VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. 13] 



Solution. — Through any given point A draw any indefinite right-hne adb in a 

 given position, meeting the moveable right-line pkl in the point d ; then call- 

 ing AD, X ; and pk or pl, y; and let q and r be quantities any how composed of 

 any given quantities and the quantity x; and let the relation between x and y 

 be defined by the equation yy -\- qy -\- r =■ O. Then, if r be a given quantity, 

 the rectangle of the segments vk and pl will be given. If 9 be a given quan- 

 tity, the sum of those segments, connected by their proper signs will be given. 

 If qq — 1r be given, the sum of the squares, pk'- -f pl'-, will be given. If 

 q^ _ 3^r be a given quantity, the sum of the cubes, pk^ -j- pl% will be given. 

 If q* — 4qqr _f- 2rr be a given quantity, then the sum of the biquadrats, 

 PK* -j- PL*, will be given. And so on in infinitum. Let there be made therefore 

 r, q, qq — Ir, q"^ — 3qr, &c. given quantities, and the problem is solved. 



In the same manner curves may be found, which shall cut off" three or more 

 segments having the like properties. Let there be the equation y^ -f- qyy 4- 

 ry -\- s =:■ O, where q, r, s denote quantities composed of any given constant 

 quantities and the quantity x; then the curve will cut off three segments. And 

 if s be a given quantity, then the solid contained by those three segments will 

 be given. If <; be a given quantity, the sum of the three will be given. If 

 qq — 2r be a given quantity, the sum of the squares of those three will be 

 given. 



Of a strange Symptom ivhich attended an Hydrops Pectoris. By Mr. Samuel 

 Doudy, F. R. S. N° 224, p. 3Q0. 



Mr. Cowper, in his letter printed in the late Transactions, of an extraordinary 

 large kidney, occasionally mentioning, that sighing and difficulty of respiration 

 are always remarkable symptoms in an hydrops pectoris, put me in mind of the 

 case of a noble peer. His shortness of breathing was very extraordinary ; for 

 he was always better in bed or lying, than sitting or standing ; quite contrary 

 to other asthmas, in which the patient is not able to lie down : the muscles of 

 the breast having a freer motion, when in an erect posture. On opening the 

 body, both the cavities of the breast were found full of water, which, when 

 standing or sitting pressed so upon the diaphragm, that respiration was per- 

 formed with difficulty ; but when lying, the floating load was so disposed, that 

 that office of nature was better performed. This seems to be so natural a 

 symptom, that it may be almost an infallible diagnostic, to distinguish an 

 hydrops pectoris from other more frequent diseases of the breast, that cause 

 a shortness of breathing. Perhaps it may not be impracticable to use the pa- 



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