EQUATION AND ITS TRANSFORMATIONS. 
773 
(2°) p = a positive integer, 
P=E=U=i(F+R'), 
Q=S=Y=-(Pd-P'); 
(3°) p = a negative integer, 
C 3 
1 
up 
7-1 cA 
II 
£ 
II 
Ph 
II 
PP 
Q = S=V=i(Q'+S / ); 
6. If we suppose the series always to terminate directly a zero factor appears in 
numerator (so that P', Q', Pd, S' are now denoted by P, Q, Pt, S), the relations are 
(1°) p not = an integer, 
P=Pt=U, Q = S = V; 
(2°) p = a positive integer, 
rO 
II 
to 
II 
II 
"po 
3 
U=i(P+E); 
(3°) p = a negative integer, 
P=R=U = I(S-Q), 
V=i(Q+S); 
The change of form of the relations, which in this mode of statement appears so 
remarkable, does not, as we have seen, occur if the series be supposed to extend to 
infinity in all cases. 
It may be observed that it is clear from the manner in which the series were 
obtained in arts. 1 and 2 that we are always at liberty to stop at the term immediately 
preceding the first term containing a zero factor in the numerator, as this finite portion 
of the series satisfies the differential equation, and that the second series obtained by 
allowing the terms to recommence and to extend to infinity also satisfies the differential 
equation. 
The phrase “term preceding the first term containing a zero factor in the numerator ” 
has been used in preference to “term preceding the first zero term” in order to 
include the cases of p=0 or p= — 1, in which no zero term occurs. 
7, It was shown in art. 5 that 
V ffQ-i) gW_ p(p-l)(p-2) a 3 * 3 \ 
P PO—k) 2! p(p-i)(p~ 1) 3! / 
1 — 
_1 92 
2 
i-A 
CCX 
1 
(p-i)(p-i) 2 4 .2! (p-W(p-iXp-i) 2 6 .3! 
T&c. 
