REPORT ON CERTAIN BRANCHES OF ANALYSIS. 301 



and therefore 



X J-, 



k - X-i 



"*" + "^1 ~ ' k^-k ' 



There are therefore necessarily two roots of the equation or two 

 values of the symbols .r, a:{, x^, . . . Xn-\, such that x + x^ and 

 X Xj are real ; and therefore it is always possible, in an equation 

 whose dimensions are impariter par, to depress them by two 

 unities, so that the reduced equation may still possess rational 

 coefficients. 



If the number of symbols involved in the original problem be 

 2^ m, then the number of their binary combinations must be 

 9, m (2^ m — 1 ) or impariter par. It will immediately follow, from 

 what we have already proved, that there are two values of the 

 sum and product of the same symbols, which are either real or 

 of the form a -\- ^ V — \ \ and consequently the symbols them- 

 selves will admit of expression under a similar form *. 



If the dimensions of the original equation be 2^ m or £'* m, or 

 any one in an ascending series of orders oi parity, it may be re- 

 duced down to the next order of parity in a similar manner : and 

 under all circumstances it may be shown that there must be two 

 roots which are reducible to the form « + /3 -/ — 1 , where « and /3 

 real or zero ; and also in any equation of even dimensions, we 

 can reduce its dimensions successively by two unities, thus pro- 

 ducing a series of equations of successive or decreasing orders of 

 parity, in which we can demonstrate the existence of successive 

 pairs of roots of the required form until they are all exhausted. 



This mode of proving the composition of equations differs 

 chiefly from that which was noticed by Laplace, in his lectures 

 to the Ecole Normals in 1795f , in the form in which the ques- 

 tion is proposed. A certain number of symbols, representing 

 magnitudes with unknoum affections, are required to satisfy 



• I-et X + x' = r (cos tf + V — 1 sin ^) 



X x' = J (cos <p + V — 1 sin (p) 



X -\- x\- — A xx' ^= R-(cos 2;/' + \^ — 1 sin 2 ^l>) 



or *■ — x' =i R (cos ^Z' + -v^ — 1 sirn^) 



r cos tf + R cos ip (jr sin ^ + R sin ip) i 



X — - • H 2 V — 1 



= r' (cos ic + -v^ — 1 sin x) 



x' = r' (cos X — V — 1 sin x)- 

 t Lemons de I'Ecole Normale, torn. ii. 



