204 REPORT—1862. 
which are rectified in a subsequent paper, ‘“‘ Remarks on the Deviation of 
Falling Bodies,” &c. t. iv. (1849), by Dr. Hart and Professor W. Thomson. 
81. The theory of relative motion is considered in a very general manner - 
in M. Quet’s memoir, “ Des Mouvements relatifs en général &e.”’ (1853). Sup- 
pose that w, y,z are the coordinates of a particle in relation to a set of move- 
able axes; let é', n', Z’ be the coordinates of the moveable origin in reference 
Pe’ dn CZ 
to a fixed set of axes, and treating the accelerations “*, “7, “% 
dt? dt?” dt’ 
were coordinates, let these, when resolved along the moveable axes, give 
u', v', w': suppose, moreover, that p,q, 7 denote the angular velocities of the 
system of the moveable axes (or axes of x, y,z) round the axes of x, y, and z 
respectively ; w’, v', w', p, g, 7 are considered as given functions of the time, 
and then, if 
as if they 
x dz ad di dr 
pet Dot Oa eet Sn ee fo ' 
is det (05; rh) dt Yat a (Py le ae 4 a dal 
_@Y , of dx dz Vig De yaa pothe siee 
vee (F 2G) +7 zat (92 —Ty)—P (PYG) Tes 
_&z dy _ dx dp __ dq a Se 7 F 
wa at2 (0G in)tyd oR te pe) —9 (TY JE ws 
it is shown that the equations of motion are to be obtained from the equation 
Im[(u—X)ca+(v—Y)éy+(w—Z)éz|=0, 
where éw, ¢y, éz are the virtual velocities of the particle m in the directions of 
the moveable axes. This equation is in fact obtained as a transformation of 
the equation 
nl (té_x Py a | Le 
onl (3 x) e+(oe Y)in+(3e )e =e 
which belongs to a set of fixed axes of &, », Z. 
82. The equations for the motion of a free particle are of course u=X, 
v=Y,w=Z. In the case where the moveable axes are fixed on the Earth, 
and moveable with it (the diurnal motion being alone attended to), these lead 
to equations for the motion of a particle in reference to the Earth, similar to 
those obtained by Gauss and Poisson. The formule are applied to the case 
of the spherical pendulum, which is developed with some care; and Foucault’s 
theorem of the rotation of the plane of oscillation very readily presents itself. 
The general formule are applied to the relative motion of a solid body, and 
in particular to the question of the gyroscope; the memoir contains other in- 
teresting results. 
83. The principal memoirs on the motion of the spherical pendulum, as 
affected by the rotation of the Earth, are those of Hansen, “ Theorie der Pen- 
delbewegung &c.”’ (1853), which contains an elaborate investigation of all the 
physical circumstances (resistance of the air, torsion of the string, &c.) which can 
affect the actual motion, and the before-mentioned memoir by Dumas, “ Ueber 
der Bewegung des Raumpendels &c.” (1855), The investigation is conducted 
by means of the variation of the constants; the integrals for the undisturbed 
problem were, as already noticed, obtained by means of Jacobi’s Principal 
Function, that is, in a form which leads at once to the expressions for the 
variation of the constants; and the investigation appears to be carried out 
in a most elaborate and complete manner. 
84. In concluding this part of the subject I refer to Mr. Worms’s work, — 
*The Rotation of the Earth’ (1862), where the last-mentioned questions — 
