402 
ON RECIPROCANTS AND DIFFERENTIAL INVARIANTS. 
[943 
For the general proof, writing for a moment 
G 0 = fiA 0 , G x = (fi + v) A 1) C 2 = (/i + 2v) A 2 , &c., 
and similarly 
C' = fi'A 0 ', C' = {il' + v')A', Gi = (// + 2v')A', &c., 
where the accented symbols refer to the values v ; m', n), we have 
[O', v ; to', n').(>, to, n)], 
G 0 'da n , * 
Goda n 
- Goda n 
* O'd^, 
+ Gi'da n , +1 
+ G 1 da n+l 
+ Gida n+ 1 
+ G 2 d cln , +2 
+ G3a n+ 2 
+ oy n+2 
+ G A Ctn' -\-2 
4" Gn‘ ^Cln+n' 
b GrAan’-if-n 
"h ^n'+i *b G n +\ ^in'+11+1 5 
or observing that in the series G 0 , C lf G 2) ..., the first term that contains a n r is G n ', 
and the like as regards the series G,', (7/, C 2 , this is 
= {(Co'da n ,) G n > 
+ {(Co'd an , + G'd an , +i )C n ' +1 
+ {(C 0 'da n , + Ci'da n , +1 + G 2 'da n ’ +2 ) G n ’+2 
— (G 0 da n ) G n } da n+n , 
— (G 0 da n + G^) G n+1 } d an+n , +1 
~ (G 0 da n + Gida n+1 + G 2 da n+2 ) G' n+2 } da n+n , +2 
and it is thus of the form 
Ofd^ + cn 
a n+n'+1 
+ C"d a , 
'n-\-n'+2 
+ &C., 
i.e. the value of is = n + n. 
I confine myself to the comparison of the first and second coefficients: substituting 
for the C’s their expressions in terms of the J/s, we ought to have 
/j,'A 0 'da n , O + n'v) A n - - fxA 0 d an (/¿' + nv) A n ' = ¡i x A 0 ", 
(,¿Ao'dan’ + 0'+ ») A'da n ' +1 ) {/* + O' + !) A A n ’+1 
- (/xA 0 da n + (/a + v) A x da n+1 ) {A + (n + 1) v'} A' n+ 1 = (X + v 1 ) A 
Now assuming m 1 = to' + to — 1, and attending to the values 
11 1 
-a-0 — — u 0 , -¿1 0 — , u 0 , -0-0 — , , 
to . to . to + to —1 
A 1 = a 0 m ~ 1 a 1 , 
Ao = a 0 m ~ 1 a 2 + ^a 0 m ~ 2 a 1 2 , 
m'+m—i