296 SIXTH REPORT — 1836. 



one term may be taken away from the general equation (1) ; that 

 general equation being changed into another of the form 



Y = y + B'y-2 + Q)y^-^ -t- &c. = 0, . . . (7.) 



in which there occurs no term proportional to y'"^~^ , the condi- 

 tion 



A' = (8.) 



being satisfiedj and Tschirnhausen discovered that by assuming 



y=zf{x) = V-\-Qix + x\ (9.) 



and by determining P and Q so as to satisfy two equations 

 which can be assigned, and which are respectively of the first 

 and second degrees, it is possible to fulfil the condition 



B' = 0, (10.) 



along with the condition 



A' = 0, . . (8.) 

 and therefore to take away two terms at once from the general 

 equation of the m* degree ; or, in other words, to change that 

 equation (1) to the form 



Y = 3/'"+C'y""' + D>"'"* + &c. = 0, . . (11.) 



in which there occurs no term proportional either to y™~ or to 

 y^~^. But if we attempted to take away three terms at once, 

 from the general equation (1), or to reduce it to the form 



Y=3/'" + D'3/'"-^+ E'^/"'-^ + &c. = 0, . . (12.) 



(in which there occurs no term proportional to 3/'""^ y™~^j or 

 y^~^,) by assuming, according to the same analogy, 



,y = P + Ua; -f Rx^ + a;3, .... (13.) 



and then determining the three coefficients P, Q,, R, so as to 

 satisfy the three conditions 



A' = 0, . . (8.) 



B' = 0, . . (10.) 



and 



C' = 0, (14.) 



we should be conducted, by the law (5) of the composition of 

 the coefficients A', B', C, to a system of three equations, of the 

 1st, 2nd, and 3rd degrees, between the three coefficients P, Q,R ; 

 and consequently, by elimination, in general, to a final equation 

 of the 6th degree, which the known methods are unable to re- 

 solve. Still less could we take away, in the present state of 



