210 REPORT—1862. 
where m is a function of p only and n of y only, while \ contains p and y. 
The geometrical signification of the equation ds*=)(da* +d"), or of the last- 
mentioned equivalent form, is that the curves 
a or \=const., B or p=const., 
intersect at right angles. ; xs ; 
The foregoing differential equation of the path, writing fu, Fv in the place 
of fa, F3 respectively, may be expressed in the form 
Spoosi+Fysin=A, 
where 7, 90°—7 are the inclinations of the path at the point (A, ») to the two 
orthotomic curves through this point. 
101. The before-mentioned equation 
(2U+C)A=fa—FB 
may be satisfied independently of C, or else only for a particular value of C. 
In the former case the law of force is much more restricted, but on the other 
hand there is no restriction as regards the initial circumstances of the motion; 
it is the more important one, and is alone attended to in the sequel of the 
memoir. * In the case in question (changing the functional symbols) we must 
have 
A=ga—aP, AU=fa—FG; 
so that the functions denoted above by fa, FB now are 2fa+4+Cga, 2F3+CaB ; 
the equation of the trajectory is 
da r dp 
V 2fa+Cpa—A VA—2F84+Cap 
and for the time the formula is 
he ga da ap dj3 
V 2fa+Coa—A WV A—2FB+CapB 
It is noticed also that taking B, e to denote two new arbitrary constants, 
and writing 
e= JdaW 2fa+Cpa—A+ fap Vv A—2FB+ Cap, 
the equation of the trajectory and the expression for the time assume the 
forms 
dO _ 
° ante! 
as is known @ priori by a theorem of Jacobi’s. 
If the forces vanish, the path is a geodesic line; and denoting by a the ratio 
of the constants A, C, we have 
da dp 
Vga—a Va—ap 
wap 
t= qet® 
and moreover 
ds=daNn ga—a+dpVa—¢p, 
which are geometrical properties relating to the geodesic line. 
102. Passing to the applications: in the first place, if a, 6 are rectangular 
coordinates of a point in plano, then writing instead of them 2, y, we have 
ds’ =da* +-dy*, which is of the required form; but the result obtained is the 
self-evident one, that the equations may be integrated by quadratures when 
U is of the form funct. a—funct. y. 
But taking instead the elliptic coordinates p, v of a point in plano,—viz., as 
