Maech 26, 1920] 



SCIENCE 



321 



we can not say that " dy is the limit of Ay." 

 First, dy is a variable and therefore can not be 

 the limit of anything; secondly, zero is the 

 limit of Ay, and therefore nothing else can be. 



A list of similar fallacies, which still persist 

 in some books (and, apparently, in some class- 

 rooms also), may be found in a paper by the 

 present writer on " The proper use of the dif- 

 ferential in calculus." ^ 



The word derivative means, of course, the 

 ratio dy/dx. 



Edward V. HuNTmcTON 



Habvaed XJniveksity 



weight and centripetal acceleration 



To THE Editor of Science: Mr. Carl 

 Hering's suggestion for a new form of dy- 

 namic compass^ ought to be challenged before 

 some one organizes a company to work the 

 idea out on a cormnercial basis. The fact is, 

 of course, that the change in weight which 

 Mr. Hering refers to occurs only when the 

 motion is in a circle having its center in the 

 earth's axis. Mr. Hering's disk is a plane 

 tangent to the earth's surface and motion in 

 this plane does not, on the basis of Newtonian 

 mechanics, affect the weight of a body. It is 

 imderstood of course, that the disk is not 

 forced to remain tangent to the earth as the 

 earth rotates. This would complicate the 

 situation by introducing the gyroscopic effect. 

 If the disk is mounted in gimbals so that the 

 earth in turning does not force a change in 

 direction of the shaft there would, as stated 

 above, be no tendency of the shaft to set itself 

 parallel with the earth's axis. 



The suggestion that the light disk with 

 equal weights at extremities of a diameter 

 would rotate in balance when in a north and 

 south plane, but out of balance in an east 

 and west plane is equally mistaken. Any 

 change in the weight of a body on the basis 

 of Newtonian mechanics must be due to an 

 acceleration of the body, part of the gravita- 

 tional force being used to produce the accel- 



1 Society for the Promotion of Enganeering Edu- 

 cation: Bulletin, Vol. 4, pp. 19-28, 1914, or Pro- 

 ceedings, Vol. 22, pp. 115-124, 1915. 



1 Science, Vol. LI., p. 46. 



eration. We may, therefore, examine the ac- 

 celerations of these bodies to see whether they 

 could produce the effect described. Each of 

 the weights on the light disk has an accelera- 

 tion composed of two components.^ One of 

 these components is directed toward the center 

 of the disk. This component is due to the 

 rotation of the disk, and may be called the 

 dish component. Since the two weights are 

 at opposite extremities of a diameter the disk 

 components of their acceleration are equal in 

 magnitude and opposite in direction, and their 

 only effect is to produce the well-known cen- 

 trifugal stress in the disk. The other com- 

 ponent of acceleration is common to the two 

 weights. It is the acceleration of the center 

 of the disk due to the earth's motion. It is 

 altogether independent of the rotation of the 

 disk. This acceleration will affect the weights 

 of the two bodies, but the effect will be the 

 same for both bodies in all positions of the 

 disk, and cannot therefore, produce un- 

 balanced rotation. 



Curiously enough there is another cause 

 that would produce a minute unbalance in a 

 disk of the sort just considered when rotating 

 in any vertical plane at any point on the 

 earth's surface. When the line of the weights 

 is in a horizontal position let the weight of 

 each be represented by w. Then neglecting 

 the weight of the disk and shaft the down- 

 ward pressure on the bearings is 2 w. Wlien 

 the line of the weights has turned through 

 90° to a vertical position one of the bodies 

 has approached the earth and consequently its 

 weight is increased. The other has receded 

 from the earth but its weight has decreased 

 less than the other increased since the attrac- 

 tion varies as the inverse square of the dis- 

 tance. Consequently the pressure on the bear- 

 ings is greater when it is horizontal. This 

 would produce a minute effect of unbalance 

 which, however, would be just as great when 

 the disk rotates slowly as when it rotates at 

 high speed. 



Burt L. ISTewkirk 



TJniveksitt of Minnesota 



2 Gimbal mounting is assumed again to eliminate 

 gyroscopie effect. 



