OF A SUEFACE, AND THEIR GEOMETRIC SIGNIFICANCE, 
337 
H = (X + H) — H = (ir y)} dt — h -{■ A. dt, 
where 
lyi-l-s 
A — S- p /- 
— -< — Sri , , 
,■=0 s =0 r ; 5 ! 
and S' implies that r and .s may not be zero together. 
Similarly 
K==/.: + Bf/G 
where 
jjhs 
V, — V V' Ji ^ 
r=0.y=U 7 : S , 
with the same signification for S' as before ; and thus, for all values of p and g, 
we have 
=z hPh + {pliP-^hk 4- qhPki-^B) dt. 
Now, as the relation 
E = E' (I fi- 2^10 + 2F'')7 ^q dt 
holds for all values of x and y, it follows that 
E {x + It, y fi- /t) = E (X + H, Y -|- K) {1 + 2^i^o “b 7 / "E ^0 
+ 2F (X 4 H, Y + K) rj^Q {x + h, y 4- k) dt. 
IIj 
Let both sides be expanded in powers of h and k ; then —= coefficient of h"‘k'^ 
rn I n ! 
in the expansion of 
V V 
E' 
4 '^ , [hPl'^ + IphP-^k^iK + ijhPh-^'^) dt} 
_p=0rx=0 2^ ■ q ■ ' 
1 + 2 2 S f ,,,, ~ dt 
----- ?• ! 6* 1 
r=0 s=0 
4- 2 
Fb 
44 {hPt(2)hP~dc'^iA-\- qhPB-^B) dt] 
Lj‘=oq=oplqi 
X X ^ . A"'' _ dt 
? =0 .<J=rO 
-t Vr+l.s • ' , 
-II ? ! ,s ! 
Remembering that the first power of dt alone is to be retained, we find this 
coefficient to be 
TT' 
_ mn 
mini 
4 SS' 
(to —r+ 
+ 222 -, 
[m — r 1)! (rt — 6-)! r ! s ! 
I 
{m — r)! (w — s )! r ! s 
+ (m -r)l{n- s 4- I)! r ! ,y t ~ ^ Vrs'E^'ra-r,n-s+i dt 
+ (m - r)! (w - s)! r ! a ! dr+i,s'P'tn-r,.-. dt ; 
the first summation SS' does not occur if )■ = m 4- I, the second summation SS' does 
VOL. CGI.—A. 2 X 
