Zi, X2, . . . Z„; Ar, A2,... An) = (16) 



The effect of such equations of restraint is to cause the determinant 

 A (0) to vanish.^ 



For, in view of (2), (5), and (7) we may write (15) in the form 



<p Gh, -h, • ■ -h) = (17) 



which implies that 



J _ d {x-[, a*2, . . .Xn) _ ^ /,Q\ 



(19) 



(20)9 



That is to say, the existence of equations of constraint causes the 

 determinant A (0) to vanish, as was stated above. 



Furthermore, the vanishing of A (0) means that at least one of the 

 roots X of (9) is zero. Since the left-hand members of (8) can not 

 contain an absolute term, it follows that at least one of the coefficients 

 a is thus no longer arbitrary, but is fixed at the value 0. 



This loss of an arbitrary constant in (8) must be in some way com- 

 pensated. This compensation is effected by the equation of con- 

 straint (15), which introduces as an arbitary constant the initial value 

 A of one of the variables X. 



Similarly it can be seen that m equations of constraint (15) cause 

 the appearance of m zero roots in (9) and of 7n arbitrary constants in 

 the form of initial values of variables X. 



8 A (0) is the determinant obtained by putting X = in the right-hand 

 member of (9). ^j, 



9 The subscript in equation (19) and (20) signifies that the values of — to 

 be talcen are those corresponding to tlie origin xi ~ X2 = . . . ^ Xn = 



