CHARACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT. 159 



and, therefore, 



*-*{(S)>+(S)v(S)*} *«>(©• 



But, substituting the values of das, &c., from (17), this becomes 



* - W { (f)(5) + OiSk) + (S)GS) } + *° (f) • 



and the first part vanishes, by (16). 

 Hence 



frx _ fry = 8*z = 8 2 Q 

 dec Sy dz §C 



which show that when the path is simultaneously a free path and a brachisto- 

 chrone, it is necessarily rectilinear. 



This might have been inferred at once, from the theorem of § 13, which shows 

 that if the free path be a brachistochrone, there can be no pressure due to the 

 motion, i.e., no curvature. But the above investigation is given as containing 

 curious additional information. It shows, for instance, that if the force be the 

 same at all points of each of a series of equipotential surfaces, the lines of force 

 are rectilinear. Also, that if the flux of heat be constant per unit of area over 

 each one of a series of isothermal surfaces, though not necessarily the same for 

 all, the propagation of heat takes place in straight lines. And, as particular cases 

 of these theorems, if the force or the flux of heat be the same throughout a given 

 space, the attraction, or the flux, therein takes place in parallel lines. 



16. Hamilton's equation for the determination of the Characteristic Function 

 (A) in the case of the free motion of a single particle is 



@y+®y+@y-*-y>. . . . m 



The comparison of this with (2) suggests a useful transformation. Introducing 

 in that equation a factor 2 , an undetermined function of z, y, z, we have 



■ 2 ' idr\* & 



If we make 

 and 



(19) becomes 



6 = <t>'(r) . . . . . . (20), 



6* 



2(H- V) 



= 2(1^-^) . . . . (21), 



(^HiFM^y^-v,). . . m 



VOL. XXIV. PART I. 2 X 



