404 
PEOFESSOE CAYLEY ON THE TEAN SFOEMATION 
be replaced by u, v ; the change in the equation-systems is so easily made that it is not 
necessary here to write them down in the new form in u, v. 
10. The case a=0 has to be considered in the discussion of order, but we have thus 
only solutions which are to be rejected ; in the proper solutions a is not =0, and it may 
therefore for convenience be taken to be =1. We have then tr=u n -i-v. The last equation 
becomes therefore 
?(2r+l>^(i+20); 
or recollecting that 0 is connected with the multiplier M by the relation ^=1+2/3, 
that is, 
and substituting for 1 + 2/3 its value, the equation becomes 
that is, the first and last coefficients are 1, — , and the second and penultimate coeffi- 
cients are each expressed in terms of v, M. The cases n= 3, n = 5 are so far peculiar, 
that the only coefficients are «, 0, or a,/ 3, y; in the next case n= 7, the only coefficients 
are a, /3, y, &, and we have in this case all the coefficients expressed as above. 
The Q.k-form — Order of the Systems. — Article Nos. 11 to 22. 
11. In the general case, n an odd number, we have O and coefficients con- 
nected by a system of 1) equations of the form 
■~U ,— V' ’ 
where U, V, . . U', V', . . . are given quadric functions of the coefficients. Omitting the 
U V 
(0=), there remains a system of %(n— 1) equations of the form ^,=y,= . . or say 
( u, v, w,.. )=o, 
| U', V, W', . . | 
which determine the ratios a : ]3 : y . . . of the coefficients ; and to each set of ratios 
there corresponds a single value of O. The order of the system, or number of sets of 
ratios, is 1) . 2 K ” -1) , =(n-\- 1).2 K ” _3) ; and this is consequently the number of 
values of O, or order of the equation for the determination of O; viz. but for reduction 
the order in O of the Q^-modular equation would be ==(« + 1) . 2 i(M_3) . In the case 
n = 3, this is =4, which is right, but for any larger value of n the order is far too high ; 
in fact, assuming (as the case is) that the order is equal to the order in v of the uv- form, 
the order should for a prime value of n be =n-\- 1, and for a composite value not con- 
taining any square factor be = the sum of the divisors of n. I do hot attempt a general 
investigation, but confine myself to showing in what manner the reductions arise. 
