745] 
ON THE SCHWARZIAN DERIVATIVE. 
149 
and, presenting themselves in connexion with the algebraically integrable cases of this 
equation, we have rational and integral functions of s, derived from the polygon, the 
double pyramid, and the five regular solids. They are called Polyhedral Functions. 
The Schwarzian Derivative occurs implicitly in Jacobi’s differential equation of the 
third order for the modulus in the transformation of an elliptic function (Fund. Nova, 
1829, p. 79, [Ges. Werke, t. I., p. 133]) and in Kummer’s fundamental equation for the 
transformation of a hypergeometric series (Kummer, 1836: see list of Memoirs): but it 
was first explicitly considered and brought into notice in the two Memoirs of Schwarz*, 
1869 and 1873. The latter of these, relating to the algebraic integration of the 
differential equation for the hypergeometric series, is the fundamental Memoir upon the 
subject, but the theory is in some material points completed in the Memoirs by Klein 
and Brioschi. 
The following list of Memoirs, relating as well to the Polyhedral Functions as to 
the Schwarzian Derivative, is arranged nearly in chronological order. 
OL ß 
Kummer, Ueber die hypergeometrische Reihe 1 + ^Ci'elle, t. xv. (1836), 
pp. 39—83 and 127—172. 
Schwarz, Ueber einige Abbildungsaufgaben. Crelle-Borcliardt, t. lxx. (1869), pp. 105—120. 
Ueber diejenigen Fälle in welchen die GWssische hypergeometrische Reihe 
eine algebraische Function ihres vierten Elementes darstellt. Do. t. lxxv. (1873), 
pp. 292—335. 
Cayley, Notes on Polyhedra. Quart. Math. Jour. t. vil. (1866), pp. 304—316; [375]. 
On the Regular Solids. Do. t. xv. (1878), pp. 127—131; [679]. 
Fuchs, Ueber diejenigen Differentialgleichungen zweiter Ordnung welche algebraische 
Integralen besitzen, und eine Anwendung der Invariantentheorie. Crelle-Borchardt, 
t. lxxxi. (1875), pp. 97—142. 
Klein, Ueber binäre Formen mit linearen Transformationen in sich selbst. Math. Ann. 
t. ix. (1875), pp. 183—209. 
Brioschi, Extrait d’une lettre à M. Klein. Math. Ann. t. XI. (1877), pp. 111—114. 
Klein, Ueber lineare Differentialgleichungen. Math. Ann. t. XI. (1877), pp. 115—118. 
Brioschi, La théorie des formes dans l’intégration des équations differentielles linéaires 
du second ordre. Math. Ann. t. XI. (1877), pp. 401—411. 
G-ordan, Ueber endliche Gruppen linearer Transformationen einer Veränderlichen. 
Math. Ann. t. xii. (1877), pp. 23—46. 
Binäre Formen mit verschwindenden Covarianten. Math. Ann. t. xii. (1877), 
pp. 147—166. 
[* Schwarz, Ges. Werlte, t. ii. , p. 351, remarks that the Derivative occurs implicitly in a niemoii by 
Lagrange, “Sur la construction des cartes géographiques,” (1779), Œuvres, t. iv., p. 651.]