%||i Mr. 6. B. Jerrard vn n Method of Tnm^fqrming Equations 



the development of whicL can be effected with all the facility 

 which attaches to the use of the multinomial theorem. If, for 

 example, we wish to expand the function in question according 

 to the descending powers of Q^ we shall instantly find 



+ 6.(aP + 7l^+ .. +^Lr. 

 I shall now, pursuing the method given in my Mathematical 

 Researches, examine anew some of the consequences which flow 

 from equation {a). 



Problem I. — To take away the second, third, and fourth terms 

 at once from the general equation of the mth degree. 



4. It appears from equation («), that in order that A'p A'^ A'a, 

 the coefficients of the 2nd, 3rd, and 4th terms of the transformed 

 equation in g, may vanish simultaneously when 



y = P + Q + Ra?2_|. _ ^I^X^ 



we must find such values of P, Q, R, . . L as will satisfy the 

 equations 



@.(0P + 1Q + 2R+ ,. +XL) = 0, 



©.(0P + 1Q + 2R+ .. +XL)2=:0, 



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



Now we know from the theory of elimination, that in fulfilling 

 these three conditions we shall, in general, be conducted to a 

 final equation of I . 2 . 3 or 6 dimensions. Here, then, we seem 

 to be stopped. But when the series for y rises to the fourth 

 power of ir, the difficulty may be eluded in the following manner. 



Mode of solution when X=4. 



6, Since in the equation for a? we are permitted to assume 

 A, =0, A2=0, it is clear that ©1 and ©1^ may both of them 

 be made to vanish. Hence on considering that the first two of 

 the equations of condition may, when \=4, take the forms 



@1Q + 

 @.(0P-f2R-r3S + 4T)=0, 



and Sl^QH 



2S1.(0P4-2R + 3S + 4T)Q + 

 @.(OP-f-21l + 3S + 4T)*=^0, 



