362 Mr. J. Cockle on the Theory of Equations 



{e) In the equation 



make m = 2 and m = \ successively. Then we find 



and 



v^ being arbitrary. Let u vanish ; ^, ^'i and ^'j vanish also. 

 And, in the Eulerian form («), v^ remains undetermined. (Com- 

 pare Jerrard, loc. cit. p. 549.) 



(/) If I add that each form of ^' — ?^" gives 

 ^=0t + 0u-^0v = 0, 

 and that f'g and ^'3 are, respectively, the_^ and/^ of Mr, Jer- 

 rard {vide loc. cit. pp. 552 and 557), our results, general and 

 particular, are seen to be confirmed. 



Section IV. 



26. On making 



$,'-'^ = [\+i%, ^'^^=-i% and S'"'=-i2r«, 

 we obtain 



Si^'^ =y, + byp + cy, + dy,=0,- 



^^^^=!/. + by« + cyp + dyy=0, 



d^''^=ys + by, + cy, + dyp = 0, |> . . . . (c) 



d^^) = y^ + bys + cy, + dy^ = 0. 



and (b) is superseded by a system in which the cycle a, 13, y, 8, e 

 runs through all its phases. These five equations, like those 

 from which they are derived, are, save in form, identical with 

 each other. 



27. Since the number of difiercnt cyclical arrangements of 

 five quantities is twenty-four, we may, by properly permuting 

 the roots in (a), construct twenty-four essentially different systems 

 corresponding to (c). 



28. It is not necessary actually to exhibit the twenty-three 

 remaining systems, but each of them possesses the properties 

 which we have proved to belong to (c), and conducts to one, and 

 only one, value of P. And, as these systems exhaust the modes 

 in which 4> can be made to vanish, the number of values of P 

 cannot exceed twenty-four. 



29. Using k, I, m, n, r to represent the suffixes a, ^, 7, h, e 

 in an arbitrary order, make 



^^''^=y,+by^ + cy^^dy=Q, 



