510 
Proceedings of the Royal Society of Edinburgh. [Sess. 
which is the relation desired. If <p x does not involve f x , cp 2 does not in- 
volve f x or / 2 , (p s does not involve f x or / 2 or / 3 , and so on,* the determinant 
in the denominator takes the value ( — l) n , and the relation becomes one of 
equality. 
The last section deals with the theorem regarding the change of vari- 
ables in multiple integrals,— a theorem which in the ten years from Jacobi’s 
memoir to Bertrand’s had been discussed by Boole f and Dienger.J 
Spottiswoode, W. (1851, 1853). 
[Elementary Theorems relating to Determinants, . . . viii-f 
63 pp., London. Second edition as an article in Crelle’s Journ., 
li. pp. 209-271, 328-381.] 
Spottiswoode has a special chapter (§ x., pp. 51-57) headed “ On 
Functional Determinants,” its contents being a selection of Jacobi’s 
theorems in unimproved form and a reprint of the first three paragraphs of 
Cayley’s paper of 1847. In the second edition (§ ix., pp. 338-343) there is 
no change, save that the extract from Cayley is left out. 
Sylvester, J. J. (1853, June). 
[On a theory of the syzygetic relations .... Philos. Trans. R. Soc. 
London, cxliii. pp. 407-548 : or Collected Math. Papers, i. pp. 
429-586.] 
The only interest of this long and important memoir in the present 
connection lies in the fact that Sylvester at page 476 of it uses for the 
first time the term Jacobian and the symbolism J (/, g). His words are 
“ J indicates the Jacobian of the given functions f , g , ... . meaning thereby 
the functional determinant of Jacobi.” 
Donkin, W. F. (1854, February). 
[On a class of differential equations, including those which occur in 
dynamical problems, Part I. Philos. Trans. R. Soc. London, cxliv. 
pp. 71-113.] 
It is only the first four pages of Donkin’s memoir that concern us, 
these being introductory and referring to properties of a set of n functions 
* Or iif n be not involved in the 0 n , neither f n nor/„_i involved in Jn-\ , and so on. 
f Cambridge Math. Journ , iv. (1843), pp. 20-28. 
X Archiv d. Math. u. Pliys., x. (1847), pp. 417-421. 
