...initial displacements from that depending on the initial velocities, it is only necessary todifferentiate with respect to the time, and to replace the arbitrary...functions) which express the initial velocities by those whichi express the corresponding initial displacements. Thus, if <f> be any normal co-ordinate satisfying...

...we have only to differentiate with respect to t, and to replace the arbitrary constants or arbitrary functions which express the initial velocities by...which express the corresponding initial displacements. Conversely, to get the disturbance due to the initial velocities from that due to the initial displacements,...

...we have only to differentiate with respect to t, and to replace the arbitrary constants or arbitrary functions which express the initial velocities by...which express the corresponding initial displacements. Conversely, to get the disturbance due to the initial velocities from that due to the initial displacements,...

...type. We may readily prove from this that in order 1 Dynamical Theory of Diffraction, Cambridge Tram. Vol. ix. p. 1, 1856. 95.] CONJUGATE PROPERTY. 129...terms of the initial values of <£ and <£ is <f> = <£„ cos nt + - <£0 sin nt ..................... (1), of which the first term may be obtained...

...velocity, but with a displacement of the corresponding type. We may readily prove from this that in order to deduce the motion depending on initial displacements...equation the solution in terms of the initial values of <f> and <£ is <f, = <f>, cos nt + - 4,, sin nt ..................... ( 1 ), ?t of which the first...

...which depends upon the initial velocities by replacing the arbitrary functions, or arbitrary constants, which express the initial velocities by those which express the corresponding initial displacements, and differentiating with respect to the time." One of the principal objects of the memoir was to determine...

...which depends upon the initial velocities by replacing the arbitrary functions, or arbitrary constants, which express the initial velocities by those which express the corresponding initial displacements, and differentiating with respect to the time." One of the principal objects of the memoir was to determine...

...prove from this that in order ' Dynamical Theory of Dijfroctton, Camhridn? Trani. VoL ix. p. 1. 1866. to deduce the motion depending on initial displacements...initial displacements. Thus, if <£ be any normal co-orjiinate satisfying the equation $ + n*$ = 0, the solution in terms of the initial values of <j,...

...differentiating with respect to the time, and replacing the arbitrary functions, or arbitrary constants, which express the initial velocities by those which express the corresponding initial displacements. Particular cases of this theorem are of frequent occurrence, but the author is not aware of any writing...

...differentiating with respect to the time, and replacing the arbitrary functions, or arbitrary constants, which express the initial velocities by those which express the corresponding initial displacements. Particular cases of this theorem are of frequent occurrence, but the author is not aware of any writing...