DIFFEKEXTIAL IXVAFIAKTS OF SPACE. 
•281 
5. Let w, V, IV become u-\- i, v + j, w + /•; and let tlie consequent values of n', v\ lo' 
be u’ + i', v' +/, iv' -b h' ; then 
i’ ■=. i {^{u + q V IV -f- h) ■— ^{u, dt 
n ml n ! 
^Imn 
dt, 
,5 7 t}i 
l\ ml nl 
/=y + sss'qqf: 7 , 
/,' = /; + SSS' n 
11 ml nl 
. here implies summation foi' all positive integer (including zero) values of /, m, 'vi, 
save only simultaneous zero values. Hence also for all integer values of p, (p r, we 
have 
rt In IH 1 'tt 
rt In })l J'>H 
+ K VS' nq-f, 
ll ml n ! 
Now 
a (u d" q V j, IV + /■') = a [u' + d, v' + j\ id + Ic) 
-f 2a (a -h q v -\-j, w + Jc) ^^00 {u + n v j, iv + d) 
+ 2h {u -f q V +J, w + ^ Vm + 0 ^ d-J, + ^>') d' 
4- 2r/ (u + q V -hi, Cioo + 0 ^’ + d’ 
Expand the first term on the right-hand side in powers of i', f, k', and substitute for 
all combinations such as i'Pfdi'' in terms of q j, k ; expand also all tlie other quantities 
in powers of i, j, k, and select the coefficient of ?’y/A. I.et 
,si 
dj {s~t)ifi’ 
> — a 4- A/ Pi > 
ally — <-Ci2v n ^ 5 
d 
then, neglecting powers of dt higher than the first and multiplying up liy a! /3! yl, 
we have 
__ i \d al-l-t, y-,idlmn H d ^ p + l-m, y-n’tlum y+l-rXhiin \ 
dOo 
dt 
-f 2SES 
dd\ /73\ / y 
J mil \ n 
i 1 d'a — l. I-i-\. j:i) 'll H” — ?; y~'iid W1) il i da~I, y^'idsl + 1 ^ it 1 ) 
where the summations are for all integer values of I from 0 to a, of m from 0 to /3, 
VOL. con.—A. 
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