186 REPORT—1862. 
tially the same as the extract thereof in ‘ Liouville’s Journal,’ referred to in 
my former Report ; but since the date of that Report there have been published 
in the ‘Comptes Rendus,’ 1861 and 1862, several short papers by the same 
author; also Jacobi’s great memoir, see list, Jacobi, Nova Methodus &c. 1862, 
edited after his decease by Clebsch; some valuable memoirs by Natani and 
Clebsch (Crelle, 1861 and 1862) relating to the Pfaffian system of equations 
(which includes those of Dynamics), and Boole “ On Simultaneous Differential 
Equations of the First Order, in which the number of the Variables exceeds by 
more than one the number of the Equations,” Phil. Trans. t. clii. (1862) 
pp. 437-454. 
Rectilinear Motion, Article Nos. 1 to 5. 
1. The determination of the motion of a falling body, which is the case of 
a constant force, is due to Galileo. 
2. A variable force, assumed to be a force depending only on the position 
of the particle, may be considered as a function of the distance from any 
point in the line, selected at pleasure as a centre of force; but if, as usual, 
the force is given as a function of the distance from a certain point, it is 
natural to take that point for the centre of force. The problem thus becomes 
a particular case of that of central forces ; and it is so treated in the ‘ Principia,’ 
Book I. § 7; the method has the advantage of explaining the paradoxical 
result which presents itself in the case Force O¢ (Dist.)—?, and in some other 
cases where the force becomes infinite. According to theory, the velocity 
becomes infinite at the centre, but the direction of the motion is there 
abruptly reversed; so that the body in its motion does not pass through the 
centre, but on arriving there, forthwith returns towards its original position ; 
of course such a motion cannot occur in nature, where neither a force nora 
velocity ever is actually infinite. 
3. Analytically the problem may be treated separately by means of the 
a 
1ax\2 
equation qea* which is at once integrable in the form (a) =049/3 Xdzx. 
4. The following cases may be mentioned :— 
Force o Dist. Thelaw of motion is well known, being in fact the same 
as for the cycloidal pendulum. 
Force ¢ (Dist.)-2, =, which is the case above alluded to. 
‘ a 
Assuming that the body falls from rest at a distance a, we have 
x=a (1—cos ¢), 
where, if n= ¢ is given in terms of the time by means of the equation 
B 
nt=p—sin ¢. 
If the body had initially a small transverse velocity, the motion would be in a 
very excentric ellipse, and the formule are in fact the limiting form of those 
for elliptic motion. 
5. There are various laws of force for which the motion may be determined. 
Tn particular it can be determined by means of Elliptic Integrals, in the case 
of a body attracted to two centres, force OC (dist.)-2: see Legendre, Exercices 
de Cal. Intég. t. ii. pp. 502-512, and Théorie des Fonct. Ellip. t. i. pp. 531- 
538. I : 
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