420 
PEOFESSOE CAYLEY ON THE TEAN SFOEMATION 
and it was in this way that the equation for the case n — 15 was calculated. Observe 
that writing u— 1, we have (0+l) 3 (0— 1)=0, or say 0 y =0 2 =0 3 = — 1, 0 4 =+l. The 
equation in v thus becomes {(v— l) 5 (w + l)} 3 (y + l) 5 (y— 1) = 0, that is (v— l) l6 (v-bl) 8 =0. 
28. The case where n has a square factor is a little different ; thus n=. 9, the values are 
wwwi vww, 
1,3 , 9 , roots ; 
but here a being an imaginary cube root of unity, the second term denotes the three 
values, 
■/¥/(«). VWffiW \/2 
the first of which is =u , and is to be rejected; there remain 1 + 2 + 9, =12 roots, or 
the equation is of the order 12. 
Considering the equation as compounded of two cubic transformations, if the value of 
v for the first of these be 0, then we have 
0 4 +2Q 3 u 3 -2Qu-u 4 =O; 
and to each of the four values of 0 correspond the four values of v given by the equation 
v 4 +2v 3 0 3 —2v0— 0 4 =O. 
One of these values is however v= — u, since the ^-equation is satisfied on writing 
therein v= — u; hence, writing 
d 4 + 2 0 3 u 3 -20u-u i =(0-0 l )(0-0 2 )(0-0 3 )(0-0J, 
we have an equation 
(v 4 -\-2v 3 0\—2v0 l — 0\)(v 4 . . — 0 4 2 )(v 4 . . — 0l)(v 4 .. —01) — 0 , 
containing the factor (y + w) 4 , and which, divested hereof, gives the required modular 
equation of the order 12, which was in fact obtained in this manner. 
Observe that writing u=l we have (0+l) 3 (0— 1) = 0, or say 0 l =0 a =0 3 = — l,0 4 =l; 
the modular equation then becomes 
{(v — l) 3 (v + l)} 3 (y + l) 3 (w— 1 )-t-(w + 1) 4 =0, 
that is 
(y — l) 10 (v + l) 2 =0. 
The Multiplier Equation . — Article No. 29. 
29. The theory is in many respects analogous to that of the modular equation. To 
each value of v there corresponds a single value of M ; hence M, or what is the same 
thing is determined by an equation of the same order as v, viz. n being prime, the 
order is =n-\- 1. The last term of the equation is constant, and the other coefficients 
are rational and integral functions of u 8 , of a degree not exceeding \{n— 1) ; and not 
only so, but they are, n= 1 (mod. 4), rational and integral functions of u 8 (l — u 8 ), and 
«=3 (mod. 4), alternately of this form, and of the same form multiplied by the factor 
(1 — 2m 8 ). 
