DERIVATIVES ASSOCIATED WITH LINEAR DIFFKKKXTIAL EQUATIONS. 898 



titiea which have a dimension-number 3 are P 3 and dP/dx, and therefore the 

 general form of 8 3 is 



(the coefficients A, ... being constants throughout) ; those which have a dimension- 

 nuraber 4 are P i? dP^/dx, d^P^/dx"-, P./, and therefore the general form of t is 



those which have a dimension-number 5 are P 5 , dPJdx, d^P^/dx 2 , 

 P S P 3 , and P 3 dPcJdx, and therefore the general form of B 5 is 



RjJ PSJ r'j FPP FPS- 

 B rf* + '~dJ +D ~dJ+*' P * P3+ P8 1ZT' 



and so on. 



18. It is evident that the product of two functions 6,, 8,, is a function of the type 

 6,+,.. A. composite function of this kind, resoluble into the product of two functions 

 with lower indices, will not be considered as properly associated with the dimension- 

 number (<r + <r'). The functions will be supposed ranged in order with increasing 

 index, and a composite function may thus be considered as included in the aggregate 

 of earlier functions. The method of determination of the invariants 8,, will appear to 

 be practically founded on the solution of a partial differential equation of the first 

 order, as is usuil with all invariantive functions of any nature ; and, as would be 

 expected when the most general possible form of S f is adopted so as to determine the 

 assumed constants, composite functions of index p will occur associated with undeter- 

 minable arbitrary constants. For simplicity of calculation, it would therefore appear 

 desirable to exclude from 6, all terms which occur disjunctively in the aggregate 

 ^<r,>- but owing to the form of the implicit partial differential equation to be 

 satisfied this is not completely possible ; what proves to be possible, as will be seen 

 later, for the adequate determination of a non-composite function 8, is that terms 



and terms, of course, of the dimension-number p, involving as factors either P 2 or 

 some derivative of P a or combinations of them, alone need be considered. 



We now proceed to what is practically the formation of the partial differential 

 equation, deriving it by a generalisation of the ordinary method of infinitesimal 

 variation which is used to obtain the characteristic differential equations satisfied by 

 concomitants of algebraical qualities. The general characteristic equation is not 

 explicitly given on account of its complicated form ; it is implicitly given in all the 

 particular cases, and its principal use is to obtain the numerical coefficients of the 

 different functions 8. 



MDCCCLXXXVIII. A. 3 E 



