14. nEPORt—1874, - 
and since 4°=1, 4'=1, 4°=2, 4?=3, by making #=0, 1, 2, 3, we find P=175, Q=45, 
R=36, and S=0, agreeing with Cayley’s value. 
The general method of treatment is now clear, and it is unnecessary in this abstract 
to proceed further. Thus for 5” we should have the relation 5° = 14+42'+435?+44' +", 
the law being evident; and it will have been seen that by the use of prime circu- 
lators (in which the sum of the coefficients is zero) the result is exhibited in the 
most convenient form, as when so expressed it may be written (A+Bo-!4Co-* 
+....)o7+&c., o being an nth root of unity; and the operation (BM*1—1)7! is 
performed at once on the circulator as it stands, since a—1 is a factor of the coef- 
ficient of w, It will also be remarked that we can always express (A,, A,,.... 
A,_}) circlor a, as a series of ‘prime circulators, one for each factor of a— having 
the same number of constants; and that whenever the complementary function so 
expressed contains a term identical in form with one which already appears in the 
equation of differences, we shall (as is known from the theory of such equations) 
have to expand or differentiate, and so obtain a term with a new form of coefficient 
to the same prime circulator. Since x"=1"—!+2"—?,...+n°, we see that, to de- 
termine the partitions of 2 into the x elements 1, 2,....2, we should require to sub- 
stitute the values of (n—1)**4, (n—2)**?,,,.2°*"-?; so that if m were large the 
work would be laborious. 
There is an interesting class of questions which arise in connexion with Arbogast’s 
derivations, and which admit of solution by the principles explained above. If we 
consider, for example, the second derivation of a‘, viz. a°c and a°b’, we notice that 
a’c in any succeeding derivation can never give rise to more than one term, while 
a°b? gives rise to a*be and ab’. The terms in any derivation therefore are of two 
kinds, viz. are (1) extinct or sterile terms which merely continue to give rise in each 
derivation to one term of the same type as themselves, or (2) active or prolific terms 
which will give rise to two terms in some subsequent derivation. Thus in the 
fourth derivation of a* the active terms are a*c’, ab*c, and b*, while ae and a*bd are 
extinct. In fact all terms are extinct except those in which the last letter is raised 
to a power, or in which the last two letters are consecutive. 
Suppose it now required to find the number of extinct terms in the xth derivation 
of a; let 2* denote this number, then, as before, 2=1'+2° and 2**?-2*=1, but 
here 2°=0 and 2'=0, Solving and determining the constants, we have 
2" =1{27—14(1,—1) per2,}. 
Let 3° denote the number of extinct terms in the zth derivation of a’, then 3° 
=1°+2'+3°, and the equation of differences is 
grr —3*=}{2e+5+(1,— 1) per 2541 ; 
eee atieerating and determining the constants from the conditions 3°=0, 
ed he) 
3° = 7p {62° +122 — 1+9(1, —1) per 2,-+-8(—1,—1, 2) per 3,}. 
If 4* denote the number of extinct terms in the xth derivation of a‘, we haye 
Arts 4214 -98+2 4 get] 
a & 
= },{6n?+60r4143 +9 (2a? —a—1) © +-&e.48(—a,-142a,-V) > +&e.t, 
whence 
ry A = 2 2a — 1 _ aa} a® 
4? ots {(Z—345A)(62?+6024143)4 362-— 7 — 5 +k&e. 
4 
+-52(—1~2a,~1) +&e.+compl. funct.} 
= ype {2u?+ 182°+39e+02(1,—1) per 2,4+32(—1,—2, 0) cirelor3,4-A 
+B(1,—1) per2,-+(C, D,—C,—D) per 4}, 
