Macfarlane — Differentiation in the Quaternion Analysis. 215 



whicli reduces to 0, because p and — are at right angles. For similar 

 reasons the coefiScients of (5) and of (6) reduce to 0. There remain the 



The term in - reduces to — — • As r^ = - p, and f -^ ) reduces 

 or r or oF \^" J 



to - 1, the first two terms in the coefficient of (8) reduce to p-^ + -^p 



do do 



■which reduces to 0. The third term reduces to 



a^p 1 



"a'^a^' 



which is equal to 



cos 6 (- sin (ji . J + cos (fi . k) 



sin 6 (- sin (f> .j + cos ^ . ^)' 

 that is, - cotan 6. 



The first term of the space- coefficient of the term in — reduces to 



0(^ 



1 • i-i Hi sin^apo* ,, ^. . , , cos^ ^ 



- -:-r7i^ ; the second to - -^— -: — - ; the third to ^-r^.p^- Hence the 



term is 



1 9 ( ) i • ■ n '^Po n I 



'«■ + sm o* . ^TT^ - cos ^ . po) 



r^ snr 6 ocfi 



r~ sin^ d(f> 

 Laplace's operator in polar coordinates is 



32 1 a^ 1 3^ 2 a 1 a 



dr^'^ r^de^'^ {r sin Bf a"^2 "^ r 3r "*" r^ ^° W' 



and the above reduced operator is the precise negative of Laplace's 

 operator. 



* By po is meant p witliout the v - 1- 

 C12 



