3840 Prof. Maxwell on the Theory of Molecular Vortices 
any set of three axes. We shall first consider the effect of three 
simple extensions or compressions. . 
Prop. 1X.—To find the variations of «, 8, y in the parallelo- 
piped z, y, z when w becomes v+ 6x; y, y+6y; and z, 2+6z; 
the volume of the figure remaining the same. 
By Prop. II. we find for the work done by the vortices against 
pressure, 
SW =p,o(ayz) — — — (atyzda + B?zxdy+y*%axydz); (59) 
and by Prop. VI. we find td the variation of energy, 
oh = —— fo («Bat BoB+ydy)ayz. . « . « « (60) 
The sum a4 BE must be zero by the conservation of energy, 
and o(vyz) =0, since xyz 1s constant ; so that 
a(3a—a~)+-8(88-8 4) +(8y—-7=)=0. (61) 
In order that this should be true independently of any relations 
between a, 8, and y, we must have 
daa, SB= =p, Sy = ay. pie ae 
Pie op. X.—To find the aan of a, 8, y due toa rotation 0, 
about the axis of # from y to z, a rotation 6, about the axis of y 
from z to 2, and a rotation @, about the axis of z from 2 to y. 
The axis of 8 will move away from the axis of « by an angle 
6,; so that 8 resolved in the direction of x changes from 0 to 
The axis of y approaches that of 2 by an angle @, ; so that the 
resolved part of y in direction z changes from 0 to y@. 
The resolved part of « in the direction of « changes by a quan- 
tity depending on the second power of the rotations, ¥ which may 
be neglected. The variations of a, 8, y from this cause are 
therefore 
du=y0,—PBO,, SB=a0,—y0,, Sy=R0,—aO,. (63) 
The most general expressions for the distortion of an element 
produced by the displacement of its different parts depend on the 
nine quantities 
Ae Ch ee d qd. dy. 
ip Oe? hs 783 Fp By, dy sag = y3 £ 82, Pa 3 
and these may always be pane rand in terms of nine other quan- 
tities, namely, three simple extensions or compressions, 
da! dy! ba! 
gee OT 3 
« 
a “ 
