266 



Mr. M. W. Crofton on the Theory of 



[Feb. 27, 



the arc differs very slightly from the horizontal distance, or ds=dx nearly ; 

 and the following is the resulting equation : — 



{ 2(^+1) n+2 - 2(n+l) ^ 



where A=-^. — and cand a are the constants, and n the index in the 



Qn 



general equation 



Examples of the application of this are given, showing the calculated ele- 

 vation for the 12-pounder muzzle-loading Armstrong gun for ranges of 

 2855 yards and 4/19 yards, the gun being 17 feet above the planes. 



The calculated elevations were 6° 56' and 14° 6', the actual elevations 

 being 7° and 15° respectively. 



It is not intended to claim more than approximate accuracy for the 

 formulae in this paper. The general formula has been shown to be 

 derived by taking mean values of n and c, whereas the actual results would 

 indicate that the value of n increases with the diameter of the projectile ; 

 and it is shown in a note that the values of n which agree best with expe- 

 riment are, 



for the small shot n=2'4, 

 for the large shot w=4, 

 corresponding to the following resistances, 



small shot R 



large shot B,=v^. 



Whether in reality the index does increase with the diameter of the shot 

 must be left to be determined by more extended experiments ; meantime 

 it may be assumed that the general formula in this paper represents with 

 tolerable accuracy the law of resistance and the loss of velocity of pro- 

 jectiles varying from 8-8 lbs. to 251 lbs. in weight, from 3 inches to 

 9 inches in diameter, and from 1500 to 600 feet per second in velocity. 



II. "On the Theory of Probability, applied to Random Straight 

 Lines.'' By M. W. Crofton, B.A., of the Royal Military 

 Academy, Woolwich, late Professor of Natural Philosophy in the 

 Queen's University, Ireland. Communicated by Prof. Sylvester. 

 Received February 5, 1868. 



(Abstract.) 



This paper relates to the Theory of Local Probability — that is, the appli- 

 cation of Probability to geometrical magnitude. This inquiry seems to 

 have been originated by the great naturalist Buffon, in a celebrated pro- 

 blem proposed and solved by him. Though the subject has been more 

 than once touched upon by Laplace, yet the remarkable depth and beauty 

 of this new Calculus seem to have been little suspected till within the last 



