Poissons Equation for certain Volume Distributions. Ill 



Petrini's generalization of Pois son's equation. 



5. Professor Petrini has formulated the following gene- 

 ralization of Poisson's equation : — 



" La fonction /\V existe toujours, meme si les derivees 

 ^sy ^2y ^sy 



2 , -^— g , et -^-g- n'existent pas separement, si on deflnit le 



symbole /\ de la maniere suivante : 



* 1= o £j tZ K L o% ox J 



A 2 =0 



ou lim r^^=0 et determinee." 



It is easily seen from (A) of Art. 2 that the generalization 

 holds for the Case I. For, let 



hi _h 2 _h 

 a"~ j3~ V 



where a, /3 are always different from zero as well as from 

 infinity. Then 



( 1 JL 1 Jl. 



A ' = ^o ~15 l0 " 7^-15 l0g 7-1. 



16*. log i) 



+ TF log — 3 V. 



15 "« 



Now the expression within the brackets is equal to 



8r-, { l0g F 



log 



i5" ,8 /,„,.'i J .,„„i\; 



|lo g i + logI|(logi+logij 



= =-=- log 1 in the limit. 

 15 & 



Thus A Tr a ,! 



