VOL. XX.^ PHILOSOPHICAL TRANSACTIONS. 335 



Of a Child that stvalloived livo Copper Farthings. Communicated Inj Dr. Edw. 

 Baynard, Fell, of the Coll. of Physicians. N° 246, p. 424. 



A child, aged about 3 years, swallowed by accident 2 copper farthings, only 

 half a year after each other. After the first farthing, he ate nothing for 10 

 days, and complained of a great pain at his stomach, and drivelled as if he had 

 been salivated; and often said he had a nauseous venomous taste in his mouth, 

 the farthing not coming from him in half a year. After the swallowing of the 

 second farthing he began by degrees to lose his limbs, his breast growing nar- 

 row, and the child consumptive ; but was afterwards perfectly cured by the Bath, 

 and his breast dilated and grew broad as before. 



Analytical Investigation of the Curve of Siviftest Descent.* By Mr. R. Sault. 

 N°246, p. 425. Translated from the Latin. 



Let AP (fig. 10, pi. 5,) be a horizontal line, p the point from whence the 

 heavy body descends through the curve line required pde, c and d two points 

 infinitely near, through which the body may fall, cd a right line connecting 

 those points ; also dc and sc, df and so, fs and gc or sh, moments of the 

 curve, of the absciss, and of the ordinate respectively. Take dr = ds, and 



CT := CB. 



Then because in small nascent lines, the time is as the path ran over directly 

 and as the velocity inversely, that is, in this case, as the root of the altitude of 



the falling body, by the hypothesis it will be as -^ — |- —^ = the least time. 



And the velocity in the points of equal altitude s and e along the curve dsc and 

 the right line dec is the same, the time through uc, which is a minimum, will 



be as — [• —-—: put therefore these times equal, and it is — 1- 



-i/OD a/of ' T ' ./on ' 



DB EC ,, . . DE — DS SC— BC liU TS 



= -; r -; — > that is, = , or — - = . 



V^aD v^aF ^Qu yar ' ^/GLX) ^q.f 



But the evanescent triangles brs, bts are equiangular to the triangles dsf, 

 Hsc; therefore — = — , and — = — . Compound these two ratios of 



DS SF HS ST "^ 



BR _ TS . , ^ Van VaF 



equality, and then = , — . And ex aequo 



DS X HS SF X ST ' SF X ^T DS X HS- 



Now as any of the elements may be supposed to flow equably, put ds = sc, 

 and the most simple expression of the curve becomes ^— ^ = — ^ every 



' ' SF DS •' 



where. That is, in the point of flexure the curve will always be in a ratio com- 



* See N° 224, p. 1 29, of this volume. 



