82 
MACFARLANE. 
The multiplier of the numerator part is not the index m, 
but the index m diminished by 2. 
The operator 
!_ 
dx i 
A i 4 _i_ 1 
dy j dz k 
applies only to a function of coordinates along fixed rect¬ 
angular axes; but it is to be observed that g 2 is formed from 
[7 by direct operations of the calculus. If we expand 
(±l + ±l + ±i\ ( 
\ 3x i dy j dz k ) \ 
we get three terms of the form 
d 1 d 1 _j_ 
dx i dy j dz 
L L) 
fo k ) 
A ( H ) 1_\1 
V dx i ) i 
dx 
and three pairs of terms of the form 
± 4 ) 1\ i , J_ (H ) i\ i. 
dx \ dy j ) i dy \ dx i ) j 
Because the axes are constant, they may be removed out¬ 
side the bracket, giving 
*(_)! and-A/l + ll 
dar* i 2 My \ ji n ij j * 
Further, because i and j are normal to one another 
ji — — ij, and each of the three pairs of terms vanishes. 
Hence 
V dx 2 i 2 ^ dy 2 j 2 ^ dz 2 k 2 ’ 
and when the function operated on is scalar, 
2 = * 4- A 4- A 
F — + + ~w 
The simplicity of this expression for p 2 depends entirely 
on the conditions that i j k are constant. 
