In this paper the solution of the well-known and hitherto refractory problem, “to investigate the law of motion of a uniform elastic plate,” is presented as an example of the application of the theory of primaries to physical research. This problem, on account of its importance in the theory of acoustics, has attracted the attention of the most celebrated mathematicians.

If we suppose a physical elastic line, flexible only in one plane, to have a number of masses fixed at intervals along it—if we investigate the law of the motions of these masses—and if, thereafter, we imagine the masses to be indefinitely subdivided and their parts distributed along the intervals, we shall arrive at the law of vibration of an elastic plate.