172 ANNUAL OF SCIENTIFIC DISCOVERT. 



The proljlom then is, o^ivon the annual numljer of deaths at 

 diil'erent ages in a fluctuating ijopulation, and the numbers of the 

 IJopulation at the same ages at tlie middle of the year; required, 

 the eorresiionding relative number of deaths and of living in a 

 stationary population governed by the samci law of mortality. 



The usual mod^s of aftecting this conversion are indirect and 

 tedious ; the method now proposed is direct and brief, reducing 

 the labor of weeks to that of hours. 



This important result is accomplished by observing that the 

 rates of annual mortality at dilferent intervals of age are equiva- 

 lent to the derivative (or differential co-eflicient) of the Napierian 

 logarithm — taken negatively — of tlie proi)orlionate numbers of 

 the living at tiiose ages, in a stationary i)oi)ul:ilion. 



Taking advantage of this fact, Mr. Elliot showed how, by veiy 

 simjile processes, desirable forms of the life-table might readily 

 be computed. 



THE nilLOSOrUY OF A TOP. 



The reason why a top stands in an erect position when it is in a 

 spinning mi>tiou is explained by the " Scientific American" in the 

 following manner : — 



*' The same explanation that we gave, some time since, of the 

 gyroscope, applies to a top. If you tie a stone to the end of a 

 string and swuig it about j^our finger, then, while it is whirling, 

 if a sheet of thin paper be held so that the stone will strike it at 

 a sharp angle in a way to turn the stone from the \)hu\(i of its 

 revolution, the stone will resist this effort to turn it from its course, 

 and will pass through the pajjcr. If a sufiicient number of stones 

 are united to form a complete wheel, and the wheel is put in rota- 

 tion, each one of the stones will resist any effort to change the 

 plane of its revolution; and thus the whole wheel will resist any 

 effort to change the plane of its rotation. When a toj) is rotating 

 in an upright position, it cannot huin towiird any side without 

 changing tlie plane of rotation of all its parts ; consequently, so 

 long as it is rapidly rotating it stands upright. 



"When the axis of the top is inclined, the force of gravitation 

 tends to draw it downwards, and thus to change the planes of rota- 

 tion of all its parts. If you will take a wheel, and incline its a^is, 

 you will see that the struggle to resist this change will move the 

 wheel forward, and will thus give to it a revolution around an 

 imaginar}- vertical axis. Even in this revolution the j^laues of 

 rotation are constantly changed, but the change is the less the 

 more nearly the axis of the top coincides with the imaginary 

 vertical axis about which it is revolving ; hence it is subjected to 

 a constant tendency to assume an upright j^osition, and the more 

 rapid its rotation, the stronger is this tendency. 



"The resistance offered by a rotating wheel or disk to any 

 change in the plane of its rotation is worthy of consideration iu 

 many applications of mechanism. ' This resistance tends to make 

 a flj--wheel run true, and, consequently, to so wear its beai'ings^ 

 as to correct any slight error in its original hanging. It increases 

 the resistance of locomotive and car wheels to the change in the 



