206 



and the problem consists in determining the coefficients of U. This 

 gives 



A ? (a/3.. 1 ', *,$,.. '',.. )=n^X' /A .... U. 



the quantity within brackets being the coefficient of 



Then if the term be selected in which z=z,= . . = 1, and if A q be 

 so transformed that it shall appear as a function linear in each of 

 the systems 



i^_ (01).. 



andif 



then op.. n A .. n .. n aj3 .. n A .. .. 



and U will be at once given by the equation 



U=(a/3.., a.jS,.., ..J a/3 .. D ,,!!,. D..A*. 



This method has been applied in the case of two variables to the 

 calculation of quadratic and cubic invariants. 



But in the case of two variables the coefficient a may be expressed 

 by a series of symbols with a single suffix, thus : 



, a t , .. BO, a,.., n , D,, .. on,,!!,.. 



Now since A is of the same degree in I, m, and in /', m 1 ', the coeffi- 

 cients of all powers and products of a , a,, . . in which the degree of 

 I, m is above or below that of /', m 1 , will vanish in the invariant. And 

 from this and some other considerations it is shown, that not only 



mn 



nU=0, >n^U=0, .. >T~ 1 n^U=o, 



but that the coefficients of the invariant may be calculated from the 

 equations arising from equating to zero each term of the last equa- 



