Mr. J. Cockle on Multiplicity of Values. 4*4*7 



Equations*, an rc-ary m-ic function. Also let 



dw dw dw , r 



and, in general, 



( d d d V . N 



Then if o-(w) be symmetric, I call w a hyposymmetric function. 

 If a r (w) be the first of the functions symmetric with respect to y } 

 I call w an ( unsymmetric function of the (r— l)th class/ When 

 none of the functions cr is symmetric, w is ' unsymmetric/ 



XVI. Let w' be what w becomes when each of the quantities 

 y receives an equal increment y' } and let 



X (y ! ) = cr{w)y' + *>) |^ + &c. ; 

 then 



w'=w+ x(y% w=w'— %( y') • 



XVII. If the substitution of —y r for y 1 change w' into w ri 

 we have 



w=w r -x(-yr)=p( r )> 



for all values of r from 1 to n, both inclusive. Hence 

 w=p(l)=p(2)=.=p(n). 



XVIII. But, when w is hyposymmetric, % and w r and p are 

 respectively commetric ; and we see that, in such case, the values 

 of w are distributed in groups each involving n equal functions. 

 The n identical expressions p{\), /o(2), &c. thus become represen- 

 tatives of n changes of the same function w, and consequently 

 we know that the number of values of a hyposymmetric function 



* The First and Second Parts of that Analysis appeared in the 32nd and 

 3/th volumes respectively of the third series of this Journal. In Part I. 

 (p. 367) the reference to "vol. ii." (of the Camb. Math. Journ.) should be 

 changed to vol. i. In Part II; art. (19.) (p. 503) reference is made to 

 page 34. of vol. 1. of the Mechanics' Magazine. The " First Solution" of 

 the page last cited should be corrected by means of the investigations in 

 paragraph XXXI. of the last of my series of papers on the Method of 

 Vanishing Groups (see Camb. and Dub. Math. Journ. for February 1853, 

 pp. 55, 56), and, for the purposes of that solutfon> though not of the 

 ' Second ' and * Third/ the given quadratics must be considered as of the 

 ninth order. Each Part of the Analysis was given in the form of a letter 

 to the late eminent geometer Thomas Stephens Davies. To the first, Pro- 

 fessor Davies appended notes of his own, which enhance such value as it 

 may happen to possess. 



My " Notes on the Theory of Algebraic Equations/' adverted to in the 

 Analysis, are comprised in three series, of which the first appeared in the 

 46th volume of the Mechanics' Magazine. The second is contained in the 

 48th and 49th volumes of the same journal, and the third and concluding 

 series will be found in its 52nd, 53rd and 55th volumes. 



My Hotcb Algebraicce are printed in the 47th, 48th, 49th, and 50th 

 volumes of the Mechanics' Magazine. 



