188 Mr Love, On the Theory of Discontinuous [May 4, 



f 1 V(l - if) , _ f 1 (1 - ^ 2 ) du 

 JaVfa'-a*) J a V(l-^ 2 )V(^-a 2 )' 



putting m = a sec (f>, the latter integral becomes 



•cos- 1 a (1 _ a 2 sec 2 <£) a sec <£ tan $d$ _ foos-^a (l - a 2 sec 2 <£) d<£ 

 o a V(l — a 2 sec" <£) tan (/> J V(l — <*>* — sin 2 0) ' 



putting sin <£ = V(l — & 2 ) sin ^ = k sin -^r, we have 



7T 



2 F cos 2 ^ c£l/r 



'o [l-Fsin*^]* 



7T . . 



or h — in the limit when k is small. 



4 



f 1 du /"cos -1 a 



AgaiD 'i«VK^?) = io SeC * # 



= k in the limit 



fi (l— u 2 ) roos- 1 a ' 



and -^-= tgr du=\ s (1 - a sec 2 <6) sec d>arf> 



JaV{u -a) Jo 



/"cos -1 a (X 2 f -l Tcos -1 a 



= .1 sec <£ cty - -s- j [sec <f> tan <£] COS a + I sec <£ (20 



1+F, /1+fc 1 7 



= as far as & 2 . 

 Thus neglecting F the height of the rim is 



To find the breadth of the disc we must write (14) in the form 



^=2 [1 + V(1 " W)] V(1-^ 2 -^ 2 )' 



and integrate from u = to u = a and double the result. We may 

 reject k altogether, and thus find the breadth 



a + \ [sin -1 a + a */(l — & 2 )]> 



or in the limit when a is very nearly 1 we may take the breadth 

 equal to - 



^ (18). 



thus the ratio height of rim to breadth of disc = ^k 2 = e say. 



