36 



G ANESH PRASAD , 



3 



!/■' (7t)\ + b\ (3ä,+ 7cd + 2jtX^) + ßjt Kk 



{irwi+il^}- (8) 



Therefore G is less than the right side in (8), for this is evidently greater 

 tlian the right side of (7). 



34. Consider d. Groing back to Arts. 28 and 29 it is easily seen that öj, 

 jo, 6^ may be tbus defined: 



Consider circles of diameter X to be most closely packed in a plane. Then, 

 if a circle of imit area be placed in any position in the plane it will cover the 

 centres of a certain number, N^, of these circles ; and for the particular position 



Let N be the greatest possible value that JS^ can have ; then 



If a circle of unit area be placed in any position in the plane it will wholly 

 Cover a certain number, N^, of the circles of diameter A: for the particular 

 position 



It is easily seen from these definitions that pd, the greatest of the values 

 that p — 0i and p — ö^ have, is less than Ax the circumference of the circle 

 of unit area. 



It need scarcely be mentioned that definitions, exactly similar to those given 

 above, hold for the general case when L is of any shape ; also it is evident that 



where l is the periphery of L. 



35. Suppose that X^<10~'^X^. Now it has already been supposed in Art. 



p = JSfx 



4 



(1) 



pd < U, 



28 that 



A > - and < 10"', 



s 



whence 



A<2-'xlO-U. 



