Theory of the Spirit Level. 



401 



r h 



being known, we fhall have R:r:: It+m : m=-^ — • If we make r=i?, m and a 



R 



will become infinite, if we make r=: — we fhall have m = b, and confequently_' a = 16 



toifes nearly. If r be made =: 50000 toifes, placing the tube in the direflion of the 

 meridian, we Ihall have at the Equator ot = o,oi56i6" and a = 0,2461749 toifes, and 

 and at the pole ot=: 0,015353", and a ==0,246 11 87 toifes. 



If, laflly, the value of R be fought, the angle A D B being known, we fliall have 



n-—b : r :: n : Rr=: — — .. This determination will always be very uncertain, on account 



of the extreme fmallnefs of A B. We have feen in the laft preceding example that a 

 difference of -j't P'Tt of a line in A B produces a difference of about 55000 toifes in 

 A C, though we have fuppofed a very great degree of fenfibility in the tube. 



(5.) In order to form a notion how this praflice may increafe or diminifh the fenfi- 

 bility of a tube, let us imagine any two portions of matter whatever to be placed one 

 upon the other, with very fine fand between them, fo that they may be worn by friflion ; 

 for example, two pieces of glafs. If thefe two pieces AB, C D, fig. 3, of which it is 

 un'necefTary at prefent to confider the breadth, be of equal length and rubbed one againll 

 the other in the direftion of .their length, there will almofl conflantly be a part, either 

 large or fmall, where one will overhang the other, which part confequently will not be 

 rubbed while the other continues to be afted upon by the friftion. The two pieces are 

 precifely in the fame {jtuation, becaufe the parts B D, A C, which are lefs rubbed than 

 the middle, are equal to each other; each of thefe pieces is therefore equally difpofed 

 to become hollow in the middle, which, however, cannot happen, becaufe it is impolli- 

 ble they fhould wear except at the places which touch, and they tend by grinding-to 

 touch each other every where in a right line or a circle, which are the only lines that in 

 the different pofitions here fuppofed can touch each other every, where. This is evi- 

 dently impolFible, provided both fhould become hollow at the fame time, and the one 

 caimot become hollow fooner than the other. 



But if the two pieces be unequal in length, as in fig. 4, the longeft of the two will 

 conftantly projeft beyond the other ; and if the fhorter one do projefl, the longer will 

 projeft ftill more at the fame time. It will therefore be more difpofed to wear away 

 and become hollow in the middle. This concavity or curvature will increafe the longer 

 the grinding is continued, while the fhOrter piece will become convex in a circular line ; 

 both having the fame radius in order that they may conftantly touch throughout*. 



(6.) In 



• TTie above rcafoning is not accurate to the extreme of flrictncf!. The figure of furfaces gronnd uponjCacJi 

 other is liable to vary from partial expanfions, if the work be carclcfsly touched by the hand, and in etjual pieces 



of . 



