PARTICULARLY THOSE OF TWO VARIABLES. 
797 
K 
Another set of four is obtained by changing x into £C+—. log r 
nri 
and y „ y+£. log q : 
thus (24) gives 
(29); 
and a fourth set of four from the same equations by increasing each of the x*a and y’s 
K A 
by —. log pr, —. log rq respectively : thus (24) gives 
n *C+i, *+i) +n ®C*+ii »+i) +n *C+i. v+ir-^+i, v+i 
x ’ p+ ])+u<s > /x+l ’ P+1 
=W t P 1 W A+ ! lP ,Wn 4 (so). 
\cr + l, cr' + lj \cr+l, cr +l) \<r-f 1, <r'+1/ \cr-fl, <r+1/ 
But as might be expected from formulae (8) and (9) with (17), these equations (28) 
(29) and (30) could be obtained by uniform increase of the numbers p and v in the 
characteristics of (24). 
9. Let the value of a function when both the variables are zero he denoted by c, 
with the same subscript number as marks the function in the current-number notation; 
for shortness, let 
S- r (x, y) he written S-,. 
&r(£> rf) „ 0 r 
and & r (x+£, y+v)) „ ^(as-ff). 
Then the following equations are obtained from (23) by assigning suitable values to 
the numbers of the characteristics and to the variables. 
3c 0 9 V 
3 Cl 2 
3c 2 3 £ 2 2 
3c 3 2 = 
3c 4 3 £ 4 3 = 
3 c 6 3 ^ 6 3 = 
3c 8 2 V s 
3c 9 3 V = 
3% 2 ^12 2: 
3 C 15 3 *ll5 2: 
:C 0 3 V 
+ c 3 2 ^ 2 2 + C 3 2 "V T c W + e« s V + V + c srV + W + 
+ c 2^2 + c 3 2 "^3 2 c 4 2 ^1 2 — c 6 2 ^6 2 + c 8 2 ”^8 2 + ~ ~ 
c 0 2 "!o 3 + c l 3 ”ll 3 + c 3 3, ^3 2 4~ 4* c 6 2 ^6 2 C 8 2 ^8 2 — ~ C U^Vi ~ 
+ c i 2 ^x 2 + c 2 2 '1 2 2 — c 4 2 $ 4 3 —c 6 2 -V—c 8 2 V c g 2 ^ 9 2 -+■ c 12 2 ^ 12 2 +Cis^is 3 
+ C 2 2 V—Cg 2 A, 2 4- + C 8 2 V VV 4" c 12 2 "^13 2—c 15 2 ^1B 3 
C 0 2 V q 3 "h 3 4- ^ — Cg-'-Jg 3 + c 4 2 ^ 2 c 8 2 ^8 2 4" c 12 2 ^12 2 4" c 15 2 ^15 2 
C 0 2 V 4" r l 2 "l l 3 3 2 c 3 2 “13 2 4" tfV C 6 2 "V 4" c y 2 “!s) 2 4" c 12 2 'll3 2 c 15 2 *^15 2 
4" Cl 2 ^! 3 c 2 2 ^2 2 c 3 2 ”^3 2 — C 4 2 ^4 2 4" 4" C 12 2 ^13 2 4“ C 15 2 "ll5 2 
® 0 2fl ^0 2 C] 3 "^] 3 ^2 2 "^3 2 4~ ^ 3 2 "^3 2 4~ ® 4 3 "1 4 3 ^6 2 ^6 2 4“ ^8 2 "^8 2 ®9 2 "^9 2 + °1?>V 
(0 
(h) 
(iii) 
(iy) 
CO 
(yi) 
(yii) 
(yiii) 
(ix) 
(O 
These are not independent: thus, adding equations (ii)--(x), equation (i) is obtained ; 
and simpler relations equivalent to these will be deduced later on. 
