216 
ON waring’s formula. 
[925 
where 
fx = 
c d e 
&c. 
rp OP* OP'- 
lAj \AJ \X/ 
Hence, by Lagrange’s theorem, 
+ \mu m 1 
ode VI' 1 
.5 + «* 4 V + -J] H2 
)T 
" 1 
1.2.3 
4- &c., 
where the accents denote differentiations in regard to u. This is 
+ \m {(m — 3) c 2 u m ~ i + (m — 4) 2cdu m ~ 5 + (m — 5) (d 2 + 2ce) u m ~ 6 + .. .} 
— £ra {(m — 4) (m — 5) c 3 u m ~ 3 + ...} 
+ &c. 
+ u m ~ 2 . — me 
+ u m ~ 3 . — md 
+ . — me + \m. m — 3. c 2 
+ u m ~ 5 . — mf + \m. m — 4.2cd 
+ u m ~ 6 . — mg + \m. m — 5 . (d 2 + 2ce) — ^m. m — 4. m - 5 . c 3 
+ &c., 
which, putting therein u= — b and multiplying each side by (—) m , is the before- 
mentioned formula for (— ) w $a m : in that formula the series being continued only so 
far as the exponent of b is not negative. 
I notice also that we cannot easily, by means of the known formula 
Sa m /3P = Sa m . SaP - 8a m+ P, 
deduce an expression for Sa m 8 p ‘- in fact, forming the product of the series for Sa m , 
Sa p respectively, this product is identically equal to the series for Sa m+P , or we seem 
to obtain 0 = Sa. m S<x p — Sa m+P ] to obtain the correct formula, we have to take each 
of the three series only so far as the exponent of b therein respectively is not negative: 
and it is not easy to see how the resulting formula is to be expressed.