Mr. Jerrard on the Transformation qf Equations. 479 



to A, and the other quantities which were considered as un- 

 known 



ka* + ka*- 1 + k^- 2 ... + k, 



it was necessary not only to make 



1 2 



K = 0, K a 0, ... K = 0, 



where, as is expressed by the indices 1, 2, ... n 9 the unknown 

 quantities rose to the 1st, 2nd, ... nth degrees respectively, 



but also to make 



o 



K = 0, 



into which none of these quantities, however numerous they 

 might be, would ever enter. By means of previous transfor- 

 mations I could in a few instances succeed in finding as many 

 equations 



K' = 0, K" = 0, ... 



as there were quantities to be detached. But when the num- 

 ber of these was increased beyond a certain very narrow 

 limit, it became a problem of greater difficulty to effect the 

 preparatory transformation than to solve the problem with 

 which I had set out. At length by means of a coalition of 

 certain of the unknown quantities A' 9 A n , A n, 9 ...l succeeded in 

 forming a development all the coefficients of which should 

 contain unknown quantities. And from this time I had no 

 further difficulty in arriving at the remarkable theorem an- 

 nounced at the end of the third part of the Researches, 

 that any number- of general algebraic functions which are of 

 n' 9 n" 9 n' n ... dimensions relatively to an assignable number of 

 unknown quantities contained in them 9 can be made simulta- 

 neously equal to zero, by means qf equations of ... 9 n' 9 ... 9 n n 9 ..., 

 w' w ... dimensions*. 



I was thus enabled to perceive that the general equation of 

 the mth degree might be reduced to the form y m + K'y m ~ n ... 

 + V' = 0, without the aid of an equation of more than 

 (n— 1) dimensions. I found too that when m 9 A, B, C, ... V 



were indeterminatef, neither would the expression — occur 



* A demonstration of this theorem, very nearly according with that 

 which had suggested itself to my own mind, has lately been sent in a 

 letter to my brother Dr. Jerrard, by V. F. Hovenden, Esq., late Fellow of 

 Trinity College, Cambridge. 



♦ In the Supplement to Part III. of the Mathematical Researches, I 

 speak of reducing the general equation of the fifth degree to De Moivre*s 

 form. But the problem which I perceived to be solved when the non- 



