VOL. LXXIX.] PHILOSOPHICAL TRANSACTIONS.' 577 



fluent z of yx contained between the extreme values of x or y ; then find the 

 fluent of zz contained between the given values of z ; then the fluent multiplied 

 into the product of the sines of the angles which the first abscissa makes with the 

 plane of the ordinates and 2d absciss, and the 2d absciss makes with its corres- 

 pondent ordinates, will be the solid content required. 



2. Fig. 11. Let the 1st absciss 2 of a solid be perpendicular to the planes of 

 the ordinates, and the 2d absciss vp •=■ x perpendicular to the ordinates them- 

 selves pm = y. First, assume the 1st absciss as invariable, and find the incre- 

 ment of the arc p^m = {.P -|- ^^)^ , then assume the 2d absciss vp as constant, 

 and let mu be the fluxion of the ordinate y or m, when the fluxion of the first 

 absciss is i = w/, where ul is perpendicular to the plane of the ordinates p'pm, 

 and / a point of the surface of the solid ; draw uh perpendicular to the arc p'm : 

 then since ul is constituted at right angles to the plane pp^m, Ih will cut the arc 



1)m at right angles ; but uh = "^" ^ ~ = - ,,-T, ~rrr ; Ih = (hu^ -I- lu^)^ = 



( ./^ .- •\- z^)^ ; the fluxion of the surface will be /A X -/ (i^ + jr^). From the 



given equation expressing the relation between the 2 abscissae z and x and ordi- 

 nates y, find, by assuming z invariable, px = y, and by assuming x invariable 

 ^i: = y' = ti; which being substituted for their values in the quantity Ih X 



^{x'-^-y^), there will result (9' + p^ 4. i)§ X i X i = Ai^ = (? +P'+ ^ 1^ 



X y >C z ■=: ^y'z \ in A and b for y and x respectively substitute their value de- 

 duced from the given equation, and let the resulting quantities be Pl'xz and fi'^i, 

 where a' is a function of x and z, and b' a function of y and z ; find the fluent 

 of A.\vz, from the supposition that x is only variable, contained between the ex- 

 treme values of J7 to a given value of z, which let be -lz-, then find the fluent of 

 Li by supposing z only variable contained between given values of z ; and it will 

 be the surface of the solid contained between those values. 



The same may be deduced by finding the fluent of ji'yz on the supposition that 

 y is the only variable quantity contained between the extreme values of y, as be- 

 fore of a? to a given value of z, which let be L'i ; then will the fluent of jJz, con- 

 tained between the given values of z, be the surface required. If the solid be a 

 cone generated by the rotation of a rectangular triangle round a side containing 

 the right angle as an axis ; hu will be a given quantity, if z be given. — If the 

 above-mentioned angles are given, but not right ones, the arc/>'m and perpendi- 

 cular I'h can easily be deduced, and consequently the increment of the surface. 



3. To define a curve of double curvature, it is necessary to have 2 equations, 

 expressing the relation between the abcissae z and x and their ordinates y, given ; 

 and if the angles which they respectively make with each other be right ones, the 

 fluxion of the arc, as given in the Proprietates Curvarum, is (5^ + ^r^ + ^^)». 



VOL. XVI. 4E 



