Mr. J. H. Cotterill on Elliptic Ribs. 23 



• I begin by calculating the parts of the coefficients which contain 

 the arbitrary constants. 



We have, so far as these constants are concerned, 



M = F j p cos (j>d(j> + }I \ psm<j)d(f) + M . 

 Jo Jo 



Let the seiniaxes of the ellipse be a and a cos /3, so that sin j3 is 

 the eccentricity; then if the initial section for which <£ = be 

 supposed at the extremity of the axis major, we shall have 



a cos 2 /3 



P= — ^3 — , where A= */Y — sin 2 /3sin 2 <j>; 



C® j. J j. 2 o C® cos $ jjl 2 a sm # 



.-.1 p cos <pd<p = a cos 2 p I 3 ~ d<p=a cos 2 /3 j 



and 



" f * ■ Jl ^JL 2 O f * Sm ^ JJL /l C0S <£\ 



I p sm (j> dcf> = a cos 2 p 1 ^ #(ft = aM a/' 



Assume cos <£' = . , then 



• ,,__ V A 2 — cos 2 (ft _ sin <fr cos ff 

 sm <p _ _ ^ , 



and therefore 



M = F a cos ft sin </>' + H a(l — cos £') + M , 



a value of M which can be easily obtained from a figure/^' being 

 in fact the eccentric angle of the section considered. 



Now we have, beginning with -r^r, the factor EI being, as 



d£ 



before, merged in U, 



dV 

 d¥, 



o=J„ M ^" # 



IT 



= #cos/3l Mp sin <j>' ckj) ; 



Jo 



and it is easily proved that pd(j) = aA!d(f) , } where 



A'= \/l-sm 2 /3cos 2 </)', 



= a 2 f 2 Msin<£'A'^'. 

 Jo 



dV_ 

 d¥ 



