210 On a Multiple Differentiation of a certain Expression. 



r -, 1 c*b*ai- 

 IPl = w '-ryi—x tt-9 



c-2 a* a* j)S 

 ,11 ,i l i 

 r -. 6* a 8 b*a*p* 

 Lpc] =ic.- Y - r - 1 . r - 7 



Uzazpa C2 



azpsc* b* 



\jh£] 



azp^b* <?2 



IpeheaJ = w . ph^ff . f!*M , 



iA,l 1,1 i 7 ll -1 



r T -r p*p 9 b*p s bzps b*a e p* 



ipcbeac] =w . 7 ' , ■ - — j — ? 



p^O^a* c-2 



r T ,-. pspsp8psa 8 p* c*a s p8 



[ pcbcaeb 1 = w . t, * -^—> 



ptatca &* 



1 1 7 S. 1 



psaab* cz 



1 



«¥6^C2 

 and the final numerical coefficient of w is evidently 



-I I I A 1 



~8'8'4 2 '2 4 ""16384' 



In this process we see that in every alternate line the letter 

 c disappears from the coefficient of w, that in every fourth 

 line, starting with the second, the letter b disappears, and that 

 in the fourth line the letter a disappears ; while p disappears 

 from the final result, and the other letters, each raised to the 

 power ^, are left in the denominator. 



It is evident that the above procedure is quite general, and 

 that, following the law of succession there indicated, one letter 

 is always absent after each differentiation. To state the law, 

 suppose (for greater convenience) the order of the n letters re- 

 versed, so that 



v= T-rj — p ex P 0^- ^ ■ ?** • • ■ ^) ; 



a*b'iCz . . . ki 



then, the differentiation with regard to p having been first 

 effected, if we take 2 n compartments and place in every alter- 



