130 Mr. R. Carmichael on Homogeneous Functions, 



Multiple Definite Integrals, it will be seen that valuable results 

 can be obtained, and some examples are furnisbed. It will be 

 observed that these general principles present the means of ex- 

 tending all multiple definite integrals, in which the variables 

 enter, as complicated functions, in the indices of known quan- 

 tities unconnected with the limits. This seems to be an im- 

 portant step, but an adequate development of its consequences 

 would much exceed the limits of the present paper. 



The instrument employed is the symbol which occurs in the 

 well-known theorem of homogeneous functions. The relation 

 which the result of the operation of this symbol upon any ho- 

 mogeneous function bears to the degree of the function, seems 

 to give ground for the appellation, Index Symbol. 



In conclusion, the writer begs to express in the most ample 

 manner his acknowledgements to the distinguished mathemati- 

 cian above named. 



1. In general, if 



Mm =/(#, V> Z> &C) 

 be a homogeneous function of the mth degree between the n in- 

 dependent variables x, y, z, &c, 



du m dv m du m , 

 oc~+y—P-+z~^-- i r &c. —mu m ; 

 dx ay dz 



or, putting the operating symbol 

 d d d 



X Tx + ^d-y + 2 dz + kC -^' 

 we have 



V .u m =?n.u m , 

 and by successive operation, 



V" .u m =mP .u m . 

 Hence the theorem 



F(V).« m =F(m). Mm , (1) 



which is an extension of the theorem 



/ (x^f) ■ x m =/(»*) • x m , or /(D) . e'" e =/(m) . e m9 , 



the first fundamental principle employ :d by Professor Boole. 

 In fact, x m is a particular homogeneous function of the »ith 



7 



degree, and x-j- is the first term of V. 



2. Now if U be any mixed rational function of x, y, z, &c, it 

 can, in general, be put under the form 



U = w + « 1 +w 2 + &c. +u m ; 

 and we have a theorem for mixed rational functions corresponding 



