ME. W. H. L. EUSSELL ON THE CAECULUS OF SYMBOLS. 
7.B 
— ^0 + + CoTT^ + ^’cc. , 
= Ai + BiT-j- CiT'* + &C. , 
and substitute in the above equation, and equate the resulting coefficients of t to zero. 
We shall thus be furnished with equations for determining the values of Ao, Bg, &c., 
Ai, Bi, &c. in all cases in which the above symbolical function is capable of resolution. 
We thus obtain the values of of the symbohcal quotient. We next 
ascertain if the symbolical quotient admits of an internal factor, and repeating the pro- 
cess we at length resolve the above symbolical function into factors of the form 
To determine the condition that shall divide the symbolical function 
1 (^) + f <P«-2(’r) + &c. -f ^<pi(7r) + <po(5r) 
externally without a remainder, 
W + + &C. + + §<Pi(7r) + <Po('^) 
e-{<p.-(x) - 1^ 7: _ ; j <p.(x)} + 
/ X 4'oi^ + n-l) ] _ f ^I/o(^ + ?^-2) / + + r, 
r^-4cr. ^o(^ + »-2) , vt o(7i+w-2)4/o(^ + n-l ) , J , „ 
where the sjunbolical quotient is 
vi/o(7r + W-l) 
^^{■!r + n — l)~'^ \4/j(7r + n— 2) 4/i(^ + w— 2)\I/,(7r + M— 1) 
_L,n-3j ?n-2^ 4/o(T + ra-2) s ■ ^|/o(7^ + n - 2Ho(7r + w - 1 ) 'll 
|ij/,(7r + n — 3) 4'i(’r + « — 3)^1/, (’r + «~2) + n — 3)4', (^ + w — 2)4'i(’r + ?2—l)^”^^^j'f’ 
The required condition is found by equating the remainder to zero : whence we have 
cf, r^'i_M[),r (.a ^oWVa + 1)4^o(^ + 2) / n , o 
^ 4^iW^'' ^'^4^i(7r)4'i(’r+ 1)^^^ ^ 4^j(7r)4'i(7r+l)4'i(’r + 2)^3( ) + 
I 4^o(^)4^o(^ + l)^o(’r + 2) ...4^o(7r + »-l) 
— 4'i(w)4'i(’''+ 1)4 'i(’!’ + 2) . . — ^ 
In the next investigation we shall suppose the symbolical function arranged in powers 
of (t) instead of powers of (e). To determine the condition that 4'i{§)‘^-\-'4^o{§) be 
an internal factor of the symbolical function 
+ <?2(?)’r" + <Pi{§)^ + M§)’ 
