1 {('} Mr, J. Cockle on Multiplicity of Values. 



<*r{y) =yr\ «i(y) =y#6+ M* 



««(y) ^yiys+y^sy e s(y) -y^+y^y 

 «4(y) = y \Vi + M5) ' «8(y) = vm + yay* 



these relations, which result in 



se(y) = u and 5 = m = n, 



indicate that m, and consequently U 4 , is a pure epimetric function. 



XI. It would seem that t|t can be determined by ascertaining 

 the number of values which, e x being permanent, any one of the 

 other functions e can take. But, 



for eg and e^ ( j and for e 3 and e 4> I \ 



are the only admissible changes. Hence, whichever of these 

 four functions we consider, the number of interchanges is the 

 same, and we have 



e = 3, i|r(e) = 2 = e-l 

 . (/)(6)=6f(6)=6(e-l)=3x2 = 6, 



and the function u has only six values, a result arrived at by 

 M. J. A. Serret (Liouville, vol. xv. p. 62). 



XII. In fact, there exist the relations* 



/25\ /34\ f2345\ 



the last (epimetric) interchange being applied to the functions e 

 successively ; and, since any change applied to u will affect all 

 but one of the five functions e, the proper epimetric interchange 

 must have the same effect. Now there is only one such inter- 

 change, that just given, consequently yjr(e) is equal to 2. 



XIII. From any other of the functions 



«r(y) or yr 2 (y P y q +y 8 yt), 



we might have obtained the relations 



and have observed that the portions p, q and s, t of the epimetric 

 interchange have no effect on ce p , a q and ce 8i ce t respectively. 



XIV. The foregoing must be regarded as the merest opening 

 of the subject, in reference to a particular case. It will, however, 

 be at once remarked, that no such pure epimetric as that just 

 considered can occur when n \&'even. I shall now proceed to 

 view the subject of multiplicity under another aspect. 



XV. Let w be a function of the n quantities y and of the mth 

 degree, or, in the nomenclature of my Analysis of the Theory of 



* «( 25 )=Wy)+^))( 2 ?)+(^,(y)+«(y))( 34 ). 



