644 PHILOSOPHICAL TRANSACTIONS. [aNNO 1^00-1- 



drawi) that Aa will be parallel to it, and on = gl, where the point g will fall ; 

 which, supposing the points o and b to be fixed, may be easily discovered by 

 the known method of Max. and Min. Prosecuting the calculation, it comes 



at length to — — — = — ^^r~» whence it is evident that —^ — = a con- 

 stant quantity ; if the absciss am be called x, and the ordinate bm, y, it will be 

 EL = d.r, and lg = dy (supposed constant throughout the calculation) ; then 



will bg'- = dx'- -\- dy% whence jj ^ 4. j,, ^yi = a constant quantity. Let a be 



any constant line, and then seeing that we observe the law of homogeneity it 



will be jj ()j 4_ jy jy^ = "H) agreeing with what was found by the celebrated 



L'Hospital, and Jo. Bernouilli. And this, by the way, Bernouilli exhibited as 

 a fine specimen among some select methods for the construction of curves by 

 differential equations, in which either ;r or 7/ is considered as indeterminate ; 

 published in the Leipsic Acts for May, 1700, and from which was deduced an 

 elegant method of constructing any curves required. 



Problem II. — To determine the line of swiftest descent. Let bc, cd, (fig. 3, 

 pi. 13,) be two infinitely small particles in the required curve. Now that the 

 curve may be such that the passage from B to d, after falling from the hori- 

 zontal line Aa, be made in the shortest time, it is necessary that in the line 

 Rs, drawn parallel to aq, the point c be so taken, that the differences of the 

 ordinates, gc, de, may be equal. 



Now the velocity in the point c is -v/lc, and the velocity in the point d is 



-v/qd : therefore is the time of descent through bc, and ■ is the time of 



descent through cd (by Prop. 34, Book I. Newton). Therefore the point c 



should be so taken that 1 — = a minimum. Suppose b and d to 



be fixed, and the constant quantities gc = de ^ m, lc = b, qd = p ; the 



indeterminates bg = u. ce = z ; we have r- ' — = a mm. 



Therefore , -^- "-- -|- ". = o. But du = — dz, because u + z = 



a constant quantity : therefore , — " = -,- ; whence it is ob- 



■ ■' liJ^Vin' 4- u- pirVm- + z- 



vious that - — " = a constant quantity. Now let the absciss al =: .r, or- 



dinate lc = 7/ ; then will bg = dx, gc = di/, bc = V^d.r^ -|- di/' ; and if a 

 be put for any constant quantity, , / — - will be = -— , whence dx 



v/a = v'_y X ^ dx"- + dy\ But in all curves d* : \^ dx- + di/" :: subtangent: 



