TRANSACTIONS OF THE SECTIONS. 13 
The complementary function here is Aa,”+B8,7+Cy,”, a,, 8,, y, being the cube 
roots of unity; and it is to be observed that we can easily express it in Cayley’s 
form as a prime circulator, for it may obviously be written in the form A3, 
+B3,,_,+C3,_9, viz. (A, B, C) circlor3,. And since 3,+8,_)+3,_» is constant 
(t. e. independent of .v), we can always, by assigning a proper value to the constant 
term, make A+B-+C=0, and so take the complementary function to be P+(Q, 
R,8) per3,, where P, Q, R, 8 are to be determined by the conditions 8°=1, 3'=1, 
3’=2, For the particular integral we find 
1 1 
U,=T- E_1 (2x+9) +4 ‘ E_1 (1,- 1) per 2au1. 
The first term is readily obtained in the ordinary way by replacing E by 1+A 
and expanding in ascending powers of A; and the second term 
1 t 
=}. wee + similar function of 8} 
—1 a* 
= =e ; 5 +similar function of 8 
[EEN SS 1,0) circlor2?. =1(2 - 9 
=#(5 +5 = 7 (1,9) circlor2, =} (2, 0) circlor2,, 
which differs only from }(1,—1) per 2, by a constant (viz. +). Thus we have 
3, =P, 2, 3)r= 7 {62° +36r+P+9(1,—1) per 2,4+(Q,R,-—Q-R) per3,t; 
and by putting 2=0, 1, 2, we find P=47, Q=16, R= —8, agreeing with Cayley’s 
result. 
From the derivations of a‘ we find 44=1°+2?+3'!+-4°; so that the differential 
equation is 
(E = 1)u,=3*t Tpgrt2 1 
= 7, {92° + 8404287 +18(1,—1) per 2, ,9+9(1,-1)per2,4, 
+8(2,-1,-1) per3,,). 
The complementary function is (A, B, C, D) circlor 4,, which a little consideration 
shows may be written in the form P+(Q,—Q) per 2,+(R, S,—R,—S) per 4,; so 
that in finding the particular integral we are not to calculate the terms of the form 
P+(Q,—Q) per2,. The algebraical part of the particular integral is easily found, 
by operating on the first three terms with (E‘—1)—1, to be 535 (2a°+30a+1352), 
The terms with period 2 (omitting the coefficient #,) give 
1 2a*—a—1 a* 
: a” 
Fay (20° -2a+a—1) 5 +Ke. = Ep ae Rp + &e., 
*which takes the form o ; so that we have a term included in the complementary 
function + term found by differentiating the numerator and denominator, the 
latter being 
Ge=1'—ar" a* Toa,” “& Fy 
=7 Lt oa Dy +&e. =2 4 . 2 +&e. =4(1,—1) per. 2. 
The term with period 3 gives 
2a,—1—a,~} a,* 2a,—1—a,~' 
wv «& 
— Cp. ‘A = —1 ay" 
“eae + be = iy + &e. =(2-+a,71) 4 &e, 
; = (2, 1,0) circlor 3,=const+(1,0,—1) per3,. 
Thus 
| 4° =P(1, 2,3, d)a= 515 {20° +802° +13524+P +(92+Q)(1, —1) per 2; 
+82(1,0,—1) per 3,+-(R, 8, —R,—S8) per 4,}5 
