36 
ME. W. SPOTTISWOODE OX AX EXTEXDED FOEM OF 
Hence 
— ydx~^dx) dx'^ 
_l/o i-_2^^2ss — 
j_/o -_2^^2ss-- 
/ c? ds\ / d dsA I / 9 
y dxj y^* tffe t?.r y y efe 
, / d r^dsA ( d dsg\ d , ( d ndsA / d ds\ d 
^y^Tx~^dx) Tx~di)^ y^Tx~^dx)y dx~ d^r^ dx 
/ ^ / d ds\ 
’ y dx J dx dxj^ 
-1 
dx 
^^^dx '^dx ) \ ^^dx dx )'^ dx 
ds 
‘ dx 
.dsc 
d ds-^ 
dx dx ] dx 
■d 
S,y-— 
dsA d 
‘s 
A. 
dx ' 
dx dx 
Q ds lAj U- tto j M. I O ^ t*. w ] I* 
= 5:77. 5, -37 5, — z ^ Si Sg x ,7F ^ X ^2 ^ ^ ^2 ^ ^2 
i. 
' </,r 
«?Sl 
dx 
A. 
dx ' 
i. 
dx 
^ ds dsi 
‘ dx dx 
d_ 
' dx 
d'^s. 
+ s:t-s, 
d d 
+ ^i 
dx ^ dx ' 
d d 
dx ^ dx 
Al 
dx 
“1“ //.». ^ rlr^ ^2 
A. 
dx ' 
' dx 
. A 
^ dx 
d 
d ^ds d d ds^ d d ^ ^ds ds,^ d 
^2 ^qp Sj S Si I 1 
dx^ 
d'^s. 
d^ 
dx 
' dx ' 
d 
dx dx 
„c?Si d d dsq 
dx dx ^ 
dx 
d 
dx 
+S, 
d d 
^ds, d d ds 
dx 
A_ 
dx 
d 
dx 
d 
, „</si efeg 
^ di^^dx dx ^ dx~ 
dx 
d 
dx^ 
d^ 
s 
.2 Si 
dx 
^dsc 
dx dx' 
d d ds 
d ds d dh 
^ dx dx ' dx dx dx dx"^ ^- 
dx 
d 
^ dx 
d_ 
dx 
^ dx ^ dx dx ‘^dx ^ dx dx dx dx 
d d p,dsg d d dso 
d ^ I o^®2 d 
Si /7,V) I flnp piqf, S) 
dx dx ^ dx ^ 
d d d , d d d . d d d 
dx 
AL 
dx ^ “dx dx dx 
dx dx ’ dx 
.ds^ds-^ 
d^s d 
^ 1Z. ^2^772 s 
dx^~ ^ dx 
d d d 
d d 
d d 
^dx 'I- ^2 z/'T’ rl'T’ '®2 rU /7.y. “t" Si Sg S^^ + Si ^^S Sg + Sa S Si + Sa pjqq Si ^,,s 
dx ^ dx' dx' 
g,ds 
^ dx 
' dx ' dx 
d d 
'dx dx dx 
^ds-^ / d 
dx dx ^ dx 
d . d d' 
^dx dx dx 
dsc, 
dx ' dx dx 
d d 
d d d d\ „«Si fdd d d\ ^^ds^f d d d d 
dx dx dx dx) dx y ^ dx^ dx~A^ dx^'^ dx) ^ dx \A dx dx dx^dx 
+ 
.ds, dsg d'^Sg d'^s,\ d 
4—^ ^ — S, 77:# — S, -r4 5 -- 
dx dx 
ds 
y dx dx 
dx^ 
d'^s 
dx^ 
dx‘^ J dx 
d^Sg\ d 
^l[^)^'Tx 
,( A^s ds^ <^^Si d'^s\ d 
' y dx dx dx"^ dx'^) dx' 
Suppose that the above law holds good for m factors ; then 
1.2 .. mss, •■ *.£ = 2(sZ-(m-l)|)(s.Z_(m-2)^') ..s,._ 
d 
' dx' 
wdiere 2 indicates that the sum of all products formed by the interchanges of the suf- 
fixes 0, 1, . . m — 1, is to be taken. Then 
