296 Mr. Challis 07t the Forms of the Arhitrary Functio7is 



Treitschke divides this genus into three fomilies, according 



to the form of the upper wings. 

 Fam. a. — Uj)per wings rounded at the summit. 

 Fam. B. — Upper wings with the summit falciform ; terminal 



margin entire. 

 Fam. C. — Upper wings with the summit falciform ; terminal 



margin dentate. 

 Fam. a. Species. Icon. 



l.V\a.t.Spinula, Hiib.* . . . Hub. Bomb. tab. 11. f. 40. (mas.) 

 Fam. B. 



2.Plat.6Vc?/Za, Hiib Ernst, V. pi. ccviii. f. 277. 



3. — Curvatula, Borkh. . . . Ernst, V. pi. ccviii. f. 276. f. g. 



4. — Falcula, Hiib.f Ernst, V. pi. ccvii. f. 276. a — e. 



5.—Hamula, HUb.f .... Ernst, V. pi. ccviii. f 278. 



6. — Unguicula, Hiib Ernst, V. pi. ccvii. f. 275. 



Fam. C. 



7 .FlaULacertula, Hub. . . Ernst, V. pi. ccix. f. 279. 

 End of Vol. V. Part III. [To be continued.] 



XLVI. 0?i the Determination of the Forms of the Arbitrary 

 Functio7is xohich occur iji the Integrals of Partial Different 

 tial Equations. By J. Challis, Fello'w of the Camb. 

 Phil. Soc.l 



THE phaenomena of motion of the simplest kind, and whidh 

 have hitherto been treated with the greatest success, have 

 required the solution of differential equations between trm 

 variables, the integrals of which are e(juations between the 

 variables of determinate form, and indicate paths of motion 

 which are always contitiuous lines. The elliptic motion of the 

 planets is the most remarkable instance of this kind. But 

 another order of phainomena is presented to us, such as the 

 motions of gases and fluids, and the small vibrations of die 

 constituent molecules of solids, which requii-e us to ascertain 

 the laws of the collective motion of an infinite number of 

 moveable points. At first sight it might appear that the mo- 

 tions are of that bizarre and irregular kind, that it is impos- 

 sible to submit them to calculation. But nothing in nature 

 is indeterminate; and it is perhaps not accidental that the 

 order in complexity of the phosnomena corresponds to the 

 classification of differential equations which pure analysis 

 ■would establish without reference to the natural facts. Tlie 

 theoretical inquiry into the motions I refer to, always con- 



* CiLix, Stephens, Syst. Cat. ii. p. 157. t Drepana, Steph. 1. c. p. 156. 

 J Communicated by the Autlior. 



ducts 



