38 
MR. W. SPOTTISWOODE OX AX EXTEXDED FORM OF 
Again, 
SS1S2 
dx^ dy 
d d^ 
dso 
d^ 
— ^ dx^'^^dxdy 
dx'"^'" dx dy 
" 760 
dx 
dxdy 
ds d"^ 
d 
^2 
ds„ 
d^ 
“ dx'"''"‘^ dxdy~^ 
‘ dx"'^'" dx dy 
■ dx^^' 
dx dy 
ds d^ 
£?Sl ^ 
„ > 
d 
d^ 
dx'"'^'"^ dx dy 
dx'^-^^dxdy^ 
dx dt/ 
whence 
^ d^ f d ^ds\^ 
y^y— [s 2s, S 2 
'\^dx dx J ^ dxdy ’ 
d^ . . 
and consequently the expression GsSiS^ J^y admits of three equivalents, as follows ; 
</3 
6s Si S 2 
dx^ dy 
/ d ^ds\/ d ds^\ d / d £)ds\ / d dsi^X d 
\ ^ dyj y * dx dx j'^dx'^^dy ~‘dyy y ^ dx dx J ^ dx 
/ d dsy\ / d tfegX ^d / d / d ds\ 
y^' dy dy / y ^ dx dx J ^ dx' y^' dy ~^dy y y ^ tfe dxj 
' y ^ dy ^dy )\dx dx) ^ dx'\ ^ dy dy )\ ' dx dx ) 
d ds{\ d / d ^^ds\/ d d 
dy dy ) dx'\dx dx) y ^ dy dy ) dx 
d ,/ d iydS]\ / d ds \ d 
^ dx dx “‘dx )\dy dy) dx 
= f s^-2~ 
dx dx 
dx 
dx 
+ ( S. 
dx 
d 
dx ~‘dx 
d ds,2 
^'^dy~dy 
d ds . ", I 
d 
^ dx 
9^ 
-‘dx 
d dsA d 
dy~ dy 
^dx 
— S 
dx 
d 
I ds 
‘ dx 
'‘"dx 
d ds^ . „ 
dx dxj^'^dy' 
S--2- 
dx "" dx 
dsr 
dx dx ) * ' ay 
dy 
d dsA ^ ^ 9^*i\ / , ^ , 
^^dx dx / y ‘ </a;' ""dx / y dx dx/^~ 
ds\ d ,( d (yds< 2 \ / d d 
dx)^'' dy' y^^ dx -‘dx J y"^* dx dx J * dy 
d^ 
d^ 
In short, in order to change the expression for GsSjS.^^ into that for OsSiS.j^ ^'g^^ 
we 
