392 MH. A. R. FORSYTH ON INVARIANTS, COVARIANTS, AND QUOTIENT- 



not have a definite number assigned to it, but will have associated with it an arbitrary 

 number m ; to dyjdx we assign a number m 1, to d^y/dy? a number m 2, and so 

 on up to d n yjdx*, to which TO n is assigned. Now, if to P r we assign a number r, 

 the number assignable to P r d"~ r y/dx*~ r is m (n r) + ( ?) = m n, and is, 

 therefore, the same for all values of r, i e., for all terms in the differential equation ; 

 and, consistently with these arrangements, the number to be assigned to dfP r /dxf 

 is p r. 



In exactly the same way and by the same rules we can similarly assign numbers to 

 the coefficients Q and derivatives of these coefficients, and, as before, leave the number 

 assigned to the dependent yariable arbitrary. 



16. If now an invariantive function of the kind spoken of in the last paragraph be 

 denoted by (x) when formed from the coefficients P, and therefore by (z) when 

 formed from the coefficients Q, the invariantive relation is of the form 



e (z) z v = (x). , ::: , 



An equation of this form can exist only if 



(i.) Every term on one side has. according to the foregoing assignation of numbers, 

 one and the same dimension-number, and similarly for every term on the 

 other side ; and 

 (ii.) The two sides have the same dimension-number for the variable x, and the 



same dimension-number for the variable z. 



The first of these conditions requires a certain kind of homogeneity in the function 

 0, examples of which will immediately be given ; the second of the conditions 

 determines the index //,. For let cr be the dimension-number of (x), which may* 

 be written 0, (x) ; then cr is also the dimension-number of (z), these two 

 numbers being respective multiples of the units implicitly assigned to x and z respec- 

 tively. Consistently with the assignation of dimension-numbers, the quantity z must 

 be considered as having a number -|- 1 assigned to it in virtue of its dependence on z 

 and a number 1 assigned to it in virtue of its dependence on x. Hence the 

 z-dimension-number of 0,, (z) z v is p. cr, and its x-dimension-number is /A, while 

 the corresponding numbers of 0, (x) are respectively and cr. The second of the 



conditions requires 



p. - cr = 0, 



both of which are satisfied by /* = cr. Hence the invariantive functions are such that 



where 0, (x} is a function of the coefficients P and their derivatives such that every 

 term in the function has one and the same dimension-number cr. 



17. The following examples will illustrate these general explanations. The quan- 



