300 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and using “ cof ” for “ cofactor of ” we readily see that since 
we have 
- cof y { r x) cof yf 2) + cof y { s n cof y \ 2) 
(cof y ( r 2) ) 2 
W. cof (y [ r 1] y?~ 2) ) 
(cof 2 / 5 ^ 1) ) 2 
Integration of both sides with respect to x then gives 
and it is seen that for the cofactor of any element in the last row of W 
except the r th there is an expression in which the function y r never ex- 
plicitly occurs. It only remains then to take the well-known identity 
where I stands for the integral above written. 
Brioschi then proceeds to establish Malmsten’s theorem of 1849. 
[Ueber die lineare Abhangigkeit von Functionen einer einzigen 
Veranderlichen. Crelle’s Journ., lv. pp. 281-299.] 
The results directly bearing on the subject specified in the title of the 
paper are summed up in five propositions (pp. 293-294), the one comparable 
with Puiseux’s of 1851 being that If 2 ± f(x)*f' 1 (x) . . . f f (x) vanishes for 
all values of x from x = x 0 to x = x v where x 0 < x l5 then the functions 
f (x), f x (x), . . . , f n (x) are for those values linearly dependent In a con- 
cluding section are brought together (pp. 297-299) the properties of the 
determinants which had been used in reaching the said results. The first 
theorem is 
2 ±f( m ) • Mm + 1) . . . f n (m + n) = 2 ±f( m ) • AA( m ) • • • • & n f n (m) ( 1') 
0 = y 1 cof yf 1} + y 2 cof yf 1 1] + ... +y r cof yf~ l} + . . + y n cof y ( £ J) , 
write therein for coiy { I~ X) the substitute thus provided, and divide both 
sides by cof yf -1> ; for, this being done there results 
0 = + yJ-% + ••• +y r + • • • 
Christoffel, E. B. (1857). 
(1) 
