XX11 



APPENDIX TO THE PREFACE. 



1. If upon EM be erected infinitely narrow parallelograms 

 BGhb and MNom and their distance Mb and altitudes MN, BG be 



Mm + Bb . 

 given, and the semi sum of their bases be also given and 



called s and their semi difference - - be called x : and if the 



2i 



lines BG, bh, MN, mo, butt upon the curve nNgG in the points 

 n, N, g, and G, and the infinitely little lines on and kg be equal to 

 one another and called c, and the figure mnNgGB be turned about 

 its axis BM to generate a solid, and this solid move uniformly 

 in water from M to B according to the direction of its axis BM : 



m M 



the summ of the resistances of the two surfaces generated by the 

 infinitely little lines Gg, Nn shall be least when gG qq is to nN n as 

 BG x Bb to MN x Mm. 



For the resistances of the surfaces generated by the revolution 



of Gg and Nn are as 





and 





, that is, if 



and 



be called p and a. as and - and their summ 

 P 9 



BG MN . , , A , a . x , 



-- 1 - is least when the fluxion thereot 

 p q 



BGxp MNxq 

 nothing, or -- f = + - - . 

 pp qq 



pp 



qq 



is 



p = Gcfi& = Bbwiid + ^^quad _ ss _ % sx + X x + cc and there- 

 fore p "2sx + 2xx, and by the same argument q = 2sx + 2xx and 



BG x 2sx - 2xx MNx2sx+2xx BG x T^x MNxs+x 

 therefore - - = - , or 



pp 



qq 



pp 



qq 



and thence pp is to qq as BG x s x to MN x s + x, that is, gG qq to 

 nN qq as BG x Bb to MN x Mm. 



