366 
[943 
943. 
ON RECIPROCANTS AND DIFFERENTIAL INVARIANTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xxvi. (1893), 
pp. 169—194, 289—307.] 
I USE the term Reciprocant to denote a function of an arbitrary variable or 
variables and its differential coefficients, not connected with any differential equation; 
and Differential Invariant to denote a function of the coefficients of a differential 
equation and the differential coefficients of these coefficients. Halphen’s differential 
invariants are thus reciprocants, but the term reciprocant is not made use of by him. 
I have entitled the present paper “ On Reciprocals and Differential Invariants ”; 
in the earlier part, (except that for preserving a chronological order, I briefly refer 
to Sir J. Cockle’s Criticoids, which are differential invariants or rather seminvariants), 
I attend almost exclusively to Reciprocants, reproducing and explaining, and in 
some parts developing, the theories of Ampere, Halphen in his first two memoirs, and 
Sylvester. 
I. 
The notion of a reciprocant first présents itself in Ampère’s “ Mémoire sur les 
avantages qu’on peut retirer dans la théorie des courbes de la considération des 
paraboles osculatrices, avec des reflexions sur les fonctions différentielles dont la valeur 
ne change pas lors de la transformation des axes,” Journ. École Polyt. t. vu. (1808), 
pp. 151—191 (sent to the Institute, Dec. 1803)*. We hâve (p. 167) for the radius of 
curvature the expression 
(1 +y' t ) 1 
* A. reciprocant, the Schwarzian derivative, occurs in Lagrange’s memoir, “ Sur la construction des cartes 
géographiques,” Nom. Mém. de Berlin, 1779, Œuvres, t. iv. p. 651, but scarcely qua reciprocant, viz. the form 
in question ■ 
presents itself in the equation 
f"'(u + ti) /J 
(u+ti)' 1 
r. F’"(u-ti) 
/F" (u-ti)Y 
f (u+ti) - 
f (u + ti), 
1 F' (u- ti) s 
\ F' ( u - ti) J