of Edinburgh, Sessio7i 1864-65. 
407 
3. On the Application of Hamilton’s Characteristic Function 
to Special Cases of Constraint. By Professor Tait. 
Hamilton’s beautiful theory of Varying Action reduces to the 
discovery of a single function any problem connected with motion 
under the action of a conservative system of forces, and with con- 
straint by any system of smooth fixed surfaces. 
It does not appear to have been applied to cases (such as the 
brachistochrone) in which the requisite constraint is the thing to 
be determined. 
/ ds 
— , the time in the brachistochrone, it is shown 
that we have, r being regarded as a function of x, y, z, 
a' _ 1 1 
^ /dr\ 
\dz/ 
2(H- F) 
where H is the whole energy, and V the potential of the given 
system of forces. If a complete integral of this equation can be 
found, we have 
| = &c. 
Also, if a, fd, be constants in the expression for t, 
dr^ 
da 
where ^ and B are two new constants, are the equations of the 
brachistochrone. 
Various properties of brachistochrones, and the corresponding 
free paths, are deduced from these equations ; the connection 
between this process and that of Hamilton is illustrated by the 
solution of problems in optics, based on the corpuscular and on 
the undulatory theories ; and the paper concludes with an applica- 
tion of the principle to cases in which the characteristic function 
is of such forms as 
or 
where f and F are given functions. 
f F (po,y,z)f{y)ds 
