ON WRONSKI’S THEOREM. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xil. (1873), 
pp. 221—228.] 
The theorem, considered by the author as an answer to the question “ En quoi 
consistent les Mathématiques ? N’y aurait-il pas moyen d’embrasser par un seul problème, 
tous les problèmes de ces sciences et de résoudre généralement ce problème universel?” 
is given without demonstration in his Réfutation de la Théorie de Fonctions Analytiques 
de Lagrange, Paris, 1812, p. 30, and reproduced (with, I think, a demonstration) in the 
Philosophie de la Technie, Paris, 1815 ; and it is also stated and demonstrated in the 
Supplément à la Réforme de la Philosophie, Paris, 1847, p. cix et seq.; the theorem, 
but without a demonstration, is given in Montferriers Encyclopédie Mathématique (Paris, 
no date), t. III. p. 398. 
The theorem gives the development of a function Fx of the root of an equation 
0=fx + x^fx -1- X»f 2 X + &c., 
but it is not really more general than that for the particular case 0 =fx +x x f x x ; or 
say when the equation is 0 = (f>x + \fx* Considering then this equation 
<px + Xfx = 0, 
let a be a root of the equation (f>x — 0 ; the theorem is 
Fx = F 
V I|f, (fr-ry 1 
1 • 2 4P I 0", (S f- FT 1 
* For in the result, as given in the text, instead of Xfx write aq/jX + x^fx + &c., then expanding the several 
powers of this quantity, each determinant is replaced by a sum of determinants of the same order, and we 
have the expansion of Fx in powers of x l , x 2 , ... .