and their Index Symbol. 137 



and at a glance we catgh the proposed analogue, 



F(V).6 m W = © m .F(V + m).W, .... (5) 



where ® m is a known homogeneous function of the independent 

 variables of the mth degree. 



This result, which seems to afford the same facilities of appli- 

 cation to the subjects of partial differential equations, and mul- 

 tiple definite integrals, as the elementary theorem to those of 

 ordinary differential equations and single definite integrals, is 

 readily proved by the substitutions 



x = e®, y=e^, &c. ; 

 since then, generally, 



V.U W= (^ + ^ + &c.) .UW = U.VW + W.VU, 



and in this particular case, 



V.@ m W = © m .(V + m).W. 



9. By the same substitutions we obtain extensions of the more 

 general theorems, of which important use has been made by 

 Mr. Hargreave in connexion with the subject of the integration 

 of linear differential equations *, namely, 



<f>(D).uco = u.cp(D) ( o+ j.<j>'(D)co + ^.<f>"(D)a>+&c. 

 and 



u<f>(D)co = <f>(D)u Q >- $ (D) .u'co + $^ . u"co-&c, 



where u and w contain 6 and D is -j^. These extensions are, re- 



do 



spectively, 



$(V).UW=U.<D(V)W+ l±L.&(y)W + y-^-.4>"(V)W + &c. 

 and 



u.a>(v)w=4>(V).uw-^l.vu.w+^Jl-v 2 u.w-&c.,. 



1 I .& 



where U and W are now functions of the n variables x, y, z, &c, 

 and V is the symbol before employed. 



In the particular case in which U is homogeneous and of the 

 wth degree, or in which 



U = ro , 



we fall back upon the case discussed in the last number. But 

 the two equivalent expansions, 



* Philosophical Transactions, 1848. 



