112 ProcrcrUiifis of the Roi/nl Irish Acadeiwj. 



Hence we have a theorem analogous to Darboux's ' reciprocal ' theorem 

 (p. 95). If the curves common to two aid of three mutvMly orthogonal 11- 

 families be citrves of extreme curvature on both, the three cmnmon curves (tvjo- 

 hij-tivo) are curves of extreme curvature on the families on which they lie. 

 The data for the conclusion are necessary and sufficient. 



All these theorems are special cases of the principles expiressed in thefollotoing 

 equivalent foi-mulae, which are true of any three D-families: — 



TO \ 



I2- Iz + Bi sin 2ri)2 - Bi sin 2iA3 = + 2 —^, 



dS23 



df) 

 I,-Ii + 2)3 sin 2^3 - Di sin 2>//i = ± 2 —^, \ if) 



as3\ 



Ii - Iz + Di sin 2rf)i - D2 sin 2\Pz = + 2 — -, 



dsa I 



where + A, + Bz, ± B3 are twice the deviations (p. 106), and ^, is the 

 angle between a chosen direction of extreme curvature on Di and the curve 

 Hilla, and j/-i is the angle between the same direction and the curve nilla, 

 while 02, 1^2, i/)3, 1^3 have corresponding meanings in cyclical order. Also we 

 have 01 - i/-! = 023, etc. The formulae (7) then follow directly from (1) on 

 p. 96, and (4) on p. 105. 



G. The Indicatkix of Form. 



The Dupin indicatrix has two functions which become separated for 

 general Il-families. It gives the curvature of geodesies through the point 

 and also the limiting form of sections of the surface by planes parallel to the 

 tangent plane, as the former planes approach the latter. For a Il-family the 

 indicatrix of curvature is one generalization of the Dupin indicatrix. But we 

 may also consider the limiting forms of families of Xl-curves lying in planes 

 parallel to the D-tangent plane, as the former planes approach the latter. 

 This family will be a generalization of the Dupin indicatrices for a one- 

 parameter family of surfaces. 



Using the special axes and taking the principal directions of curvature for 

 axes of X and y (61 + a^ = 0), we find the intersections of 



{aix + aiy + Oiz) dx + {- a^ + biy + b^) dy + dz = 



with z = z', where z' is a constant ' of the second order of small quantities,' 

 X, y, dx, dy being regarded as small quantities of the first order. If we 



