312 



Mr. A. R. Forsyth. Invariants, fyc, [Jan. 12, 



and the expressions of the simplest invariants are 



Q _ p op' J 6 p" 8 5n + 7 p2 



e - P _ 5 p '+^p "_ 5 p «' J£ 7 JL±}Z P e 

 ©5-^6 2 4 + 7 3 7 2 7 w+1 2 3 ' 



similar expressions being obtained for 9 6 and 7 . The expression for 

 each of the n — 2 invariants of this class is shown to consist of two 

 parts, one of which is linear in the coefficients and their derivatives, 

 the other of which is not linear but every term contains as a factor 

 either P 2 or some derivative of P 2 . 



It is then proved that there is an implicitly general form of the 

 equation for which both Q T and Q 2 vanish ; this form, taken as the 

 canonical form, is obtainable (as is known from earlier investigations) 

 by the previous determination of the multiplier of the dependent 

 variable and by the determination of the independent variable from 

 the equation 



^•I = »TI Ps! ' 



d?6 3 



or its equivalent, dx 2 ~^ n + l ^ 2 ^ = ^' 



where z r = 0~ 2 . 



For this canonical form of equation the expressions of the foregoing 

 n — 2 invariants are given in the form 



r = <r-S JrC) 



e, = Qo-+i* r 2 (-iy*, )(r -^=- r , 



where »i >0 - is unity and, for values of r greater than 1, 



0-1) (<r-2)2(>- 3)2 . . . (a- r + (*-r) 



&r, or — 



2 . 3 . . . r (2<r-3) (2.T-4) . . . (2<r-r-l)' 



These invariants are called priminvariants. 



The proof of these results occupies the second section of the memoir. 

 The first section is devoted to a short historical sketch of the growth 

 of the subject, reference being made to the investigations of Cockle, 

 Laguerre, Brioschi, Malet, and, especially, of Halphen, all of whom 

 have, so far as concerns the theory of forms, discussed either semin- 

 variants only or, with the single exception of 3 for the wtic, in- 



