158 PROFESSOR TAIT ON THE APPLICATION OF HAMILTON'S 



Now, by (8) and (11), 8(~? j &c, are proportional to the direction-cosines of the 



resultant force, which therefore lies in the common plane of two consecutive 

 elements of the curve. 



15. The equation of the surfaces which are orthogonal to the path is 



r=C; 



and that of equipotential surfaces 



V=C 1 . 



That these may coincide we must have 



T = 0(V), 



where <p is any function whatever. 

 Hence 



/^rml 2 (f dY \\ f dY \ 2 ■ f dY \ 2 \ 1 



|^ (V) } \\dx) + W) "(^)> 2(H-V)- 



If we write 



V =Jj 2 (H - V) cp' (V) dV = * (V) , . . (15). 



this becomes 



m+QHT)'^- ■ ■ ■ ■ < 16) - 



A complete primitive of this equation is, of course, 



V = Ix + my + nz — p , 

 where p is any function of I, m, n, and 



V- + m 2 + n 2 = 1. 



The general primitive, equated to a constant, is therefore obviously the equation 

 of a series of surfaces such that the normal distance between any two consecutive 

 members of the series is everywhere the same. It is evident from (15) that the 

 surfaces thus found are identical with the isochronous and equipotential surfaces, 

 when these coincide. The equations of their orthogonal trajectory, that is, of the 

 free path which is also a brachistochrone, are therefore, 



( d E\ § (<®!\ § (^E) s 



8x 8y Sz \dx) " \dy ) ^ \dzj 



= 3F = 8C, (17). 



tej \dy) Vfa) \dx) + \dy) + \dz ) 



Hence, 



t<ffi 



\dx 



*.=*>(£[).*»■. 



