294 BROOKS — ORTHIC CURVES. [May 20, 



in fishes but in Amphibia and reptiles, would suggest that the cause 

 of the transformation of longitudinal stripes into spots on the lum- 

 bar and sacral regions of lizards is the result of the same specializing 

 growth-force. It may perhaps be regarded as a surviving remnant 

 of the segment- forming force, which has affected the pigment bands 

 in a manner identical in the vertebrates and insects. This trans- 

 formation of stripes into spots, and the fusion of two dorsal 

 tubercles into a median one, may be, then, the sign of some latent 

 or surviving amount of force concerned in the origin and forma- 

 tion of segments, which crops cut in the larval stages of insects and 

 in young lizards, resulting in this opisthenogenetic mode of origin 

 of spots from bands. 



ORTHIC CURVES; OR, ALGEBRAIC CURVES WHICH 

 SATISFY LAPLACE'S EQUATION IN TWO 

 DIMENSIONS. 



BY CHARLES EDWARD BROOKS, A.B. 



{Read May 20, 1904.) 



I propose a study of the metrical properties of algebraic plane 

 curves which are apolar, or, as it is sometimes called, harmonic, 

 with the absolute conic at infinity. If we disregard the right line, 

 the simplest orthic curve is the equilateral (conic) hyperbola, and 

 the name equilateral hyperbola is sometimes extended to orthic 

 curves of higher order. Doctor Holzmiiller,^ who devotes a section 

 to curves of this kind, calls them hyperbolas ; and M. Lucas^ calls 

 them '' stelloides." M. Paul Serret, in a series of three papers in 

 Comptes Rendus,^ uses the word *' equilatere " for a curve with 



1 Einfiihrung in die Theorie der Isogonaleii Verwandschaften und der 

 Cofi/ormen Abbildungen, Gustav Holzmiiller, Leipzig, 1882, p. 202. . . . 



■^"Geometric des Polynomes," Felix L.uc2iS, Journal de V Ecole Poly tech- 

 nique, 1879, t. XXVIII. 



^ Comptes Rendus, 1895, t, 121. Sur les hyperboles equilateres d'ordre 

 quelconque, p. 340. 



Sur les faisceaux regulieres et les equilateres d'ordre n. p. 372. 



Sur les equilateres comprises dans les equations 



0= V"~'A7i" = ^n, 



O = Si^n-l/, 7^n = y7n + IHl. 

 P- 438. 



