from which, on eliminating jo, r, s, there will finally til^uFt '^ 



a cannot therefore vanish without inducing a relation among 

 the coefficients A3, A4, . . A,„ . 



Having thus shown that S will not assume the form 



. . (l-l)H' 



it immediately follows that not only P and R^ but also Q and y 

 will in general be determinate in value. 



T proceed now to show how the problem may be solved when 

 the series for y does not extend beyond the third power of a?, iti 



^ .j,^ Mode of solution when X=3. 



7. Here L = S. The equations of condition will accordingly 

 become , , „ .u ^' .^ 



. @.(0P + lQ + 2R + 3Sr=0, p^^«f^^^ 



@. (0P + 1Q + 2R + 3S)3=0. J 



.Assuming, as before, Ai = 0, A2=0, let ^ ..^^^ ^00^ 



@l.(0P + 2R + 3S) = O*; ' ^ -^ ^ ^ 



then, on eliminating P and R, the second of the equations {e) 

 will pT*esent itself in the form 



2DQ + «S2 + 2bDS + cD2 = 0; . , . . (e^ 



in which /tD is indeterminate, and b, c are rational functions of 

 A3, A4, . . A^, the coefficients of the equation in r. 



It may be proved very readily that both b and c will in general 



be different from zero. • .,^g 



, In effect, if we observe that ^ 



^,^^' @.(0P + 1Q + 2R + 3S)2= :Z 



0Q2 + 2DQ + @ . (0P-f-2R + 3S)S -m^ 



we shall find, if P =p^ -tppj R =rS + ?'p, • ■ 



a = @ . (0/9 + 2r + 3*)^ ^^^^ 



b = @.(0;? + 2r + 35)(Ojo^ + 2r^), . 



c = @.(0p, + 2r,)2; 

 p, r, s being already known, and p^, r^ being respectively equal 



Whence it appears, that, unless certain assignable relations 

 exist among the coefficients of the equation in x, both b and c 

 will be composed of finite non-evanescent factors. 

 * ^ is the Hebrew letter Mem. 



Phil Mag, S. 4. Vol. 5. No. 33. May 1853. 2 B 



