302 Proceedings of the Koyal Society of Edinburgh. [Sess. 
Lastly, denoting the cofactors of f {n) , fp , in 
y + f ' d fl ' d % _ m d fn 
— * dx ox 2 ’ ’ ' dx n 
by W 0 , W x , . . . , W n he announces the set of results 
2 ± w 0 w;wj . . . wy = w” , 
2 ± WoWiWj . . . Wfe 1 ' = ( - 1 )"W"-y„ , 
2,±w 0 w;w; . . . wir_t’ = (- i) 2 ”W»- 5 .2 ±f~Y n ’ . 
2 ± w„w;w: . . . wir-t 1 = ( - -2 ±/«- 2 /»-)/: , 
2 ± w„w; = ( - i)'-«»w .2 ±f,fs ■ ■ ■ / ,n - !| , 
w 0 = (-i 
In regard to these we may note in passing that as W 0 , W 1? . . . , W n are 
themselves Wronskians, the name “ Compound Wronskian ” would not be 
inappropriate for the determinants on the left. Also, that in form the 
results bear a resemblance to those included in Jacobi’s theorem regarding 
any minor of the adjugate of a general determinant. 
Hesse, O. (1857). 
[Ueber die Criterien des Maximums und Minimums der einfachen 
Integraie. Crelle’s Journ., liv. pp. 227-273: or Werlce, pp. 
413-467.] 
In the course of his investigation Hesse pauses (p. 249) to enunciate the 
result which we have numbered (4) in dealing with Christoffel’s paper. He 
also gives Christoffel’s fifth result, having arrived at it, however, by a 
different method, namely, by the repeated use of the immediately preceding 
result. Thus, in the case of the 3rd order, the first part of his procedure 
would be 
U IV u 
V V v" 
W IV w" 
