Macfarlane — Differentiation in the Quaternion Analysis. 213 



"WTien the terms subsequent to the first are expanded by means of 

 the principle of differentiation by parts, we obtain 



1 a^J 1 j. 9(J 1 sv i 



r 8r3^ 9/D r'^ dd dp 



^P ^P 



1 ?!y 1 1 9( ) 1 



/• drd(ji dp r^ dtf) dp 



d^P d^P 



1 3^ j_ 1 a( ) 1 ap 2. 



r dddr dp r dr p^ dO dp 

 ^W ■ dd 



1 d\) _j__ \_ a(j _2_ iV 1 



"*" V^ d&d^ dp dp ^ r' dcj> (dp V d6d(}> dp 



dcf) 



1 8XJ J_ 1 3( ) 1 ap J_ 

 r 9^8r 3p r dr p^ 3^ 3p 



1 ?y -j_ 1 8( ) 1 ^'p L. 



"^ ^ 3^ 3p3p "^ r^ 3^ /3pY 3</»9^ ^_P_ 



E.I. A. PROC, SEK. III., VOL. VI. 



