n(n— l)...(w— m + 1) _ v \dx/ y \dy 



132 Mr. R. Carmichael on Homogeneous Functions, 



a theorem analogous to Professor Boole's 



a?D(afl)-l) . . . (#D-n+l)=#»D». 



As an example of the operation of this latter symbol, let the 

 subject be x n , and 



a?I>(a?Dr-l) . . . (a?D — n + 1) . # n = 1 . 2*3 . . . n . #". 

 To which we have the corresponding theorem for homogeneous 

 functions, by (1), 



V(V — 1) . . . (Vy— m+l).u m = 1.2.3... m.Un. 

 As a second example, we shall seek a general proof of a theorem 

 first given by Euler, namely, 



, a (iY yjjf *>{i) y 



u n =Z - = = — . .. u n , 



m a. ft 7 



where a. + ft + y + &c. = rn. Now as 



£."£.£. 



(i+«) v -a+«) <, *.a.+«) f# .a+«j"*-.., 



expanding and equating the coefficients of a m on both sides, and 

 then condensing by the formula above, 



A-X vp(±y zrfiy 



V(V-l)...(V-m + l) TT * \dx) J \dy) \dz) 



—^ . U = 2 — — _ J • _ . . . U, 



m a ft y 



and when \]=u n , we get Euler's theorem. 



4. Again, as the symbol xD furnishes solutions of the class 

 of ordinary differential equations represented by 



. d"u _, „ dPy „ 



in the form 



y=F(^D).X + F(^D).0, 

 where 



f A.^D(^D-l)...(^D-« + l)l _i 

 F(a?D)= \ + B.a?D(a?D-l)...(*D-£+l) \ ■ 



{+ &c. J 



in like manner we obtain the solutions of the particular class of 

 partial differential equations represented by 



/ d a z d a z «(«-l) , . d*z \) 



\*-dZ +aX "^d^dy + iX^i^ + • ) j 



V^dxP+t* 1 * y dx^dy + 1.2 ^ y dxP-*dy* + -')l 



+ See. 



