56 Eev. Kobert Harley. Prof. MaleCs Invariants [Dec. 18, 

 and therefore, making [i±= — ( _^ J {{n— Y)(n— 2) — 1}, we have 



Q,'+-^QiQi'-^+6Q 1 Q r (-\|{(»-lX»-2)-l)(! 1 ! 



n — 1 n — 6 \n — 1/ 



= 1_ |~p" + JL_P p ' -^iJjp + 6P1P0 

 T s|_ 1 n-1 1 1 n-3 3 1 2 



or, since T=(?^Y, we may write 



Q,« + _ 6 Q 1 Q 1 '_?(!iZpQ, + 6Q 1 Q s -( 2 Y {(n _i )(n _2)_l}Q 1 3 



Q»" 



Pl - + 6 p P ,_^^) P +6P P / 2 y {(K _ 1)(B _ 2) _ 1}p 3 



n — 1 n — 6 \n—H 



_ _ _ _ 



P« 



From this form, which holds for all values of 11 greater than 3, we 

 can readily deduce Professor Malet's function J 2 ; for, making n = 4t, 

 we get 



QT + SQjQ, '-6Q. + 60,0,-^ P/' + 2P 1 P 1 '-6P„ + 6P ] P S -^P 1 » 



Q? = P? 



.... (J 2 ) 



Professor Malet (see p. 775) writes J 2 in the form 



dOP^-SGP^-lS^L-lOSP^ + lOSPg 



dx dx~ 



p? ~ 



which differs from the dexter of (J 2 ) by a numerical factor only. 



15. Prom the foregoing inquiry it appears that the first class of 

 functions H, Gr, 1 are given explicitly in the memoirs to which I 

 referred Professor Malet ; and that the second class I 1? I 2 , J 1? may be 

 derived from general forms contained in those memoirs by simply 

 assigning particular numerical values to n. Further, it is shown in 

 the last two articles that by an obvious extension of Sir James 

 Cockle's method we may obtain forms which include the functions J, 

 J 2 , J 3 as particular cases. Professor Malet's Classes of Invariants are 

 thus completely identified with Sir James Cockle's Criticoids. 



16. Professor Malet calls these functions invariants, but they would 

 seem to partake more of the nature of seminvariants or critical 

 functions. If we agree to call an expression critical when it remains 



