of the Fifth Degree. 357 



for /3, y, S, e respectively ; so that if the change be made on the 

 right and the substitutions on the left of the equations invol- 

 ving V (i. e. ^v), those equations will still be satisfied. Hence 

 we may write 



\ = l-\- i\ \=-i^- i, % = - ?• and ^vs = 0. 



In thus passing from 'v to ^v we have altered the order of the 

 equations (a) without otherwise affecting them. In the new 

 orderj however, i^ fills the place which i held in the old. 



Thirdly, let 



Vy = 0. 



The change, in (a), of i into P is equivalent to the substitution 

 of 8, /3, e, 7 for /3, <y, 8, e respectively. Hence, proceeding as in 

 the second case, we may write 



3^4 = 1 + i, \ = - i3 - i\ \ - - »4 and \=0. 



Fourthly, let v =0. 



Change i into i^, and /3, y, 8, e into e, S, y, /3 respectively, and 

 proceed in other respects as in the second and third cases. We 

 may write 



V.=l+i3, \=i-i^-.i^, \=-i2 and %=0. 

 Lastly, let X=—fx,a, then 



^v being connected with h by the equation 



All the values obtained are finite. 



11. Hence, the five equations 



2/a + (1+ i^)yp - {i-\-i^)yy-i%= 0, ^ 



2/a-(i' + i>^+(l+i)yi-»V. = 0. > . • . (b) 



cannot, so long as the system (a) remains unaltered, conduct to 

 more than one value of /»j, or, as we may designate it, P. 



Section II. 



12. The substitution in Y of y^, y^, yy, &c. for ya, y^ yc &c. 



respectively is expressed by Yi^"^**''' . . ) or, more simply, by 



^Vo^y^Vy ' 



