## Integrating Sentence Examples

**Integrating**by parts, we find v i.**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.- (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. - 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. - 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.**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.- 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 analytic method sought to express the moon's motion by
**integrating**the differential equations of the dynamical theory. - Only by taking infinitesimally small units for observation (the differential of history, that is, the individual tendencies of men) and attaining to the art of
**integrating**them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.