OCTOBKK 12, 1883.1 



SCIENCE. 



499 



whether liquids or gases, to be either compres- 

 sible or incompressible, according to certain 

 known laws ; and we shall omit considerations 

 of fluid friction, although we admit the consid- 

 eration of friction between solids." 



The ne.xt chapter (v.) comprises pp. (3 to 

 100, and its especial object is set forth in the 

 iutroductor3- section (454), as follows: -'We 

 naturally divide statics into two parts, — the 

 equilibrium of a particle, and that of a rigid or 

 elastic body or system of particles, whether 

 solid or fluid. In a very few sections we shall 

 dispose of the first of these parts, and the rest 

 of this chapter will be devoted to a digression 

 on the important subject of attraction." In 

 other words, this chapter is devoted, with the 

 exception of a couple of pages, to an extended 

 treatment of attraction according to the law of 

 the inverse square of the distance as applied 

 to gravitation, electricity, and magnetism. 



After a brief investigation of the usual for- 

 mulae for the attraction of the spherical shell, 

 circular disk, thin cylinder, circular arc, etc.. 

 the main subject of the chapter is reached, 

 which is the modern mathematical theory of 

 potential ; which theory is the principal means 

 now employed in the discussion of questions 

 relating to the distribution of attracting matter, 

 and the forces caused by it. This theory, due 

 as it is to the anal3-tical discoveries of Lapl.ice. 

 Green. Gauss, and others, might, ueverthelcss. 

 have long remained comparatively barren of 

 fruitful results in physics, had it not been for 

 the genius of Faraday, who, though unskilled in 

 the use of analysis, had a most powerful grasp 

 of geometric and physical relations. In the 

 words of another,' •' Faraday, in his mind's 

 ej'e, saw lines of force traversing all space, 

 where mathematicians saw centres of force at- 

 tracting at a distance ; Faraday saw :i medium 

 where they saw nothing but a distance ; Fara- 

 day sought the seat of the phenomena in real 

 actions going on in the medium, thej- were sat- 

 isfied that they found it in a power of action at 

 a distance." He conceived of lines of gravi- 

 tational force as holding the planets in their 

 orbits. These lines radiated tln-ough all space 

 from the attracting body as a nucleus, regard- 

 less of the existence or non-existence of bodies 

 upon which the attraction could be exerted. 

 Furthermore, Faratlay thought of each attract- 

 ing body as surrounded at dillercnt distances 

 by successive level surfaces, — like that of the 

 ocean, for example, or the upper limit of 

 the atmosphere ; which surfaces cut the lines 

 of force everywhere at right angles. This was 

 not only true of gravitating matter, but each 



' rrefiicc of MaxwelPii Klcctrlcity and magnetifltn. 



electrified body also had its s3-stem of lines of 

 electrical force, and its corresponding system 

 of level surfaces ; and each magnet had its 

 magnetic system as well. The geometry of 

 these lines and surfaces is the basis of Fara- 

 d.ay's reasoning in his ' Experimental re- 

 searches,' and is the geometric truth hidden 

 in the analytic discoveries clustering around 

 Laplace's, Poisson's, and Green's theorems. 



That we may call these relations more clear- 

 ly before the mind, consider for a moment the 

 so-called ' equation of continuity ' of an iueom- 

 pressible fluid ; which equation is divined from 

 the geometric truth, that the quantit3' of such 

 a fluid, which flows into any assumed closed 

 surface, taken entirel^y within it, is equal to that 

 flowing out, or that the total ^ot« is nil. This 

 is preciselj- expressed by the equation 



fFdS-0, 



(1) 



in which d ^ is the area of the element of the 

 assumed closed surface, F is the normal flow 

 per square unit at thiit element, and the limits 

 of integration are so taken that it extends over 

 the entire surface. There is also another form 

 of the equation of continuitv, expressing the 

 kinematic truth, that, in an incompressible 

 fluid, the variations of the component veloci- 

 ties in the directions a;, y, z, balance ; i.e., their 

 algebraic sum is nil, which m.ay be written 

 thus : — 



du dv du> 

 dx dy dz 



■0, 



(2) 



in which m, v. to, are the component velocities 

 in the directions .c, ?/, z, respectively. 



Now, it is not difficult to picture to the mind 

 tiie motions occurring within the mass of an 

 incompressible fluid ; such as water, for exam- 

 [ile. In whatever waj- it nia^y be moving, we 

 can think of strc.am-lines along which the dif- 

 ferent parts of it flow. A number of these 

 lines, side b\- side, can be taken to form a 

 stream, and can be thought of as bounded by 

 a kind of tubular surface ; which surface might 

 be regarded as the boundary of tiic stream, 

 which isolates it from surrounding streams. If 

 tiie stream has the same velocity at every point 

 along the tube, then its cross-section must be 

 uniform ; but, where the velocitj' is less, the 

 cross-section is proportionately increased, and 

 vice versa. This follows from the fact that 

 tlie same quantity must pass each cross-section 

 per unit of time. A tube in which a unit of 

 volume passes a given cross-seetion per unit 

 of time is called a unit-tube. Now, the forces 

 of attraction in free space, caused by any dis- 

 tribution of matter, electricity, or magnetism, 



