380 Mr. J. Cockle on Transcendental and 
the n—2 conditions 
AA, +B, +..+v0,4+G,=0, 
AA, +uHB,+..+vC,+G6,=0, 
NAn—2 + MBy_2+ ++ +VOn2+ Gn-2=0, 
we shall arrive at a linear differential equation, 
d”®— a d™—24 d?ax 
da"- de gees B+ ae 
5 ad oe 
da 
For, when the above conditions are satisfied, 2"-!, 2*-?,.. x?, 
all the powers of x in short, save a itself, disappear; and 
v,..#, A, L, M are each of them known functions of a, inasmuch 
as A, B,..C, G are known functions of a. 
Thus the roots of any equation whereof the coefficients are 
functions of only one parameter may be expressed in terms of 
algebraic, circular, or logarithmic functions, and of integrals of 
algebraic functions. These integrals depend upon the quantity M. 
To a form involying only one parameter, Mr. Jerrard has shown 
that the general quintic may be reduced. Its resolvent sextic 
may also be reduced to the same form. 
Mr. Jerrard’s memorable discoveries also show that the general 
sextic may be regarded as involving two parameters only. The 
general sextic leads us to the consideration of the equation 
dfe=Fr.da+fr.db=0, 
where a and b are the independent parameters, of which the co- 
efficients may be considered as functions, and 6 is the character- 
istic of the Calculus of Variations. 
If, in art. 62 of my “ Observations,” &c. (vol. xvii. p. 342), we 
take the suffixes of @ to the modulus 6, the equations become 
0,04 + 0.09 + O:05=Ys=P(x5)s 
8109+ 9205+ O;0,=Y1="(2)), 
0,03 + 0284+ 0;09=Ya=7 (22); 
6,4, + 0,03+ 0,05=¥3=7(2#3); 
@, 19 + 6365+ 8,0;=y4=7(2,). 
And this system has a certain relation to the formula 
6,2 Oar a 6424200245 a 424300244) 
which, taking the suffixes to the modulus 6, is, for all mtegral 
values of a, equal either to y, or to yy. But all the values of y 
are not thence evolved ; and in order to obtain a convenient repre- 
sentation of the system, I avail myself of certain cyclical forms 
