222 J. C. Watson on the Elements of the Orbit of a Comet. 
and similarly for w, w’, and w’. If these three numbers are exactly 
represented by the expression 
r z\3 
«+#(52)+1(aa) ° 
where J-+-z is the general value of the argument ;—since the values of 1, 
u', and w” will be such that the third differences may be neglected, this 
formula m may be assumed to express exactly any value of the function 
corresponding to 2 value of the argument not differing much from 4, or 
between the limits z= -04 and z=-+04. 
To find the coefficients «, 8, and 7, we have? 
Argument. Function. Ist diff. 2d diff. x Function. 1st diff. 2d diff. 
gi Aa) Ha 40a) Sregot Abas 
4 J(4 ) J(4+464) I" (4) “0 b+ 
4434 fi4+04) $84 ap pty Ot 
whence by comparison we find 
a=f(4); B=$}/ (4 - $84)+/'(44404){ 5 and y=4/"(4)- 
Now in order that = middle place may be exactly represented i in longi- 
ll hay 
tude, we shall 
(aa) +°(za) += 
27 
from which we find 
m=" = (6- JB? = #? = 4a) =p, (18) 
by? ie 
4r-p.0d=—0. ( 
In the same manner, the condition that the middle place shall be exactly 
represented in latitude BEM 
or 
2—p!.dd= (18) 
In order that the orbit shall oseally represent the middle place, it 7 
quires that both conditions shall sati simultaneously, but it 
rarely, if ever, happen, that this can be e and we must therefore 
Having thus determined the most probable value of z, we compl pins 
ancaee caer of elements, with the geocentric distance Ate corresponding 
to 
The applisation of these formule is not limited to the case of three 
observations. With an approximate value of 4 we may compu 
elements from the extreme observations, and co mpare any num 
intervening places, each of which will furnish two equations of co It 
for the determination of z. Should it be found that the residuals res? 
ing from the final elements exceed the limits of the probable errors 
observations, the orbit cannot be a parabola, and it will be necess# 
determine the excentricity. 
Ann Arbor, Mich., December, 1862. 
* The coefficient 8 should not be confounded with the latitude g previously ose 
