748 



SCIENCE 



[N. S. Vol. XLIV. No. 1143 



other body or medium B; and that other body 

 or medium B while exerting a force on A, is 

 experiencing an equal and opposite force due 

 to A ; whenever the existence of a force on A 

 is discovered we should immediately seek out 

 the body or medium B which is the other party 

 to the transaction; whenever a force is men- 

 tioned, the body or medium exerting the force 

 should be clearly in mind. 



Considered from this point of view, the 

 answers to the above questions regarding a 

 rotating liquid would run somewhat as follows : 

 The forces acting on a water particle in the 

 free surface are (1) its weight, due to the 

 earth, (2) a force due to the liquid in con- 

 tact with it, and (3) a force normal to the 

 surface, due to the atmosphere. The resultant 

 of these is a centripetal force since the accel- 

 eration is centripetal. If we can prove that 

 the second force is normal to the free surface, 

 then it follows immediately from the force 

 triangle that the normal to the surface makes 

 an angle with the axis of spin whose tangent 

 is equal to the ratio of no 2 to g, and that the 

 section of the free surface is parabolic. 



The proof we need is the following: Sup- 

 pose a closed, cylindrical can, full of liquid 

 and with its bottom horizontal, is uniformly 

 rotating around the vertical axis of symmetry. 

 On any co-axial cylindrical surface within the 

 liquid with a radius r there is a pressure be- 

 cause of the rotation equal to ipr 2 ^ 2 per cm. 2 ; 

 at any height y above the bottom there is also 

 a hydrostatic pressure due to gravity equal to 

 P- — pgy. The equation for a surface of con- 

 stant pressure within the liquid is therefore 



ipr'a 2 -f- P — pgy = constant, 

 r 2 u- — 2gy = constant. 



But the force on any particle due to the sur- 

 rounding liquid is, of course, normal to the 

 surface of constant pressure at that point. If 

 we now suppose the can opened on top and all 

 the liquid within a surface of constant pres- 

 sure removed, the pressure formerly exerted by 

 the removed liquid would be supplied by the 

 atmosphere and the remaining liquid would 

 continue to rotate exactly as before. Thus 

 the free surface of our rotating liquid must 



coincide with a surface of constant pressure, 

 and the force on a surface particle due to the 

 liquid in contact with it (including surface 

 tension), being normal to the surface of con- 

 stant pressure, is normal to the free surface. 

 In a similar manner the more general proposi- 

 tion may be proved that the free surface of 

 any liquid whose particles remain _ at a con- 

 stant distance from each other during any 

 motion, is normal to the force with which the 

 liquid acts on the surface particles at each 

 point, and is not, as often stated, normal to 

 the resultant force acting on them. 



When a student finds in an elementary text 

 the statement that " when a body is accelerated 

 we may consider the force of reaction as one 

 of the forces acting upon the body," and is told 

 that one of the forces acting on one of the 

 masses of an Atwood's machine, m v is " the 

 reaction of the mass m x against its upward 

 acceleration " [which is equivalent to the 

 statement that a body when accelerated acts 

 upon itself with a force ma, so that the result- 

 ant force is always zero] — when a student 

 tries to reconcile such assertions with the 

 laws of motion, is it surprising that he be- 

 comes confused and discouraged? 



Why not use force only in the single definite 

 sense implied in the laws of motion? 



The fact that the two authors quoted are 

 unusually experienced and successful teachers 

 suggests that they are not the only ones who 

 are making the path of freshmen unnecessarily 

 difficult. I have taken the liberty of using 

 them as " horrible examples " in this respect 

 because their text-books are for the most part 

 admirably clear, and because I know them to 

 be men who are big enough not to resent well- 

 meant criticism. 



If there is any question as to the wisdom of 

 the conclusion suggested above, let us thrash 

 the matter out now. To avoid misunder- 

 standing, let me add that in using the phrase 

 " force due to — " for the sake of brevity, no 

 relation of cause and effect is implied in any 

 critical philosophical sense. 



Gordon S. Fulcher 

 University op "Wisconsin, 

 November 3, 1916 



