48 



Rev. Robert Harley. Prof. Malet's Invariants [Dec. 18, 



important part they play in the theory of algebraic equations; and 

 Mr. James Warren has also, in two papers,* given many illustrations 

 of their properties. 



7. In the article " Correlations of Analysis," cited above (see foot- 

 note*, p. 47), the following is the method employed for determining 

 the critical differential functions analogous to the critical functions of 

 algebra : — Let / Y be substituted for y in the linear differential 

 equation 



p -3fi 1 )" r ^ 



and let the transformed equation be 



(l, Qi, Q 2 > • • • Q ^| y ' 1 7 Y=0; 



then we have, by Leibnitz's theorem, 



Q„/=(l, P„ P 2 , . . . p-$J^ i)7. 



and therefore, making m=l, 2, 3, &c, successively, developing, and 

 denoting differentiations by accents, 



Q 2 /=/" + 2P 1 /' + P 2 / > 



Q 3 /=/'" + 3P 1 /" + 3P 2 /' + P 3 /, &c. 



The elimination of/, f, /", &c, by the process employed in dealing 

 with differential equations of the second order, gives 



Q 2 -Qi 2 -Qi'=p 2 -Pi 2 -p/ -.(H) 



2Q 1 3 -3Q 1 Q 2 + Q 3 -Q 1 "=2P 1 3-3P 1 P 2 + P 3 -P 1 " . . . (G) 



- 6Q n 4 + 12Q n 2 Q 2 — 4Q n Q 3 - 3Q 2 2 + Q 4 - Qr 



= -6P 1 4 +12P 1 2P 2 -4P 1 P 3 -3P 2 2 + P 4 -Pi'" .... (I) 



which are identical with Professor Malet's functions H, Gr, I respec- 

 tively. The method by which Professor Malet derives these functions 

 does not seem to me to differ in any essential particular from the 

 method followed by Sir James Cockle, as indicated above. In order 

 to determine the function I, the process is here carried a step further 

 than it is carried by its inventor in his earlier investigations ; but it 

 is proper to mention that the explicit form of that function is after- 



* Warren. " Illustrations of the Theory of Critical Functions," " Quarterly 

 Journal of Mathematics," vol. vi, pp. 231-237. " Illustrations of the Theory of 

 Seminvariants or Critical Functions," Part ii, ibid., pp. 372-382. 



