622 
attraction exerted at that point by any system of masses , the dis- 
placement is effected without rotation. For if Fg = C be the poten- 
tial surface, we have Scndg a complete differential — i.e., in Cartesian 
co-ordinates %dx + vjdy + £dz is a differential of three independent 
variables. Hence the vector axis of rotation i -f &e., 
\dy dz) 
vanishes by the vanishing of each of its constituents, or V. <3 cn = Q. 
Conversely, if there be no rotation the displacements are in the 
direction of, and proportional to, the normal vectors to a series of 
surfaces. 
For 
0 = V.rfgV. <3 o-'= (Sc/g<l )<T' — < Sa-'dg. 
Now, of the two terms on the right, the first is a complete differ- 
ential, since it may be written — (see (7) ), and therefore the 
remaining term must be so. 
Thus, in a distorted system, there is no compression if 
S. <3 o- =0, 
and no rotation if V. <] cn = 0 ; and evidently merely transference 
if <7" = a = a constant vector, which is one case of <3 cr — 0. 
In the important case of cr'= e<3 Fg there is evidently no rota- 
tion, since <3 cn= e<\ 2 Fg is evidently a scalar. In this case, then, 
there are only translation and compression, and the latter is at each 
point proportional to the density of a distribution of matter, which 
would give the potential Fg. For if r be such density, we have 
at once <1 2 Fg = 4‘zr (see (3) 1 ). This suggests a host of physical 
analogies which we cannot enter upon at present. 
V. Keeping still to the meaning of cn as the vector of displace- 
ment, as we have seen that <1 c- = s + /, where s is the condensa- 
tion of the particles near the extremity of g, and / the doubled vector 
axis of rotation of the group — we may apply the vector operation a 
second time. Thus, 
<1 2 cn= <3 s + <p. 
Now, our former results enable us to assign meanings to these 
expressions. <3 s is the normal-vector to any of the surfaces of 
equal condensation. The scalar and vector parts of <3 1 represent 
