# Integrating Sentence Examples

**Integrating**the expression for an angle of wrapping 0, we obtain the relation log € Ti/T2= µ9, where T 1 and T2 are the end tensions.- The analytic method sought to express the moon's motion by
**integrating**the differential equations of the dynamical theory. **Integrating**by parts, we find v i.- A recording drum or
**integrating**apparatus may be arranged on the pulley frames. - Thus the contribution to the total impulsive pressure exerted on the area dS in time dt from this cause is mu X udtdS X (11 3 m 3 /,r 3)e hm (u2+v2+w2 )dudvdw (I o) The total pressure exerted in bringing the centres of gravity of all the colliding molecules to rest normally to the boundary is obtained by first
**integrating**this expression with respect to u, v, w, the limits being all values for which collisions are possible (namely from - co too for u, and from - oo to + oo for v and w), and then summing for all kinds of molecules in the gas. **Integrating**(27) again, (31) y =g(zTt2t 2) = zgt(T -t); and denoting T-t by t', and taking g= 32f/s2,) y =16tt', (32 which is Colonel Sladen's formula, employed in plotting ordinates.- Di g d tan i g dt - v cos i ' and now (53) dx d 2 y dy d2xdx Cif dt 2 dt dt2 _ - _ gdt' and this, in conjunction with (46) dy _ d y tan i = dx dt/dt' (47)di d 2 d d 2 x dx sec 2 idt = (ctt d t - at dt2) I (dt), reduces to (48)
**Integrating**from any initial pseudo-velocity U, (60) du t _ C U uf(u) x= C cos n f u (u) y=C sin n ff (a); and supposing the inclination i to change from 0, to 8 radians over the arc. - If we proceed instead by the method of
**integrating**the equation H -h =6(v-w)dp/d6, we observe that the expression above given results from the integration of the terms -dh/R0 2 +w(dp/d9)/R9, which were omitted in (25). - We will suppose that P is a function of r only; then
**integrating**(~) we find ~ v2 = fPdr+const., (4) - Whence,
**integrating**with respect to t, 3/4M (~2 +5i) + 3/4162 =f(Xdx+Ydy +NdO) +const. - Q,,) tobe generated instantaneously from rest by the action of suitable impulsive forces, we find on
**integrating**(II) with respect tot over the infinitely short duration of the impulse ~-=Q. **Integrating**with respect to f from f =z to f=a, where a is a line very great compared with the extreme range of the molecular force, but very small compared with either of the radii of curvature, we obtain for the work (1,G (z) - 111(a))dw, and since (a) is an insensible quantity we may omit it.**Integrating**the first term within brackets by parts, it becomes - fo de Remembering that 0(o) is a finite quantity, and that Viz = - (z), we find T = 4 7rp f a, /.(z)dz (27) When c is greater than e this is equivalent to 2H in the equation of Laplace.**Integrating**by parts, we get J l'(z)d z = zI, G (z) + 3 z 3 I I (z) 3 f z3Cb(z)dz, fzqi(z)dz = J z21 '(z) + k z41 I (z) + a fz4(1)(z)dz.- After the establishment of universal gravitation as the primary law of the celestial motions, the problem was reduced to that of
**integrating**the differential equations of the moon's motion, and testing the completeness of the results by comparison with observation. - The mean brightness varies as z3 (or as r3), and the integral found by multiplying it by zdz and
**integrating**between o and co converges. - If the subject of examination be a luminous line parallel to n, we shall obtain what we require by
**integrating**(4) with respect to 77 from - oo to + oo. - E is then the co-ordinate relatively to 0 of any focal point 0' for which the retardation is R; and the required result is obtained by simply
**integrating**(5) with respect to from - cc to +oo. **Integrating**by parts in (II), we get J e = ikr d7 pc-11 / d (e r - ay= rJ Z d y - r / 1 dY, in which the integrated terms at the limits vanish, Z being finite only within the region T.- Taking two planes x = =b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P' at opposite ends of the diameter PP', is pdy (U - Ua 2 r2 cos 20 +mr i sin 0) (Ua 2 r 2 sin 2 0+mr 1 cos 0) + pdy (- U+Ua 2 r 2 cos 2 0 +mr1 sin 0) (Ua 2 r 2 sin 2 0 -mr 1 cos 0) =2pdymUr '(cos 0 -a 2 r 2 cos 30), (8) and with b tan r =b sec this is 2pmUdo(i -a 2 b2 cos 30 cos 0), (9) and
**integrating**between the limits 0 = 27r, the resultant, as before, is 27rpmU. - (to)
**Integrating**over the base, to obtain one-third of the kinetic energy T, 3T = 2 pf '3 4R2(3x4-h4)dx/h 3 = pR2h4 / 1 35 V 3 (II) so that the effective k 2 of the liquid filling the trianglc is given by k 2 = T/Z p R 2 A = 2h2/45 = (radius of the inscribed circle) 2, (12) or two-fifths of the k 2 for the solid triangle. - = -dQ+1dg2, and
**integrating**round a closed curve (udx+vdy+wdz) =0, and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is differential irrotational and due to a velocity function, the circulation is zero round all reconcilable paths. - J -1k di - d 3a dX +2(a2+X)d (a -) =o, and
**integrating**(a 2 + X) 3 /2ad? - The equation to these lines in terms of v and 0 is obtained by
**integrating**dE=sd0+(Odp/de - p)dv = o . - The equation to the lines of constant total heat is found in terms of p and 0 by putting dF=o and
**integrating**(it). - In thiscase the ratio of the specific heats is constant as well as the difference, and the adiabatic equation takes the simple form, pv v = constant, which is at once obtained by
**integrating**the equation for the adiabatic elasticity, - v(dp/dv) =yp. - Putting d0/dp=A/0 2 in equation (15), and
**integrating**on the assumption that the small variations of S could be neglected over the range of the experiment, they found a solution of the type, v/0 =f(p) - SA /30 3, in which f(p) is an arbitrary function of p. Assuming that the gas should approximate indefinitely to the ideal state pv = R0 at high temperatures, they put f(p)=Rip, which gives a characteristic equation of the form v= Re/p - SA /30 2 . - Denoting by So, so, these constant limiting values at p=o, we may obtain the values at any pressure by
**integrating**the expressions (27) and (28) from co to v and from o to p respectively. - (29) (30) The expression for the change of entropy between any two states is found by dividing either of the expressions for dH in (8) by 0 and
**integrating**between the given limits, since dH/B is a perfect differential. - Apparatus is added to some dynamometers by means of which a curve showing the variations of P on a distance base is drawn automatically, the area of the diagram representing the work done; with others,
**integrating**apparatus is combined, from which the work done during a given interval may be read off directly. - Now taking equation (72), and replacing tan B, as a variable final tangent of an angle, by tan i or dyldx, (75) tan 4) - dam= C sec n [I(U) - I(u)], and
**integrating**with respect to x over the arc considered, (76) x tan 4, - y = C sec n (U) - f :I(u)dx] 0 But f (u)dx= f 1(u) du = C cos n f x I (u) u du g f() =C cos n [A(U) - A(u)] in Siacci's notation; so that the altitude-function A must be calculated by summation from the finite difference AA, where (78) AA = I (u) 9 = I (u) or else by an integration when it is legitimate to assume that f(v) =v m lk in an interval of velocity in which m may be supposed constant. - He concludes that the
**integrating**principle of the whole - the Spirit, as it were, of the Universe - must be something akin to, but immeasurably superior to, the " psychism " of man. - The relation between the equilibrium pressures P and P' for solution and solvent corresponding to the same value po of the vapour pressure is obtained by
**integrating**the equation V'dP' = vdp between corresponding limits for solution and solvent. - Putting in these values and
**integrating**we have, neglecting terms involving 0', P=12.06 0-0.021 O s where P is the osmotic pressure in atmospheres. - The Mean Specific Heat, Over Any Range Of Temperature, May Be Obtained By
**Integrating**The Formulae Between The Limits Required, Or By Taking The Difference Of The Corresponding Values Of The Total Heat H, And Dividing By The Range Of Temperature. - Substituting L= Lo+ (S-s)(6-Bo), and
**integrating**between limits, we obtain the result log e p=A+B/o+C log e 6,..