NATURE 
165 
THURSDAY, JUNE 24, 1886 
MR. MINCHIN’S TREATISE ON STATICS 
A Treatise on Statics, with Applications to Physics. 
By G. M. Minchin, M.A. Vol. II. 3rd ed. (pp. §12 + vi.). 
Clarendon Press Series. (Clarendon Press, Oxon, 
1886.) 
HIS new edition of this work has been separated 
into two volumes. The first volume (351 pp.), 
dealing with “ Equilibrium of Coplanar Forces,” aims at 
the standard of Undergraduate Honours ; it is noticeable 
for the frequent use of graphic methods and for a long 
discussion on funicular polygons (now forming so im- 
portant a help in graphic applications to engineering) ; 
this was published in 1884. The second and longer 
volume is a masterpiece of constructive skill in the 
adaptation of modern methods ; it is particularly notice- 
able for the introduction of the theory of screws and of 
astatic equilibrium, also for an extensive selection of 
excellent examples, and for the free use of hyperbolic and 
elliptic functions in solutions: the reading required is 
thus considerable ; it is, in fact, intended for those who 
seek Honours. The work is so much improved in this 
edition that it merits an extended notice. The second 
volume is divided into only seven chapters, each of which 
is an essay on its special subject. The numbering of 
chapters and articles is continuous with Vol. I., whilst the 
pagination is distinct. 
Chapter XIII. (the leading chapter, 64 pp.) deals with 
Non-Coplanar Forces, and contains the usual proposi- 
tions (16 pp.) about compounding and resolving forces 
and couples, about resultants, equilibrium, and central 
axis: then follow (48 pp.) the theory of screws, cylin- 
droids, complexes, and degrees of freedom ; the construc- 
tions given for the cylindroid are neat : in one the surface 
is traced by the blades of a pair of scissors, which open 
horizontally at a uniform rate, whilst the rivet falls 
vertically ; this gives a vivid idea of the surface. 
Chapter XIV. (34 pp.) treats of Astatic Equilibrium, 
which is defined to be a balance amongst forces of fixed 
magnitudes and directions at definite points of a body 
subject to displacement. This is treated by quaternions, 
the Cartesian method being found cumbrous. It is shown 
that a system of forces can always be astatically balanced 
by a set of ¢/ree forces in any given directions (and even 
by three equal rectangular forces) applied at three points 
lying in a plane fixed in the body, and also by two 
_ forces if these points lie in a line, or by one force if they 
coincide (the general proof of all this is easy by ele- 
mentary methods). This subject has some practical 
application in electrical measurements, for which an 
astatic magnet-pair is much used, and in seismometry, 
for which it has been sought to make pendulums astatic 
for small displacements (see Milne’s new work on “ Earth- 
quakes,” p. 26). 
Chapter XV. (91 pp.) treats of Virtual Work. The 
term “ work-coefficient” has here been with great ad- 
vantage introduced to replace the lengthy and ¢xcorrect 
term “generalised component of force.” Lagrange’s 
method is treated of at length ; its advantage is shown to 
VOL, XXxIV.—No. 869 
consist in reducing all problems to the case wherein the 
displacements are independent, by introducing internal 
forces to represent the constraints. One disadvantage is 
its undue length, most marked in simple cases. Another 
is a decided risk of error in estimating the work of the 
internal forces ; instances of error due to thisin Lagrange’s 
researches are shown, ey. the cases (1) of an inextensible 
surface wherein Lagrange assumes (incorrectly) that 
6¢S = 0 fully expresses the inextensibility ; and (2) of an 
extensible surface wherein he assumes (incorrectly) the 
work of internal deformation to be simply proportional 
to dS; and (3) of an elastic wire wherein Lagrange over- 
looks the distortion. A brief summary of Jellett’s re- 
searches on inextensible surfaces is given, and it is 
shown that such a surface is quite determinate (and there- 
fore not deformable) if any bounding edge of it be fixed, 
except it be anticlastic or developable, which latter 
admit of deformation when certain edges only on them 
are fixed. The surface-tensions of liquid-films are in- 
vestigated (12 pp.), and the experimental way of pro- 
ducing several such forms is given, and their stability 
discussed. 
Chapter XVI. (45 pp.). On Strings and Springs. —The 
properties of strings in general, also on rough and smooth 
surfaces, are discussed, with some cases of the extensible 
string; next those of plane elastic rods and plane 
springs ; lastly, those of a twisted wire (20 pp.): this last 
is important in electrometers. The interesting A2netic 
analogies are shown (1) of a plane elastic rod with the 
simple pendulum, and (2) of a bent and twisted uniform 
wire with a heavy mass moving about a fixed point, viz. 
that the differential equations in the analogous problems 
are similar. 
Chapter XVII. (123 pp.), on Attraction, is divided into 
four sections. 
Sec. I. (29 pp.). On Attraction in General.—It is 
explained that the law of gravitation implies that the 
attracting particles must be very small compared with 
their distance. Notice is most usefully drawn to this 
limitation several times in the sequel, ¢.g. it is shown that 
the Cartesian expressions see to give indeterminate 
attraction for very close points ; also that for attractions 
more rapid than 1/7” the attraction on an internal point is 
really infinite. 
Sec. II. (40 pp.). On Potential.—In the definition the 
usual idea of motion from infinite distance has been 
dropped, and the definition runs as the work done in 
bringing a tiny mass from a position of zero attraction, 
&c. (not from infinity): this is much better. The con- 
tinuity of the gravity-potential and of its first derivatives, 
the discontinuity of its second derivatives, the absence of 
maxima or minima thereof in empty space, and the insta- 
bility of equilibrium under gravitation to several masses 
are shown. ‘The application of the method of inversion 
is given; and, amongst many examples, Thomson’s solu- 
tion of the attraction of a spherical sheil whose density 
e (distance)*. 
Sec. III. (13 pp.). On Ellipsoids.—After the usual 
investigation of their attraction, it is shown that the sur- 
faces of prolate and oblate spheroids are not equi- 
potential: various problems interesting in the figure of 
the earth are given. 
Sec. IV. (42 pp.). Spherical Harmonics,—Green’s equa- 
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