August, 1887.] ELLIOTT SOCIETY. 171 



on KO is double the square on KE, and hence KO is equal to the diagonal of 

 the square on KE, as stated in the preceding section. 



9. The calculus gives these results more readily though less easily accessible 

 to most students. Supposing the intensity of disturbance to vary inversely as 

 the Titii power of the distance from O, it will vary directly as the n^i^ power 

 of the sine of angle of emergence at the surface, which angle call A. Then the 

 intensity of the horizontal disturbance will vary directly as cos A sinii A. Dif- 

 ferentiating this expression and treating for a maximum, we shall find cos^ A= 



1 



n+i ^^ 



leu uu 



then cos2 A=— , 



uuo i» oi JLuaji 



When 



71=1, 



A=45°.00' 



(( 



'*2, 



3' 



" 54° 14' 



it 



" 3, 



u . 1 

 4 ' 



♦' 60"00' 



(( 



-4, 



5' 



" 63°26' 



&c. &0. &c. 



Now if Kc be drawn and KO taken as radius Ko will be sine of angle KOC, 

 and cosine of KCO, the angle of emergence at 0; also if KO be put equal to 1, 

 KM will numerically be equal to the square of that cosine and will equal half 

 the radius, and so will exactly equal the value of the cosine in the first line of 

 above table ; the geometrical and algebraical processes giving of course the same 

 result. In like manner, if Ke be drawn, Ke will be the cosine of angle of 

 emergence at E, to same radius, and KR will be the square of that cosine ; the 

 calculus shows above in second line of table, that when n=:2, that is, when law 

 supposed is the inverse square of the distance, then cos2 A =-3-; and this was 

 the value used for KR, in order to find the point e, then E and q, and lastly the 

 line Qq. Proceed in like manner for the other values of n. 



AUGUST 25th, 1887. 

 The President in the Chair. 

 The following papers were read : 



