136 Mr. J. Cockle on the Method of Symmetric Products. 

 that in x by the relation 



y>- = Qo + Ql'^r + Qa^r + Qa^r + Q44, 



which is substantially the most general transformation that can 

 be adopted^ and let us, in the first instance, make Qq vanish. 

 The critical nature of tt justifies us in this course. 



66. We shall find 



there being six relations of this form. 



67. Although a want of hyposymmetry in e', e", &c. will 

 probably complicate the process by which we arrive at the equa- 

 tions corresponding to (/), such equations exist in all cases. The 

 argument of (63) will apply to i, e", &c. 



68. The functions e are, all of them, six-valued. The inves- 

 tigation of their properties vnll be aided by that of relations like 

 those which follow, and which seem to indicate that epimetric 

 and other unsjonmetric expressions admit of systematic discussion, 



69. Let S be the sign of symmetric and S of epimetric sum- 

 mation ; so that, in (39), we may make 



2' . 2/1^(2/22/5 + 2/32/4) =" = Sl(2/l''?/22/3)- 



70. Change the y's within the brackets into their correspond- 

 ing squares, and depress those without to the first power, and 

 we have 



S' . 2/1(2/2^2/5^ + 2/3 V) = s, (2/1 2/2 V) • 



71. We may take 



s,=s,('2):=S3(!«)=s,(")=s,(i=)=s„(«) 



as giving the law of suffixes*. 



73. The following relation, in which a singular case is given 

 by b = 2a, holds, 



73. We also have, omitting identical suffixes, 



-J- 2I4 • X^ Wc^ X^ -f- 2^ • X-^~Xc^^X^ ^^ yjj 



a result which admits of generahzation, and connects epimetrics 

 of various forms f. 



t The product of S, . Xi-x-^x^ into Sj . x^x^xi is the six-valued function 

 2 . XiX2Xs-\-'S, . XiX.{x^XiXi^-\-'S, . xi^x.^x^x^+ 



2/ . X^ \Xq "'2^4 ~r'**'5 X2X^~\~Xn **'5^4~r**4 *^3"'^/~l 



2 . Xi (3^3"'3^2"'?6 ~r 'f^i'Xj'X^ 4- x.2'X^x^ + x^x•,^X2), 

 in which the sequences occur iu known cycles. An outline of the history 



I 



